Fire Data Analysis 

Handbook

Third Edition 

FA-266/November 2021

Fire Data Analysis 

Handbook

Third Edition

FA-266/November 2021

Mission Statement

We support and strengthen fi re and emergency medical 

services and stakeholders to prepare for, prevent, mitigate 

and respond to all hazards.

Foreword iii

Foreword

The fire service exists today in an environment constantly 

inundated with data, but data are seen of little use in the 

everyday, real world in which first responders live and work. 

This is no accident. By themselves, pieces of data are of 

little use to anyone. Information, on the other hand, is very 

useful indeed. What’s the difference? At sporting events, 

people in stadiums hold up individual, multicolored squares 

of cardboard to form a giant image or text, which could be 

recognized only from a distance. This is a good analogy for data 

and information. The individual squares of cardboard are like 

data. They are very numerous and they all look similar taken 

by themselves. The big image formed from the organization 

of thousands of those cards is like information. It is what can 

be built from many pieces of data. Information then is an 

organization of data that makes a point about something.

The fire service of today is changing. More and more, it is not 

fighting fires as much as it is doing emergency medical services 

(EMS), hazmat, inspections, investigations, prevention and 

other nontraditional but important tasks, which are vital to the 

community. Balancing limited resources and justifying daily 

operations and finances in the face of tough economic times 

is a scenario that is familiar to every department.

Turning data into information is neither simple nor easy. It 

requires some knowledge of the tools and techniques used 

for this purpose. Historically, the fire service has had few of 

these tools at its disposal and none of them has been designed 

with the fire service in mind. This book changes that. It was 

designed solely for the use of the fire service. The examples 

were developed from fire data collected from departments all 

over the nation. This book also was designed to be modular 

in form. Many departments’ information needs can be met by 

using only the first few chapters. Others with a more analytical 

and statistical background may want to go further. The point 

is, it’s up to the reader to decide. This handbook is a tool, like 

a pumper or a ladder, to help do the job.

The U.S. Fire Administration
iv Fire Data Analysis Handbook
Table of Contents v

Table of Contents

Chapter 1: Introduction .................................................... 1

Why data analysis? ................................................................... 2

National Fire Incident Reporting System .............................. 3

Data entry and data quality.................................................... 6

Statistical packages for computers........................................ 8

How to use this handbook ...................................................... 9

Books on statistics and data analysis.................................... 10

Chapter 2: Histograms ...................................................... 13

Data as a descriptive tool........................................................ 13

Types of variables..................................................................... 13

Developing a histogram........................................................... 20

Cumulative frequencies........................................................... 22

Summary ................................................................................... 23

Chapter 3: Charts............................................................... 25

Introduction .............................................................................. 25

Bar charts .................................................................................. 26

Column charts........................................................................... 28

Line charts................................................................................. 31

Pie charts ................................................................................... 32

Dot charts .................................................................................. 33

Pictograms................................................................................. 35

Summary ................................................................................... 36

Chapter 4: Basic Statistics ................................................ 37

Measures of central tendency ................................................ 37

Properties and uses for measures of central tendency...... 38

Measures of dispersion ........................................................... 39

Normal distribution and standard score .............................. 41

Properties and uses for measures of dispersion................. 42

Skewed distributions................................................................ 43

Central limit theorem............................................................... 44

Chapter 5: Analyses of Tables .......................................... 45

Introduction .............................................................................. 45

Describing categorical data..................................................... 45

The chi-square test................................................................... 47

2-way contingency tables ........................................................ 54
vi Fire Data Analysis Handbook

Percentages for 2-way contingency tables ........................... 55

Joint percentages...................................................................... 56

Row percentages ...................................................................... 57

Column percentages ................................................................ 58

Selecting a percentage table................................................... 59

Testing for independence in a 2-way contingency table..... 60

Constructing a table of expected values............................... 61

Calculation of chi-square for a 2-way contingency table.... 62

Chapter 6: Correlation ...................................................... 67

Introduction .............................................................................. 67

Scatter diagram ........................................................................ 67

Correlation coefficient ............................................................. 69

Calculating the correlation...................................................... 71

Other types of correlations..................................................... 74

Appendix: Critical Values of Chi-Square......................... 77

Reference............................................................................ 79
Chapter 1: Introduction 1

Chapter 1: Introduction

The primary objective of this handbook is to describe 

statistical techniques for analyzing data typically collected by 

fire departments. Motivation for the handbook stems from 

the belief that fire departments collect an immense amount 

of data and it should not go unused. With basic analytical 

training and an understanding of statistical techniques, a fire 

department can gain a better understanding of the nature of 

fires in the area they serve and effectively present data and 

information to help save lives and property. 

Consider the incident reports that fire departments complete. 

You document information such as the type of situation found, 

action taken, time of alarm, time of arrival, time completed, 

number of engines responding and number of personnel 

responding. For fires, the list grows even longer, including area 

of fire origin, form of heat of ignition, type of material involved 

and other related factors. Additionally, if a civilian or firefighter 

is injured, there are other reports to complete.

Fire departments have a legal requirement to document these 

incidents. Victims, insurance companies, lawyers and many 

others want copies of reports. Fire departments maintain files 

to retrieve individual reports. 

The reports can, however, be beneficial to fire departments by 

providing insight into the nature of fires and casualties in their 

jurisdiction. Basic information is probably already available. 

Typically, the number of fires handled last year, the number of 

fire-related injuries and the number of fire deaths are tracked. It 

is another story, however, if more probing questions are asked:

ĵ How many fires took place on Sundays, Mondays, etc.?

ĵ How many fires took place each hour of the day or month 

of the year?

ĵ What was the average response time to fires?

ĵ How much did response times vary by fire station areas?

ĵ What was the average time spent at the fire scene?

ĵ How much did the average time vary by type of fire?
2 Fire Data Analysis Handbook

This handbook describes statistical techniques to turn data 

into information to answer these types of questions and many 

others. The techniques range from simple to complex. 

For example, the next 2 chapters describe how to develop 

charts to provide more effective presentations about fire 

problems. These charts may be useful for city or county 

officials to explain the activities and needs of your fire 

department. “Chapter 4: Basic Statistics” discusses measures 

of central tendency (mean, median and mode) and measures 

of dispersion (range, variance and standard deviation). 

“Chapter 5: Analyses of Tables” explains the chi-square statistic 

and its use in analyzing table data. “Chapter 6: Correlation” 

discusses the Pearson correlation coefficient and additional 

correlations. These are all techniques that can tell you more 

about the nature of fires and casualties. 

One way to become more comfortable with data analysis is 

to work with real data. For this handbook, we obtained data 

from fire departments in several large metropolitan areas. 

Working with real data makes it easier to understand the 

different techniques.

Why data analysis?

You may still question why we should go to all this trouble to 

analyze data. Many decisions do not require analysis, such as 

decisions on personnel, grievance proceedings, promotions 

and even decisions on how to handle a fire. It is certainly true 

that fire departments can continue to operate in the same way 

they always have without doing a lot of analysis.

On the other hand, there are 3 good reasons for looking closely 

at the data: 

1. Gain insights into fire problems. 

2. Improve resource allocation for combating fires. 

3. Identify training needs. 

The most compelling reason is that analysis gives insight 

into fire problems, which in turn can affect operations in the 

department. For example, you may find that the average 

time to fires in an area is 6 minutes, compared to less than 
Chapter 1: Introduction 3

2 minutes overall. This may be helpful in requests for more 

equipment, more personnel or justifying another fire station.

Another example of improved resource allocation, statistical 

analysis of emergency medical calls can determine the impact 

of providing another paramedic unit in the field. Increasing the 

number of EMS units from 4 to 5 may, for example, decrease 

average response times from 5 minutes to 3 minutes — a 

change that may save lives.

The analysis can also be used to identify training needs. Most 

firefighting training is based on a curriculum that has been 

in place for many years. Analysis of your fire data can allow 

you to see how training matches characteristics of fires in a 

particular jurisdiction. This is not to say that other training 

is not important. However, knowing more about the fires in 

an area can lead to improvements in training. Additionally, 

analysis of firefighter injuries may indicate a need for new 

types of training.

In summary, this handbook will help you deal with the volume 

of data collected on fire incidents. By using the techniques 

presented here, you will be able to improve your skills in 

collecting and analyzing data, as well as presenting the results.

National Fire Incident Reporting System

The National Fire Incident Reporting System (NFIRS) was 

established more than 45 years ago to collect and analyze 

data on fires from departments across the country. More 

than 24,000 fire departments from all 50 states, the District 

of Columbia and the Native American Tribal Authority report 

their fires and losses to the NFIRS. This makes the NFIRS the 

largest collector of fire-related incident data in the world. Fire 

departments report over 1.2 million fire incident responses 

each year to the NFIRS.

Incident data collection is not new. In 1963, the National Fire 

Protection Association (NFPA) developed a dictionary of fire 

terminology and associated numerical codes to encourage 

fire departments to use a common set of definitions. This 

dictionary is known as NFPA 901, Standard Classifications for 
4 Fire Data Analysis Handbook

Fire and Emergency Services Incident Reporting. The current 

NFIRS 5.0 data standard represents the merging of the ideas 

and definitions from NFPA 901 and the many suggested 

improvements from the users of the NFIRS 4.1 coding system.

Version 5.0 of NFIRS consists of 11 separate modules that allow 

fire departments to report any type of incident they respond to. 

The Basic Module (Module 1) is required. It includes incident 

number and type, incident date, alarm time, arrival time, time 

in service, and type of action taken. 

Modules 2 through 5 are required if applicable. If you respond 

to a fire, you complete the Fire Module (Module 2). It includes 

property details, cause of ignition, human factors, equipment 

involved and other information. 

If you respond to a structure fire, you complete Module 3, the 

Structure Fire Module. It includes such things as structure type, 

main floor size, area of fire origin, and presence of detectors 

and automatic extinguishment equipment. 

If there were civilian or fire service casualties, you complete 

Modules 4 or 5, respectively. 

The remaining modules are optional at the local level. They 

include EMS (Module 6), Hazardous Materials (Module 7), 

Wildland Fire (Module 8), Apparatus or Resources (Module 9), 

Personnel (Module 10), and Arson (Module 11).

Usually, the state fire marshal’s office in each NFIRS state has 

the responsibility for collecting data from its fire departments 

or overseeing the NFIRS submissions. It also manages system 

access for its fire departments. Some states maintain a statelevel incident database. These data files are combined with 

data from other fire departments into a statewide database. 

The state then submits the data files to the NFIRS national 

database at the U.S. Fire Administration (USFA), National 

Fire Data Center (NFDC), by using the USFA file upload 

tool. Or, per state policy, local fire departments can submit 

their files directly to the national database using the USFA 

file upload tool. Departments and states that do not use a 
Chapter 1: Introduction 5

vendor product may still participate in the NFIRS by using its 

no-cost, web-based data entry tool. Today, there are very few 

departments that have no electronic capabilities, but some 

remain. Their paper reports are sent to their state office which 

then enters the reports into the NFIRS.

All states and fire departments within them have been invited 

to participate in the NFIRS on a voluntary basis. Most of the 

data are collected electronically through third-party vendor 

software. The NFDC maintains the NFIRS specification for 

vendors so they may prepare products to meet the NFIRS 5.0 

data standard. Data on individual incidents and casualties are 

preserved incident by incident at local, state and national levels.

The NFDC, among other organizations, can analyze NFIRS 

data at the national level to help develop public education 

campaigns, make recommendations for national codes 

and standards, guide allocations of federal funds, ascertain 

consumer product failures, identify the focus for research 

efforts, and support federal legislation.

Every fire department is responsible for managing its 

operations in such a way that firefighters can do the most 

effective job of fire control and fire prevention in the safest way 

possible. Effective performance requires careful planning; this 

can only happen if accurate information about fires and other 

incidents is available. Patterns that emerge from the analysis 

of incident data can help departments focus on current 

problems, predict future problems in their communities and 

measure their programs’ successes.

The same principle is also applicable at the state and national 

levels. The NFIRS provides a mechanism for analyzing incident 

data at each level to help meet fire protection management 

and planning needs. In addition, NFIRS information is used by 

labor organizations to analyze such matters as workloads and 

firefighter injuries.
6 Fire Data Analysis Handbook

Data entry and data quality

Data quality is an area of great importance. The following 

criteria are used in monitoring data in the NFIRS during the year: 

ĵ The data are complete. 

ĵ The data are accurate. 

ĵ The data are current. 

These criteria are monitored by creating reports that show 

the number of reporting fire departments, the number of 

incidents by state, the number of invalid incidents and the 

number of unreleased incidents. The USFA provides the 

reports to the state NFIRS program managers and works 

with them to resolve any data issues. USFA provides technical 

assistance (e.g., telephone support) to states to help address 

any data quality and data reporting needs.

Audits of the data are performed during the year to identify 

any inconsistencies. The audits focus on 3 criteria: gaps in 

reporting, critical errors in the data and outliers in the data. 

In particular, the USFA works closely with states to monitor the 

quality of data coming from third-party vendor software. The 

USFA assists states in monitoring vendor data quality issues 

or contacts vendors directly to discuss an issue at a state’s 

request. Examples of data quality issues that are reviewed are 

questionable, high dollar-loss incidents and questionable, high 

numbers of fire deaths. 

Quarterly, USFA staff queries the database for questionable 

values (i.e., outliers) and verifies the values with state- and 

local-level NFIRS program managers. These important steps 

ensure that the data meet the USFA’s 3 criteria before the data 

are released in the NFIRS Public Data Release format.

One assumption throughout the handbook is that data on 

fire incidents and casualties are available for analysis. Manual 

analysis is possible, but the tedious calculations quickly 

overwhelm the ability to perform analysis in any meaningful 

manner. Today, many analytical and statistical software 

packages exist to help process and analyze data quickly and 

accurately. Even spreadsheet software, such as Microsoft 
Chapter 1: Introduction 7

Excel, can be used to complete analysis on smaller sets of data 

at the department level. 

Most fire departments enter their data into either third-party 

vendor software that is purchased by the state or local fire 

department, or the free web-based tool supplied by the USFA 

to states. If third-party vendor software is used, it must be 

compatible with the NFIRS standard. A list of active-status 

vendors is available from the USFA, but it is the responsibility 

of the individual states to ensure that a vendor’s software 

meets the qualifications. If the USFA web-based tool is used, 

it must be supported by the state.

A word of caution: Any third-party vendor software should 

contain an error-checking routine. Data quality is always a 

concern, and in data science, the principle “garbage in, garbage 

out” certainly applies to fire department reports. The software 

should, for example, check each item to make sure a valid code 

has been entered. Whenever the software encounters an error, 

it should provide the user with the opportunity to correct the 

error before it becomes part of the database. For example, 

alarm times obviously cannot have hours greater than 23 and 

minutes greater than 59. Data entry software should check 

hours and minutes for valid numbers and allow corrections to 

be made immediately.

The data collected to describe an incident are the foundation 

of the system. Therefore, editing and correcting errors is 

a system-wide activity involving local, state and federal 

organizations. All errors resulting from the edit/update 

process need to be reported to fire departments, and the 

submission of corrections from fire departments is essential. 

This is especially important for fatal errors, which prevent the 

data from being included in the NFIRS database.

At the local level, fire departments need to establish data 

quality procedures if they intend to take full advantage of 

their data. There should be a system in place to double-check 

the collection and data entry. Field edits and relational edits 

can be built into the system that will reveal unacceptable and 

unreasonable data. Data management personnel can use 

these techniques to improve and validate the data. 
8 Fire Data Analysis Handbook

In summary, data entry software should include code-checking 

routines to identify errors in individual items in the report and 

errors reflected through inconsistencies between items. Because 

data entry software cannot be expected to find all errors, fire 

departments also need to implement data quality procedures to 

ensure that correct data are entered into their systems.

Statistical packages for computers

In this handbook, we present many different types of 

analyses. “Chapter 3: Charts,” for example, discusses 

several types of charts, including bar charts, column charts, 

histograms, line charts and dot charts. Other chapters show 

how to calculate statistics, such as means and variances, and 

how to do more advanced calculations, such as chi-square 

tests and correlation coefficients.

For a good understanding of the analysis, it is important to 

know what is involved in the statistical calculations, but it is not 

recommended to do data analyses by hand. There are several 

good analytical and statistical software packages available for 

data analyses. If you intend to apply the techniques in this 

handbook, you should acquire and learn how to use a fullfeatured statistical analysis software application. Excel (and 

Microsoft Access) may get you started but can quickly get 

overwhelmed by larger datasets and complex analysis.

The following are a few examples of full-featured software 

packages:

Software 

package Website

IBM® SPSS® www.ibm.com/products/spss-statistics

R www.r-project.org

Stata www.stata.com

SAS® www.sas.com

NCSS www.ncss.com

JMP® www.jmp.com
Chapter 1: Introduction 9

How to use this handbook

Data analysis is not an easy process. It requires careful data 

collection, attention to detail, access to statistical programs 

and skills in result interpretation. These are not impossible 

tasks, but they require time and patience on your part for 

success. Equally important, you need experience. In the long 

run, you can only develop capabilities in analysis by applying 

techniques from this handbook on actual data sets.

As a final note, one way of thinking about analysis is to 

consider it a 4-stage process. 

ĵ Stage 1: Collect the data, which is what the NFIRS does. In 

and of themselves, the data are meaningless. 

ĵ Stage 2: Organize and summarize into information that 

can be analyzed.

ĵ Stage 3: Analyze according to whatever problem or issue 

is being considered. This yields a better understanding of 

the information.

ĵ Stage 4: Use the information to make decisions. 

Our ultimate objective is to make better and more informed 

decisions at the local fire department level. Data have no utility 

in a vacuum, and fire reports stay as data if we do nothing 

with them. Analysis turns data into information. We move, 

for example, from knowing individual alarm and arrival times 

to knowing average travel times. Our review of travel times 

increases our knowledge about what is going on with fire 

incidents, which results, in turn, in more informed decisions 

within fire departments.

Chapters 2 and 3 are devoted to descriptions of different 

t ypes of charts and graphs. “Chapter 2: Histograms” 

describes histograms, which are probably the easiest charts 

to understand. Chapter 3 expands to other types of charts: 

column charts, pie charts and dot charts. 
10 Fire Data Analysis Handbook

In Chapter 4, several basic statistics are introduced, including 

means, medians, modes, variances and standard deviations. 

Chapter 5 discusses analysis of tables, which is particularly 

important since fire data often come to us as summaries in 

the form of tables. In Chapter 6, correlations and variable 

relationships are discussed. In these chapters, the goal is to 

present how to perform the calculations associated with these 

subjects as well as how to interpret the results.

In developing these chapters, we recognized that readers will 

have varying backgrounds and capabilities. Therefore, while 

a certain understanding of the principles behind the various 

techniques is presented, in most cases a practical application 

approach is used. The subject material becomes more difficult 

as the handbook progresses. The first few chapters are easier 

to understand. More technical subjects, such as chi-square 

analysis and correlation, are more difficult and may require 

knowledge of basic algebra. Even in these chapters, however, 

emphasis has been placed on understanding results rather 

than concentrating on theory.

We made every effort to simplify what can be a very complicated 

topic. While there are many mathematical and statistical 

symbols normally involved with the formulas and calculations 

used in this handbook, none are used here. This is meant to be 

a handbook, not a statistical textbook. It is written so anyone 

can pick it up and be able to do basic statistical analysis of data. 

For those who want more in-depth discussions of the subject 

matter, a list of texts is included.

Books on statistics and data analysis

The following is a sampling of books on data analysis techniques 

as well as some specific statistical topics handled or referred 

to in this book. Most are basic or intermediate in scope, but all 

have more detail than can be presented in this handbook.

The Art of Data Science: A Guide for Anyone Who Works with Data

by Roger Peng and Elizabeth Matsui (Skybrude Consulting, 2016).

Too Big to Ignore: The Business Case for Big Data (1st Edition) by 

Phil Simon (John Wiley & Sons, Inc., 2013). 
Chapter 1: Introduction 11

Practical Statistics for Data Scientists: 50+ Essential Concepts 

Using R and Python (1st Edition) by Peter Bruce and Andrew 

Bruce (O’Reilly Media, Inc., 2017).

Storytelling with Data: A Data Visualization Guide for Business 

Professionals (1st Edition) by Cole Nussbaumer Knaflic (John 

Wiley & Sons, Inc., 2015).

The Data Detective: Ten Easy Rules to Make Sense of Statistics by 

Tim Harford (Riverhead Books, 2021).

Envisioning Information (4th Edition) by Edward R. Tufte 

(Graphics Press, 1990).

Analyzing Tabular Data: Loglinear and Logistic Models for Social 

Researchers by Nigel Gilbert (UCL Press, 1993).

Data Analysis: An Introduction by Michael S. Lewis-Beck (SAGE

Publications, Inc., 1995).

From Numbers to Words: Reporting Statistical Results for the 

Social Sciences by Susan E. Morgan, Tom Reichert and Tyler R. 

Harrison (Allyn and Bacon, 2002).

Misused Statistics (2nd Edition) by Herbert F. Spirer, Louise 

Spirer and A. J. Jaffe (M. Dekker, 1998).

Say It With Charts: The Executive’s Guide to Visual Communication

(4th Edition) by Gene Zelazny (McGraw-Hill, 2001).

Schaum’s Outline of Theory and Problems of Beginning Statistics

by Larry J. Stephens (McGraw-Hill, 1998).

Sorting Data: Collection and Analysis by Anthony P. M. Coxon 

(SAGE Publications, Inc., 1999).

Statistics (3rd Edition) by David Freedman, Robert Pisani and 

Roger Purves (W. W. Norton & Co., Inc., 1997).

Statistics: Concepts and Applications by Amir D. Aczel (Irwin, 1995).

Statistics and Data Analysis: An Introduction (2nd Edition) by 

Andrew F. Siegel and Charles J. Morgan (John Wiley & Sons, 

Inc., 1998).
12 Fire Data Analysis Handbook

Statistics: The Exploration & Analysis of Data (4th Edition) by Jay 

L. Devore and Roxy Peck (Brooks/Cole, 2001).

Your Statistical Consultant: Answers to Your Data Analysis 

Questions by Rae R. Newton and Kjell Erik Rudestam (SAGE 

Publications, Inc., 1999).
Chapter 2: Histograms 13

Chapter 2: Histograms

Data as a descriptive tool

“A picture is worth a thousand words” is an old saying that 

applies to numbers as well as words. The task of reaching 

conclusions from numbers is challenging, particularly when 

we are looking for trends and patterns in the data. It is for this 

reason that we turn our attention to histograms and other 

charts in this chapter and Chapter 3. These tools will assist 

you in understanding fire data, since the human mind seems 

to comprehend pictures quicker than words and numbers. 

However, before we delve into describing histograms and 

other charts, we define the types of variables used to create 

these charts.

Types of variables

For purposes of analysis, fire department variables can be 

divided into 2 types: qualitative variables and quantitative 

variables. 

Qualitative variables are variables that are classified into 

groups or categories. For example, fires can be broken into 

structure fires, vehicle fires, refuse fires, explosions, etc. 

Qualitative variables are also known as categorical variables 

(data) since they are not measured in quantity but segregated 

into groups. Examples of categorical data in the fire service 

would include property use, cause of ignition, extent of flame 

damage, etc. Most categorical variables that will be used in fire 

data analysis will be found in the NFIRS modules.

Quantitative variables always take on numerical values that 

reflect some type of measurement. Quantitative variables 

can be discrete or exact or can be analog or continuous. 

An example of a discrete variable would be the number of 

days in the month or year (1 through 30 or 1 through 365), 

but no fractions of days, whereas time, in hours, minutes, 

seconds and infinite fractions of seconds, would be analog or 

continuous. Other examples of quantitative variables would be 

the number of fires in a district over a period of time (discrete), 

the response time from alarm to arrival on the scene (analog), 

and the dollar losses of fires (discrete).
14 Fire Data Analysis Handbook

There is a distinction between a variable and data. A variable 

is a characteristic that varies or changes. For example, days of 

the week vary from Sunday through Saturday; months vary 

from January through December; and types of fires vary, such 

as structure fires, vehicle fires, residential fires, etc. A variable 

is said to be independent if its variation does not depend on 

another variable. Additionally, a variable is considered to be 

dependent if its values depend on another variable.

Whenever observations are made on a variable, data are 

created to be analyzed. Each time an NFIRS report is completed, 

data for the variables listed are created. For example, by listing 

the day of week, hour of day, month, type of situation found 

and values for all other applicable variables in the NFIRS Basic 

Module, data are created. The data then can be summarized in 

a variety of ways, such as tables, graphs and charts.

The techniques used to summarize data found in Chapters 2 

and 3 include:

Histograms

Bar

charts

Line

charts

Column

charts

Pie

charts

Dot

charts

Pictograms

This chapter describes histograms, while Chapter 3 is devoted 

to the other techniques. With these graphic aids, we can 

answer several basic questions. When are fires most likely to 

occur? What are the primary causes of residential fires? Vehicle 

fires? How many civilian injuries occurred last year by month? 

What are the ages of civilian casualties? What percent of the 

fire incidents have travel times less than 4 minutes? How many 

structure fires resulted in dollar losses greater than $50,000?

A histogram is a column graph where the height of the 

columns indicates the relative numbers, counts, frequencies 

or values of a variable. The values may be numeric, such as 

travel times, or nonnumeric, such as days of the week. The 

column may also be used to show the relative frequency

(proportion) or percentage of counts in each category. The 

relative frequency of a category is the count in a particular 

category divided by the total count. The following examples 

show how to organize and display fire data into histograms.
Chapter 2: Histograms 15

Example 1. One of the most fundamental ways to describe the 

fire problem is to show how fires are distributed by month, day 

of week and hour of day. Figure 2-1 shows a frequency list of 

fires by hour of day for Canton, Ohio, for 1 year. A list or array 

of numbers such as this is almost always the starting point for 

a descriptive analysis, but the numbers by themselves are not 

very useful. It is difficult to get a “feel” for what is happening 

by scanning a list of numbers.

To grasp what the numbers say in Figure 2-1, we can develop a 

frequency histogram, as shown in Figure 2-2. Similarly, Figures 

2-3 and 2-4 show histograms by day of week and month of 

year. Study these figures for a few minutes and draw your 

own conclusions about what they represent. Don’t dwell on 

individual numbers, but instead look for patterns. Ask yourself 

3 questions:

1. Where are the low points and high points in the histogram?

2. What groups of times (hours, days or months) have similar 

frequencies?

3. Is there anything in the histogram that runs counter to 

your experience?

Figure 2-1. Fires by hour of day — Canton, Ohio

Time period Number Time period Number

Midnight - 1 a.m. 15 Noon - 1 p.m. 31

1 - 2 a.m. 15 1 - 2 p.m. 33

2 - 3 a.m. 13 2 - 3 p.m. 39

3 - 4 a.m. 13 3 - 4 p.m. 35

4 - 5 a.m. 13 4 - 5 p.m. 46

5 - 6 a.m. 11 5 - 6 p.m. 39

6 - 7 a.m. 16 6 - 7 p.m. 30

7 - 8 a.m. 11 7 - 8 p.m. 50

8 - 9 a.m. 17 8 - 9 p.m. 32

9 - 10 a.m. 17 9 - 10 p.m. 29

10 - 11 a.m. 19 10 - 11 p.m. 28

11 a.m. - Noon 19 11 p.m. - Midnight 24
16 Fire Data Analysis Handbook

Answers to these questions provide the first insights into your 

fire data and any conclusions drawn from it.

While these histograms suggest several conclusions, the key 

ones are:

1. Canton has 2 distinct hourly patterns. The hours from noon 

to midnight overall have almost twice the fires than the 

hours from midnight to noon. The hours of 7 p.m. to 8 p.m. 

and 4 p.m. to 5 p.m. have more fires than any other hours 

in the day.

2. The fewest fires occur in the time period from 2 a.m. to 

6 a.m. and from 7 a.m. to 8 a.m.

3. Sunday sees the most fires with a continuous decline until 

Friday, which sees the fewest fires per day.

4. May has the most fires with June, July and November tied 

for second. The fewest number of fires occur in February.

With these histograms, we begin to see a picture of the fire 

problem in Canton. Histograms allow for an easy descriptive 

and analytical procedure without having to think too much 

about the numbers themselves. Graphical displays should 

always strive to convey an immediate message describing a 

particular aspect of the data.

Figure 2-2. Fires by hour of day — Canton, Ohio

0

10

20

30

40

50

60

11 p.m. - Midnight

10 - 11 p.m.

9 - 10 p.m.

1 - 2 p.m.2 - 3 p.m.3 - 4 p.m.4 - 5 p.m.5 - 6 p.m.6 - 7 p.m.7 - 8 p.m.8 - 9 p.m.

Noon - 1 p.m.

11 a.m. - Noon

10 - 11 a.m.

9 - 10 a.m.

1 - 2 a.m.2 - 3 a.m.3 - 4 a.m.4 - 5 a.m.5 - 6 a.m.6 - 7 a.m.7 - 8 a.m.8 - 9 a.m.

Midnight - 1 a.m.

Hour of day

Number of incidents

1515 131313 11 11

16 17 17 1919

31 33

39 39

30

35

46 50

322928

24
Chapter 2: Histograms 17

Figure 2-3. Fires by day of week — Canton, Ohio

60

70

80

90

100

Saturday

Friday

Thursday

Wednesday

Monday

Tuesday

Sunday

Day of week

Number of incidents

96

91 90

83

68

76

87

Figure 2-4. Fires by month — Canton, Ohio

20

30

40

50

60

70

80

Jan Feb Mar Apr JulJunMay Aug Sep Oct Nov Dec

Month of year

Number of incidents

36

30

46 47

68

74

61 61

45 47 43

61

44

Example 2. Ages of civilian casualties. Suppose a fire chief 

is interested in developing a fire prevention program aimed 

at reducing civilian injuries and deaths. Descriptive data on 

civilian casualties is available from the NFIRS reports, and 

there are a number of different descriptions that could be 

developed from the data. One of the most basic is descriptive 

data on the ages of civilian casualties.

Figure 2-5 shows the ages of civilians injured or killed in fires in 

Denver, Colorado, for 1 year. This distribution is considerably 

different from the previous histograms primarily because it 

does not have the same “smoothness.” However, the 5-year 
18 Fire Data Analysis Handbook

age groups show some interesting patterns. For example, the 

age group 36 to 40 accounts for the most civilian casualties, 

followed in frequency by 26 to 30 and 46 to 50, respectively. 

Also of interest is how the frequency takes a rather sudden 

drop for the 16 to 20 and 56 to 60 age groups. Spikes in the 

data occur at the 26 to 30 and 36 to 40 age groups. 

The figure also reveals several gaps in the data for ages 6 to 10 

and 76 to 80 as a result of no reported casualties in these age 

groups. Due to these gaps at either end of the distribution, 2 

outliers are created in the under 5 and 81 to 85 age groups.

Figure 2-5. Ages of civilian casualties — Denver, Colorado

0

2

4

6

8

10

12

14

11-15

16-20

21-25

26-30

31-35

36-40

41-45

46-50

51-55

56-60

61-65

66-70

71-75

81-85

1-5

Ages of civilian casualties

Number of casualties

2 2

5

6 6

9

2

12

5

8 7

3

5

3

2

Notes: Age was not provided for 7 casualties. 52% of the casualties were between 26 and 

50 years old.

Spikes are high or low points that stand out in a histogram. 

Gaps are spaces in a histogram reflecting low frequency of data. 

Outliers are extreme values isolated from the body of data.

In histograms and other charts, it is sometimes useful to 

include comments and conclusions with the chart. In Figure 

2-5, a note was provided that 7 casualty records did not 

include age information and were therefore not included in 

the histogram. Other notes provide summary information on 

the data such as the percent of casualties between the ages 26 

and 50 years old. Anyone studying the histogram could reach 

the same conclusion, but the summary saves time and effort.
Chapter 2: Histograms 19

Example 3. Response times to fires. Response times to 

fires are one of the most important data sets to study in fire 

departments. Many fire departments have objectives for average 

response times to fires and try to allocate personnel to achieve 

these response times. Figure 2-6 shows a frequency distribution 

for response times to fires in Boston, Massachusetts.

Figure 2-6. Response times — Boston, Massachusetts

Response time Frequency

Less than 1 minute 129

1 to 2 minutes 206

2 to 3 minutes 759

3 to 4 minutes 1,406

4 to 5 minutes 1,312

5 to 6 minutes 747

6 to 7 minutes 384

7 to 8 minutes 206

8 to 9 minutes 110

9 to 10 minutes 62

10 to 11 minutes 18

11 to 12 minutes 15

12 to 13 minutes 15

13 to 14 minutes 10

14 to 15 minutes 5

15 to 16 minutes 6

16 to 17 minutes 2

17 to 18 minutes 0

18 to 19 minutes 1

19 to 20 minutes 1

Total fire calls 5,394

Notice in this example that the times are clustered at the low 

end of the distribution as would be expected since response 

times to fires are generally low for most fire departments.
20 Fire Data Analysis Handbook

Figure 2-7 provides a frequency histogram for this distribution. 

In this figure, we have combined the last few points into a 

category of 10 minutes or more. A histogram with the same 

shape as in this figure is said to be skewed to the right or 

skewed toward high values. What is meant by these terms 

is that the distribution is not symmetrical, but instead has 

a single peak on the left side of the distribution with a long 

tail toward the right. In fire departments, on-scene time data 

(from time of arrival to time back in service) and fire dollar loss 

data also reflect values skewed to the right.

Figure 2-7. Response times — Boston, Massachusetts

0

200

400

600

800

1,000

1,200

1,400

1,600

<1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 >10

Number of minutes

Number of incidents

129 206

759

1,406

110

1,312

747

384

206

62 73

Developing a histogram

Making a histogram is relatively straightforward:

1. Choose the number of groups for classifying the data. In 

most cases, 5 to 10 groups are sufficient, but there are 

exceptions, such as histograms by hour of day. Sometimes 

the groups are natural, as in our exhibits by day of week and 

month. With other data, developing appropriate intervals 

will be necessary as was done in Figure 2-5 with the ages 

of civilian casualties.

2. Determine the number of events (fires, casualties, etc.) for 

each of the groups.

3. For data such as ages and response times, intervals usually 

need to be defined. For these intervals, convenient whole 

numbers should be chosen. That is, try to avoid the use of 
Chapter 2: Histograms 21

fractions in the groups and always make the intervals the 

same width. In Figure 2-5, intervals of 5 years were used 

for grouping the data. Data such as day of week do not 

require this step since their intervals are naturally defined.

4. Determine the number of observations in each group. 

Statistical software packages are particularly useful in this 

step since they usually include routines for tabulating data.

5. Choose appropriate scales for each axis to accommodate 

the data. Again, most statistical packages will do this with 

a default setting.

6. Display the frequencies with vertical bars.

Do not expect to get a histogram — or any other type of chart — 

exactly right on the first try. You may need several tries before 

you get a satisfactory histogram.

The histograms presented in the previous section offer good 

examples of different characteristics for describing the data. 

In Beginning Statistics with Data Analysis, a text by Mosteller et 

al. (1983), the following definitions of histogram characteristics 

are presented:

1. Peaks and valleys. The peaks and valleys in a histogram 

indicate the values that appear most frequently (peaks) or 

least frequently (valleys). Figure 2-2 shows clear peaks and 

valleys for incidents by hour of day.

2. Spikes and holes. These are high and low points that stand 

out in the histogram. In Figure 2-5, for example, there is a 

spike for the 36 to 40 age group and a hole for the 16 to 

20 age group.

3. Outliers. Extreme values are sometimes called outliers 

and are points that are isolated from the body of the data. 

In Figure 2-5, there are 2 outliers: the under 5 and the 81 

to 85 age groups.

4. Gaps. Spaces may reflect important aspects of a histogram. 

In Figure 2-5, there are gaps in the 6 to 10 and the 76 to 80 

age groups.
22 Fire Data Analysis Handbook

5. Symmetry. Sometimes a histogram will be balanced along 

a central value. When this happens, the histogram is easier 

to interpret. The central value is both the mean (average) 

for the distribution and the median (half the data points 

will be below this value and half above).

Cumulative frequencies

There are 2 other types of distributions which will be 

important in later chapters: the cumulative frequency

and the cumulative percentage frequency. A cumulative 

frequency is the number of data points that are less than or 

equal to a given value. A cumulative percentage frequency 

converts the cumulative frequency into percentages.

Example 4. With the data in Figure 2-6, we can calculate the 

cumulative frequency and cumulative percentages for the 

response time data from Boston, Massachusetts, found in 

Figure 2-8.

Figure 2-8. Cumulative response times — Boston, 

Massachusetts

Response Frequency Cumulative 

frequency

Cumulative 

percent

Less than 1 minute 129 129 2.4

1 to 2 minutes 206 335 6.2

2 to 3 minutes 759 1,094 20.3

3 to 4 minutes 1,406 2,500 46.3

4 to 5 minutes 1,312 3,812 70.7

5 to 6 minutes 747 4,559 84.5

6 to 7 minutes 384 4,943 91.6

7 to 8 minutes 206 5,149 95.5

8 to 9 minutes 110 5,259 97.5

9 to 10 minutes 62 5,321 98.6

10 or more minutes 73 5,394 100.0

Total 5,394 100.0
Chapter 2: Histograms 23

The first entry under the “cumulative frequency” column is 

129, which is the same as in the “frequency” column. The 

second entry shows 335, which is 129 + 206, the sum of the 

first 2 entries in the “frequency” column. By adding these 

2 numbers, we can say that 335 incidents have response 

times less than 2 minutes. The next entry is 1,094 (129 + 206 + 

759) and means that 1,094 incidents have response times less 

than 3 minutes. The cumulative frequencies continue in this 

manner with the last entry in the column always equal to the 

total number of incidents in the analysis.

The last column, labeled “cumulative percent,” merely converts 

the cumulative frequencies into percentages. This step is 

accomplished by dividing each cumulative frequency by 5,394, 

which is the total number of incidents. The column shows that 

2.4% of the incidents have response times less than 1 minute, 

6.2% less than 2 minutes, 20.3% less than 3 minutes, etc.

In general, cumulative percentages describe data in “more than” 

and “less than” terms. We can conclude, for example, that about 

half the calls have response times of less than 4 minutes and 

about 95% have response times less than 8 minutes. Response 

times exceed 10 minutes in only about 1% of the calls.

Summary

A list of numbers is frequently the starting point for analysis. 

If the question of interest is for specific information, then the 

list of numbers serves the purpose. For example, Figure 2-1 is 

useful if we are asked about how many fires occurred between 

2 and 3 a.m., or if we want to know the exact difference 

between the busiest and the least busy hour. On the other 

hand, Figure 2-1 is not very useful for determining the 6 busiest 

hours of the day.

Histograms provide a much better method for getting the feel 

of a list of numbers and answering several questions about 

relationships. The patterns in a histogram are especially 

important, such as high and low frequencies and trends 

indicated by spikes, outliers and gaps. Histograms give quick 

graphic representations of the data that otherwise would be 

hidden and hard to dig out of a table of numbers.
24 Fire Data Analysis Handbook
Chapter 3: Charts 25

Chapter 3: Charts

Introduction

In this chapter, we extend beyond histograms to other types 

of charts. Histograms are only one of many different ways of 

presenting data. As an analyst, you must decide which type 

of chart best portrays the results you want to represent. A 

histogram may serve as the best vehicle in some cases, but 

other types of charts should be considered, such as bar charts, 

line charts, pie charts, dot charts and pictograms. Each of 

these types of charts are discussed in this chapter.

There are 2 questions to keep in mind throughout this process:

1. What are the main conclusions from your analysis?

2. What is the best way to display the conclusions?

As with the previous chapter, several sets of real fire data 

are presented. You should study each example carefully and 

draw your own conclusions about the results. You may, in fact, 

disagree with what the book emphasizes, or you may identify 

an aspect of the data that was overlooked. In either case, the 

point is to think about how you would present your viewpoints 

in a graphical format to a given audience. The audience may 

be an internal group of managers, an outside association or 

group of citizens, or even your own city or county council. The 

audience itself influences the type of chart that is selected.

Therefore, the first step is to determine the key results from 

the data. Once they have been identified, select the best type 

of chart to convey them. Often it is helpful to try different 

charts to determine the best presentation for a particular 

audience and data set.

Each of the following sections describes a different type of 

chart. At the end of the chapter, you will find guidelines on 

selecting a type of chart suitable for different conclusions.
26 Fire Data Analysis Handbook

Bar charts

A bar chart is one of the simplest and most effective ways to 

display data.

In a bar chart, a bar is drawn for each category of data allowing 

for a visual comparison of the results. For example, the figures 

in Figure 3-1 give the causes of ignition (from NFIRS 5.0 codes) 

for 12,600 structure fires reported in Chicago, Illinois, for 1 year.

Interest in a list of this type usually centers on how the items 

compare to each other. What is the leading cause of ignition 

in structure fires? How do unintentional causes compare to 

intentional ones? How many causes are never determined?

Some results can be determined relatively easily from the 

list of numbers. For example, “cause undetermined after 

investigation” is clearly the leading cause of ignition followed by 

“intentional,” “equipment failure” and “unintentional,” all close 

in number. The remaining 3 — “not reported,” “other” and “act 

of nature” — account for less than 1% combined. While these 

comparisons can be made from the list, they require mental 

manipulations and are not easily made or retained in full.

Figure 3-1. Cause of ignition for structure fires — 

Chicago, Illinois

Cause of ignition Number Percent

Intentional 2,771 22.0

Unintentional 2,583 20.5

Equipment failure 2,654 21.1

Act of nature 29 0.2

Cause, other 40 0.3

Not reported 55 0.4

Cause undetermined 

after investigation 4,468 35.5

Total 12,600 100.0
Chapter 3: Charts 27

A bar chart overcomes these problems by presenting the data 

in frequency order as displayed in Figure 3-2. The horizontal 

dimension gives the percent, while the vertical dimension 

shows the category labels. The bars are presented in numerical 

order, starting with “undetermined” as the most frequent. 

Each bar also contains the number of fires for that cause of 

ignition as additional information to the reader.

It should also be noted that the category “cause under 

investigation” had no cases reported, but this fact is mentioned 

in a footnote since it is a listed option in the NFIRS module. 

Also in a footnote are the complete titles of 2 of the categories 

that were abbreviated in the table listing.

As a general rule, the horizontal dimension in a bar chart is 

numeric, such as percentages or other numbers, while the 

vertical dimension shows the labels for the items in a category. 

It is not always necessary to include numbers in each bar, 

especially if there is an accompanying table or list, but they can 

be useful to readers unfamiliar with the data. If the numbers 

are omitted from the chart, a total number should be provided 

either in the title or a footnote.

Figure 3-2. Cause of ignition for structure fires — 

Chicago, Illinois

Percent

29

40

55

2,583

2,654

2,771

0 10 20 30 40

Act of nature

Cause - other

Not reported

Unintentional

**Equipment failure

Intentional

*

Undetermined 4,468

*”Undetermined after investigation.”

**Or “heat source failure.”

Note: No cases reported under “cause under investigation” category.
28 Fire Data Analysis Handbook

A clustered bar chart shows 2 categories in the same chart. 

In Figure 3-3, for example, the causes of ignition for structure 

fires in Chicago are shown in a residential versus nonresidential 

format. The figure shows that fires that are undetermined 

after investigation comprise over 40% of the nonresidential 

fires and only 16% of the residential ones. Interestingly, the 

chart also shows an almost exact ratio of 40% and 15% for 

unintentional causes of residential and nonresidential fires, 

respectively. In addition, while the percentages are close for 

residential and nonresidential fires under the “unintentional” 

and “equipment failure” categories, the numbers differ by 3 

to 4 times due to the large difference in total fires between 

residential and nonresidential. The clustered or paired bar 

chart clearly shows the differences in ignition causes for these 

2 types of structure fires.

Figure 3-3. Cause of ignition residential versus 

nonresidential fires — Chicago, Illinois

0 10 20 30 40 50

Nonresidential

Residential

Act of nature

Cause - other

Not reported

Unintentional

Equipment failure

Intentional

Undetermined

Percent

16

13

31

9

18

37

1,041 1,522

2,166

488

637

2,134

410

4,058

Column charts

In Chapter 2, several column charts were displayed. For 

example, Figures 2-2, 2-3 and 2-4 showed Canton, Ohio fires 

by hour of day, day of week and month, respectively. These 

are all examples of time series presented as column charts.
Chapter 3: Charts 29

Column char ts of this t ype are par ticularly useful in 

demonstrating change over time. Where is the series 

increasing, decreasing or staying about the same? If the 

analysis shows change over time, then column charts are 

particularly beneficial in presenting the changes.

As an example, the figure from Chapter 2 on fires by hour of 

day is repeated in Figure 3-4. By looking from left to right, you 

can visualize the change. The horizontal scale shows the hours, 

but it is not really needed to be able to see the overall changes. 

The numbers of reported fires are low in the early morning 

hours, then increase in the afternoon and evening hours.

Figure 3-4. Fires by hour of day — Canton, Ohio

0

10

20

30

40

50

60

11 p.m. - Midnight

10 - 11 p.m.

9 - 10 p.m.

1 - 2 p.m.2 - 3 p.m.3 - 4 p.m.4 - 5 p.m.5 - 6 p.m.6 - 7 p.m.7 - 8 p.m.8 - 9 p.m.

Noon - 1 p.m.

11 a.m. - Noon

10 - 11 a.m.

9 - 10 a.m.

1 - 2 a.m.2 - 3 a.m.3 - 4 a.m.4 - 5 a.m.5 - 6 a.m.6 - 7 a.m.7 - 8 a.m.8 - 9 a.m.

Midnight - 1 a.m.

Hour of day

Number of incidents

1515 131313 11 11

16 17 17 1919

31 33

39 39

30

35

46 50

322928

24

Column charts show frequency distributions that allow for easy 

identification of trends and other characteristics, particularly 

with time series data. The horizontal scale defines the natural 

groupings for the chart and the columns give the frequencies.

Another good application of column charts is to show 

comparisons across sets of data. Figure 3-5 lists the causes 

of ignition from Figure 3-3. Due to their small numbers for 

illustrative purposes, the “not reported,” “causes - other” and 

“act of nature” categories have been combined into “other.” 

Comparisons between the venues are not easy because the 
30 Fire Data Analysis Handbook

totals differ so much. Nonresidential fires total just under 

10,000 while residential fires have 2,619. A simple way to 

overcome this problem is to develop percentages.

By converting the residential and nonresidential figures to 

percentages, as shown at the bottom of the figure, you can 

make a better comparison. The percentages for both add up to 

100%. While there are many conclusions that could be drawn 

from these percentages, the key ones are:

ĵ “Intentional,” “equipment failure” and “other” account 

for about the same percentages in both residential and 

nonresidential fires.

ĵ “Unintentional” fires account for 40% of the residential 

fires, while 41% of the nonresidential fires fall into the 

“undetermined” category.

Figure 3-5. Comparison of causes of ignition in residential 

versus nonresidential fires — Chicago, Illinois

Cause of ignition Residential Nonresidential

Intentional 637 2,134

Unintentional 1,041 1,542

Equipment failure 488 2,166

Cause undetermined 

after investigation 410 4,058

Other 43 81

Total 2,619 9,981

Cause of ignition Residential Nonresidential

Intentional 24.3% 21.4%

Unintentional 39.7% 15.4%

Equipment failure 18.6% 21.7%

Cause undetermined 

after investigation 15.7% 40.7%

Other 1.6% 0.8%

Total 100.0% 100.0%
Chapter 3: Charts 31

To display this result, stacked column charts were developed, 

as shown in Figure 3-6, using the percentages for each cause 

of ignition. The columns have the same height since they both 

total 100%. The colors highlight the differences among the 

causes of ignition. The results just discussed should be clear 

from the figure.

Figure 3-6. Comparison of causes of ignition by 

percent — Chicago, Illinois

0

20

40

60

80

100

120

Intentional

Unintentional

Equipment 

failure

Other

Undetermined

Residential Nonresidential

Note: “Other” causes include “not reported” and “act of nature.” 

Line charts

Effective presentation of time series data also may be 

developed from line charts. Figure 3-7 shows a line chart of 

fires by hour of day for Canton, Ohio, previously displayed as 

a histogram in Figure 2-2. The line chart immediately highlights 

the jump in fires from a sharp rise in the early afternoon until a 

peak at around 8:00 p.m. Many statisticians believe that a line 

chart is the clearest way for showing increases, decreases and 

fluctuations in a time series.
32 Fire Data Analysis Handbook

Figure 3-7. Fires by hour of day — Canton, Ohio

0

10

20

30

40

50

60

11 p.m. - Midnight

10 - 11 p.m.

9 - 10 p.m.

1 - 2 p.m.2 - 3 p.m.3 - 4 p.m.4 - 5 p.m.5 - 6 p.m.6 - 7 p.m.7 - 8 p.m.8 - 9 p.m.

Noon - 1 p.m.

11 a.m. - Noon

10 - 11 a.m.

9 - 10 a.m.

1 - 2 a.m.2 - 3 a.m.3 - 4 a.m.4 - 5 a.m.5 - 6 a.m.6 - 7 a.m.7 - 8 a.m.8 - 9 a.m.

Midnight - 1 a.m.

Hour of day

Number of incidents

Pie charts

A pie chart is an effective way of showing how each 

component contributes to the whole. In a pie chart, each 

wedge represents the amount for a given category. The entire 

pie chart accounts for all of the categories.

For example, Figure 3-8 shows the causes of ignition for 

structure fires in the Chicago Fire Department for 1 year 

divided into “undetermined,” “equipment failure,” “intentional,” 

“unintentional” and “other.” The percentages are included 

with each wedge label. Although the percentage numbers are 

not necessary, they aid in comparisons of the wedges. The 

pie chart emphasizes the fact that the largest percentage 

of fire causes is undetermined. In addition, “intentional,” 

“unintentional” and “equipment failure” all account for about 

the same percent of the causes.
Chapter 3: Charts 33

Figure 3-8. Cause of ignition for fires — Chicago, Illinois

Undetermined

35.5%

Other

1.0%

Equipment failure

21.1%

Unintentional

20.5%

Intentional

22.0%

In developing pie charts, you should follow these rules:

ĵ Convert data to percentages.

ĵ Keep the number of wedges to 6 or less. If there are more 

than 6, keep the most important 5 and group the rest into 

an “other” category.

ĵ Position the most important wedge starting at the 12 o’clock 

position.

ĵ Maintain distinct color differences among the wedges.

While pie charts are popular, they are probably the least 

effective way of displaying results. For example, it may be hard 

to compare wedges within a pie chart to determine their rank. 

Similarly, it takes time and effort to compare several pie charts 

because they are separate figures.

Dot charts

Dot charts or scatter diagrams emphasize the relationship 

between 2 variables. For example, the 10-year trend in other 

residential fires over a decade was generally a decrease from 

a high of 15,000 to a low of 11,000. During these years, a 

decrease in fire deaths also occurred. You expect deaths to 

decrease with a decrease in fires. This relationship is depicted 

in Figure 3-9.
34 Fire Data Analysis Handbook

Figure 3-9. Fires and deaths

Fires (thousands)

Deaths

10 11 12 13 14 15 16

20

40

60

80

100

120

Outlier

Year 5 Year 1

Year 3

Year 2 Year 4

Year 8

Year 9 Year 10

Year 6

Year 7

The figure is a dot chart for fires versus deaths for the 10 years. 

Fires are along the horizontal or x-axis, while deaths are along 

the vertical or y-axis. The pattern is the important aspect of 

a dot chart, rather than the individual dots. The horizontal 

scale (x-axis) should reflect the independent variable, while 

the vertical scale reflects the dependent variable. That is to say 

that a decrease in fires (the independent variable) has a positive 

correlation with a decrease in fire-related deaths (the dependent 

variable). Recall from earlier in this text that a variable is said 

to be independent if its variation does not depend on another 

variable. Additionally, a variable is considered to be dependent 

if its values depend on another variable.

Another useful application of scatter diagrams is to identify 

outliers in the data. In Chapter 2, outliers were defined as 

points that are isolated from the body of the data. In Figure 

3-9, there is a general pattern showing a decrease in deaths 

over time as fires decrease. While the decrease in fires pattern 

is maintained for Year 5, however, deaths rise to the highest 

count over the 10-year period. Therefore, Year 5 has many 

more fire-related deaths than expected based on its amount 

of fires. This outlier from the general pattern can be useful in 

revealing an area of further data analysis that would account 

for this discrepancy from the rest of the data.
Chapter 3: Charts 35

Pictograms

The final type of chart takes advantage of pictures to display 

data. Data by geographical areas, such as counties, census 

tracts or fire districts can be presented on maps showing 

the boundaries of the areas. Figure 3-10, for example, 

shows firefighter deaths by region for 1 year. Each region is 

broken down by career, volunteer, and, if applicable, wildland 

department.

Figure 3-10. Firefighter deaths by region

South

Career - 14

Volunteer - 13

Total - 27

North Central

Career - 5

Volunteer - 15

Total - 20

Northeast

Career - 5

Volunteer - 25

Wildland - 2

Total - 32

West

Career - 8

Volunteer - 2

Wildland - 5

Total - 15

The key is that presentation in this manner is more effective 

than any listing of the death rates. It can be easily seen that:

ĵ Career deaths in the South are 2 to 3 times more than in 

other regions.

ĵ Volunteer deaths in the Western region are a fraction of 

those in the rest of the country.

ĵ The Northeast has the most total deaths largely due to a 

high number of volunteer deaths that is almost double the 

next largest region.

You can easily imagine other pictograms for state and local 

data. At the state level, data from individual counties may be 

collected. A pictogram provides a good way of depicting the 
36 Fire Data Analysis Handbook

county data by taking a state map showing county boundaries 

and developing a figure similar to Figure 3-10. Similarly, for a 

local jurisdiction, such as a city or a county, there may be data 

for individual fire districts. A jurisdiction map with fire district 

boundaries may be an effective way of presenting the data.

Summary

In this chapter, 6 types of charts were presented: bar 

charts, column charts, line charts, pie charts, dot charts and 

pictograms. The primary purpose of using any chart is to 

indicate conclusions more quickly and clearly than is possible 

with tables or numbers. It may be necessary to try several 

types of charts before the most appropriate one is found, 

but in a chart, simplicity is the key. The message is what is 

important, so the chart form should not interfere with it.

As a quick reference guide on chart selection, the following is 

recommended:

ĵ Use a bar chart with categorical data when the objective 

is to show how the items in a category rank. Most fire data 

are categorical, such as cause of ignition, property use, 

area of origin, nature of injury, etc. These are reflected in 

the NFIRS modules.

ĵ Use a column or line chart for data with a natural order, 

such as hours, months or age groups. The chart will reflect 

the general pattern and indicate points of special interest, 

such as spikes, holes, gaps and outliers.

ĵ A pie chart is beneficial when the objective is to show how 

the components relate to the whole. It is recommended that 

the number of components be kept to 6 or less and that 

the forming of several pie charts for comparison purposes 

be avoided.

ĵ A dot chart or scatter plot depicts the relationship between 

2 variables. Generally, these variables are continuous rather 

than categorical. The pattern between the 2 variables is the 

important aspect for a dot chart.

ĵ A pictogram is a pictorial representation of the data. 

Breakdowns by geographic areas, for example, are 

effectively shown by a pictogram.
Chapter 4: Basic Statistics 37

Chapter 4: Basic Statistics

Data can be summarized in a variety of ways by using tables, 

graphs and charts. In this chapter, ideas about summarizing 

data will be extended by introducing basic descriptive 

measures to include measures of central tendency (i.e., mode, 

mean and median) as well as measures of dispersion (i.e., 

range, variance and standard deviation).

Measures of central tendency

Measures of central tendency provide a single summary 

figure that best describes the central location of an entire 

distribution. The 3 most common measures of central 

tendency are the mode, the mean and the median. Each of 

the measures are defined, and the individual properties and 

uses for each measure are also discussed.

The mode is the value that occurs most frequently in a 

distribution. It is, therefore, easily recognized since no 

calculations are necessary.

The mean is also known as the arithmetic mean or average. 

However, since the term average is sometimes used 

indiscriminately for any measure of central tendency, it 

should be avoided. It is defined as the sum of all values in 

a distribution divided by the total number of values. For 

example, suppose that travel times to 9 incidents are 3 

minutes, 2 minutes, 4 minutes, 1 minute, 2 minutes, 3 minutes, 

3 minutes, 4 minutes and 3 minutes. Adding these travel times 

gives 25 minutes in total, and dividing by 9 yields a mean travel 

time of 2.78 minutes.

The third measure of central tendency is the median, which is 

defined as the middle value (50th percentile) of a distribution. 

To determine the median, the data must be ordered. Using 

the 9 travel times from the above example, they would look as 

follows if arranged in order: 1, 2, 2, 3, 3, 3, 3, 4, 4. The median 

is the fifth, or middle, value, which is 3 minutes. There are 4 

data values below and 4 data values above. In other words, 

50% of the values lie on either side of the median, placing it at 

the 50th percentile.
38 Fire Data Analysis Handbook

If there had been an even number of data values, then the 

median would have been the mean of the 2 middle values. 

For example, if the on-site times for 10 fire incidents were 12, 

15, 17, 25, 27, 29, 32, 35, 37 and 42 minutes, then the 2 middle 

values would be 27 and 29. Totaling them and dividing by 2 

(calculating the mean value) results in a median value of 28. 

Again, the median splits the values with 5 below and 5 above.

Properties and uses for measures of central 

tendency

The mode is the only measure of central tendency that can be 

used for qualitative data. This is really its only redeeming quality 

other than to serve as an additional qualifier for a distribution. 

The mode by itself is an unstable measure of central tendency. 

Equal size samples taken from a distribution are likely to have 

different modes. Further, on many occasions, distributions have 

more than 1 mode (bimodal), which adds to the confusion.

The median is a better choice than the mode for a measure 

of central tendency. Unlike the mode, it cannot be used with 

qualitative data but with quantitative variables. The median 

on-scene time for fires or the median dollar loss for fires can be 

determined. However, the “median type of fire” or the “median 

cause of ignition” has no meaning since these are qualitative 

variables. Responding to how many values lie above and below, 

but not to how far away, the median is less sensitive than the 

mean to the presence of a few extreme values.

Generally, the mean is the most frequently used measure of 

central tendency. Unlike the mode and the median, the mean is 

responsive to the exact position of each value in a distribution. 

It serves as a fulcrum point, balancing all of the values in a 

distribution. Consequently, the mean is very sensitive to 

extreme values (outliers) in a distribution. 

For data from skewed distributions, the use of the median is 

a better choice than the mean because it is not influenced by 

large outliers. When a measure of central tendency needs to 

reflect the total of the values, the mean is the best choice since 

it is the only measure based on this quantity. Another of the 

more important characteristics of the mean is its stability over 
Chapter 4: Basic Statistics 39

samples drawn from a distribution. This becomes especially 

important when further statistical computation is done.

Measures of dispersion

While measures of central tendency provide a summary of 

the values in a distribution, measures of dispersion provide 

a summary of the variability or spread of the values in a 

distribution. Measures of dispersion express quantitatively 

the extent to which the values in a distribution scatter about 

or cluster together. The 3 main measures of dispersion are the 

range, variance and standard deviation. As with the measures 

of central tendency, they will first be defined and then their 

properties and uses will be discussed.

The range is the most basic measure of dispersion. Its 

definition is simply the difference between the lowest and 

highest value in a distribution. For example, with the 10 on-site 

times used in the median discussion, the lowest value is 12 

minutes and the highest is 42 minutes. Therefore, the range 

is 30 minutes.

Another measure of the variability of a distribution is the 

variance. To calculate the variance, it is necessary to first 

obtain what is known as the deviation values of a distribution. 

The deviation values are the difference between the values 

in a distribution and its mean. Since the mean is the balance 

point of the values in a distribution, the total of the deviation 

values would be 0. Therefore, to calculate the variance, it is 

necessary to square the deviation values to eliminate the 

negative values.

To illustrate the calculation of a variance, the 9 travel times 

used in the example for the mean are used. In Figure 4-1, the 

mean of 2.78 has been subtracted from each individual travel 

time and the result squared.
40 Fire Data Analysis Handbook

Figure 4-1. Calculation of variation

Travel time Travel time - mean

(2.78) Squared

1 -1.78 3.17

2 -0.78 0.61

2 -0.78 0.61

3 0.22 0.05

3 0.22 0.05

3 0.22 0.05

3 0.22 0.05

4 1.22 1.49

4 1.22 1.49

Total 0.00 7.57

Variance .95

The middle column displays the amount of deviation from 

the mean for each point. The first deviation is -1.78 (1 minute 

minus 2.78 minutes), indicating that this travel time is 1.78 

units from the mean and is to the left of the mean (since the 

sign is negative). Note that the sum of the middle column is 0; 

that is, the sum of the deviations from the mean is 0. In fact, an 

alternative definition for the mean is that it is the only number 

with this property.

In the right column is the square of each deviation. The sum 

of the squared deviations is 7.57, and the variance is obtained 

by dividing this sum by 8, which is 1 less than the total number 

of values. The reason for subtracting 1 from the total number 

of values will be discussed shortly. The variance from this 

calculation is then 0.95. Since the variance is small compared 

to the mean, it indicates that the values are close to the mean.

In discussing the calculation of the variance, the sum of the 

squared deviations was divided by the number of values 

minus 1. This was done to correct for a statistical error that 

results when using inferential statistics. If the distribution is 

the entire amount of values being considered, then dividing by 

that number is perfectly legitimate. However, if the distribution 

is merely a sample of a larger distribution, which it usually is, 
Chapter 4: Basic Statistics 41

then a better representation of the entire population of values 

can be obtained by subtracting 1 from the sample distribution. 

The final measure of dispersion is the standard deviation. It 

is obtained by taking the square root of the variance. In the 

current example, the standard deviation is 0.97 since this is 

the square root of 0.95. This means that the spread (variability) 

around the mean is not very large (in this case less than 1.0 

compared to a mean travel time of 2.78 minutes). Therefore, 

the mean is a good descriptor of the data in this example.

Normal distribution and standard score

Unless there is a compelling reason otherwise, statisticians 

usually assume a normal distribution for any given set of 

values. As shown in Figure 4-2, a normal distribution is equally 

spread out in the general shape of a bell. In fact, it is known 

as the bell curve. In a normal distribution, the mean, the 

median and the mode are the same. Half the values are above 

the mean and half below. Most of the values, 68%, fall within 

1 standard deviation on either side of the mean; within 2 

standard deviations, 95%; and within 3 standard deviations, 

99.7% of the distribution is represented.

Figure 4-2. Normal distribution

High Mean

Relative frequency

Low

-4 -3 -2 -1

Z

95.44%

99.74%

-1 standard

deviation

+1 standard

deviation

68.26%

43210
42 Fire Data Analysis Handbook

By using a standard score, it is possible to compare values 

from different distributions on an equal basis. A standard 

score is a derived score that describes how far a given value 

in a distribution is from some reference point, typically the 

mean, in terms of standard deviation units. One of the most 

commonly used standard scores is the z-score. Transforming 

the values of a distribution to z-scores changes the mean to 

0 and the standard deviation to 1, but does not change the 

shape of the distribution. For example, in the travel times used 

in Figure 4-1, a z-score of 1 would be equivalent to a score of 

3.75 minutes. That would be calculated by adding the mean of 

2.78 to the standard deviation of 0.973. In another distribution 

of travel times with a different mean and standard deviation, 

a z-score of 1 would be totally different. However, using the 

z-scores, they could be compared equally without distorting 

the original distributions.

Properties and uses for measures of dispersion

The range is ideal for preliminary work or in circumstances 

where precision is not an important requirement. However, it is 

not sensitive to the total condition of the distribution since only 

the 2 outermost values determine its calculation. Therefore, the 

range is of little use beyond the descriptive level.

Since the variance is the mean of the squares of the deviation 

values of a distribution, it is responsive to the exact position 

of each value in a distribution. It can, therefore, be very 

important in inferential statistics because of its resistance to 

sampling variation. However, it is of little use in descriptive 

statistics because it is expressed in squared units.

The standard deviation, like the mean and the variance (from 

which it is derived), is responsive to the exact position of 

every value in a distribution. Because it is calculated by using 

deviations from the mean, the standard deviation increases or 

decreases as the individual values shift away from or toward 

the mean. Like the mean, it is influenced by extreme scores, 

especially with distributions that have a small amount of values. 

As the number of values increase, each individual value has 

less ability to shift the mean and the standard deviation. If the 

mean and the standard deviation of a distribution are known, 

a fairly accurate picture of the distribution can be obtained.
Chapter 4: Basic Statistics 43

Once again, using the travel time example from Figure 4-1, the 

mean is 2.78 and the standard deviation is 0.973. Assuming a 

normal distribution, 1 standard deviation from the mean in 

both directions should cover 68% of the values. In this case, 

values between 1.807 and 3.753 include 2, 2, 3, 3, 3 and 3. Since 

there are 9 values in the distribution, the 6 values that fall 

within 1 standard deviation from the mean account for 67%. 

Considering its small size, that is an extremely accurate picture 

of the distribution. It is also a good example of how powerful 

the combination of the mean and the standard deviation 

can be. Each are the best measures of their type (central 

tendency and dispersion) and both are used extensively in 

more sophisticated statistical calculations.

Skewed distributions

Even though statisticians assume a normal distribution based 

on a first impression, not all distributions are normal or 

symmetrical. As stated before, in a normal distribution, the 

mean, median and mode are all the same. However, this is 

not the case with skewed distributions. As shown in Figure 

4-3, distributions can be skewed positively or negatively. In a 

positive skew, the extreme scores are at the positive end of 

the distribution. This exhibits the “tail” on the right side and 

pulls the mean to the right. Since the median and the mode 

are less responsive to extreme scores, they remain to the left 

of the mean. So, in a positively skewed distribution, the mean 

has the highest value with the median in the middle, and the 

mode is the smallest value. Conversely, in a distribution with 

a negative skew, the extreme values and the “tail” are at the 

negative end, the mean is the smallest value with the median 

in the middle, and the mode is the largest value.
44 Fire Data Analysis Handbook

Figure 4-3. Skewed distributions

Mean

Median

Mode

Mean Mode

Symmetrical

distribution

Positive

skew

Negative

skew

Median

Mode Mean

Median

Central limit theorem

To perform statistical tests and analyses, statisticians rely on 

their assumption of a normal distribution. However, as we 

have seen, this is not always the case. Fortunately, there is a 

rule which allows them to make this assumption even when the 

distribution is not normal. The central limit theorem states that 

the sampling distribution of means increasingly approximates 

a normal distribution as the sample size increases. That is a 

distribution whose individual values are the means of samples 

drawn from the main distribution (population). The central limit 

theorem allows inferential statistics to be applied to skewed and 

otherwise normal distributions.

The central limit theorem is very powerful, and in most 

situations it works reasonably well with a sample size greater 

than 30. Therefore, it is possible to closely approximate what 

the distribution of sample means looks like, even with relatively 

small sample sizes. The importance of the central limit theorem 

to statistical thinking cannot be overstated. Most hypothesis 

testing and sampling theory is based on this theorem.

While there is a mathematical proof for the central limit 

theorem, it goes beyond the scope of this text. It is discussed 

here to show that there is a solid statistical base for assuming 

a normal distribution for the statistical tests used in inferential 

analysis of fire data. With the proper sample size, the results 

will be valid even if the population distribution is not normal.
Chapter 5: Analyses of Tables 45

Chapter 5: Analyses of 

Tables

Introduction

As was discussed previously, most fire data are qualitative 

(categorical) in nature. Examples of categorical data in the fire 

service would include property use, cause of ignition, extent of 

flame damage, etc. Since this type of data cannot be expressed in 

terms of the mean, median and standard deviation, the number 

of each category can be used and listed in a table format. It 

might be found, for example, that arson fires account for 56% of 

all structure fires, equipment failure for 23%, and so on.

This chapter provides techniques for analyzing tables 

developed from categorical data. This includes the 

development and interpretation of percentages for categorical 

data and the use of a nonparametric statistical test known as 

the chi-square. The chi-square is used to determine whether 

the percentage distribution from a table differs significantly 

from a distribution of hypothetical or expected percentages.

A nonparametric statistical test is one that makes little or 

no assumptions about the distribution. As stated previously, 

statisticians assume a normal distribution in their calculations. 

However, categorical data by nature are not described in 

this manner, i.e., mean, standard deviation, etc. Therefore, 

statistical tests that have certain parameters to their use 

would not be appropriate for this type of data. The chi-square 

was designed to be used without these parameters, and as 

such, is ideal for categorical data.

Describing categorical data

Summarizing a categorical variable is usually done by 

reporting the number of observations in each category and 

its percentage of the total. For example, consider Figure 5-1 

for types of situations found in the fires of Lincoln, Nebraska. 

These percentages are simple to calculate and easy to 

understand: 24.9% of the fires are structure fires, 26.9% are 

vehicle fires, and so on. As described in Chapter 4, the mode 
46 Fire Data Analysis Handbook

is the category with the largest number of data values. In this 

example, the mode is vehicle fires, totaling 175 fires.

Figure 5-1. Type of situations found — Lincoln, 

Nebraska, fires

Type of fire Number Percent

Structure fires 162 24.9

Outside of structure fires 44 6.8

Vehicle fires 175 26.9

Tree, brush, grass fires 166 25.6

Refuse fires 88 13.5

Other fires 15 2.3

Total 650 100.0

By way of comparison, Figure 5-2 shows the nationwide 

picture of types of situations found for fires. From a national 

perspective, structure fires accounted for 28.7% of the total, 

closely followed by tree, brush and grass fires at 27.3% and 

vehicle fires at 20.2%.

Figure 5-2. Types of situations found — nationwide fires

Type of fire Number Percent

Structure fires 523,000 28.7

Outside of structure fires 64,000 3.5

Vehicle fires 368,500 20.2

Tree, brush, grass fires 498,000 27.3

Refuse fires 226,500 12.5

Other fires 143,000 7.8

Total 1,823,000 100.0

Looking at these figures would prompt the question of 

whether the distribution of fires in Lincoln differs from 

the national picture. Some differences can be noticed by 

comparing percentages. For example, 26.9% of the Lincoln fires 

were vehicle fires, compared to 20.2% nationwide. Similarly, 

2.3% of the Lincoln fires were other fires, compared to 7.8% 
Chapter 5: Analyses of Tables 47

nationwide. Therefore, it would seem that the distribution of 

fires in Lincoln deviates from the national picture. However, 

a statistical test can be made to test this difference more 

precisely. The next section provides such a test.

The chi-square test

The chi-square test (pronounced kī; rhymes with pie) is a 

statistical test designed to be used with categorical data. Like 

most statistical tests, it is stated in precise statistical language 

by defining a hypothesis to be tested. The use of the term null 

hypothesis is commonly seen. The null hypothesis merely 

states that there is no difference between the 2 distributions 

being compared. In this case, the null hypothesis would be 

that there is no statistical difference between Lincoln and 

the national percentages in the categories of fires in Figures 

5-1 and 5-2. This is usually the way it is stated, that there is 

no difference. If a difference is found, the null hypothesis is 

rejected. It is sort of like innocent until proven guilty!

Although the chi-square test is conducted in terms of 

frequencies, it is best viewed conceptually as a test about 

proportions. To illustrate these ideas, it will be easier at this 

point to use a format that does not include fire data. Instead, 

consider a simple experiment where a die is thrown over 

and over again. The resulting data values are the number of 

dots showing after each throw. The number of dots varies 

between 1 and 6; that is, there are 6 possible outcomes. If a 

“fair” die is thrown a large number of times, one would expect 

each number of dots to show up one-sixth of the time. The 

chi-square test can be used with a certain degree of assurance 

to determine if, in fact, the die is “fair.”

Suppose, for example, that a die is tossed 90 times and the 

results are as shown in Figure 5-3.
48 Fire Data Analysis Handbook

Figure 5-3. Results of die throws

Dots visible Number Percent

1 16 17.8

2 17 18.9

3 12 13.3

4 14 15.6

5 17 18.9

6 14 15.6

Total 90 100.0

If the die is a “fair” die, one would expect to have 1 dot turn 

up exactly 15 times (one-sixth of the total), 2 dots visible 

exactly 15 times, and so on. The actual results differ from these 

expected results as shown in Figure 5-4.

Figure 5-4. Actual and expected results

Dots visible Actual number Expected 

number

1 16 15

2 17 15

3 12 15

4 14 15

5 17 15

6 14 15

Total 90 90

To summarize, a die has been tossed 90 times and obtained 

the results shown in Figure 5-3. The null hypothesis is that the 

die is “fair,” which means that there is no difference between 

the actual and the expected number of times each number of 

visible dots appears. The actual results are not the same as 

the expected, either because the die is not “fair” or because 

of variations inherent in throwing a die only 90 times. The 

chi-square test will determine whether the actual results differ 

significantly from the expected results.
Chapter 5: Analyses of Tables 49

In statistical terms, significance does not mean something 

meaningful or important; rather, statistical significance refers 

to the claim that a result from data generated by testing or 

experimentation is not likely to occur randomly or by chance 

but is instead likely to be attributable to a specific cause. In 

other words, significance means that the results of a test are 

likely real and not caused by luck or chance.

The following are the steps in performing the chi-square test:

1. Calculate the expected number for each category by 

multiplying the expected or population percentages by 

the total sample size. This calculation has already been 

performed as shown in Figure 5-4 with the “Expected 

number” column.

2. For each category, subtract the expected number from the 

actual number and then square the result.

3. Divide the results from step 2 by the expected number.

4. Sum the results from step 3. This is the calculated chi-square 

statistic. The larger this number, the more likely there is a 

significant difference between the actual and expected values.

5. Find the degrees of freedom, which are defined as the 

number of categories minus 1. In the die example, there 

are 5 degrees of freedom.

6. Obtain the critical chi-square value from the table in 

the Appendix by selecting the entry associated with the 

appropriate degree of freedom. Note: The table includes 

levels of significance from 0.05 to 0.001. Commonly, the 

0.05 level is used for most determinations. This indicates 

that results exceeding the critical value will be statistically 

significant 95% of the time. The other levels are used 

depending on how critical the results may be. For example, 

the more stringent 0.001 level is used in drug testing where 

lives may depend on the results. 

7. If the computed chi-square statistic is greater than the 

critical value obtained from the table, the null hypothesis 

is rejected. Otherwise, the null hypothesis is accepted. 
50 Fire Data Analysis Handbook

Rejecting the null hypothesis means there is a significant 

difference between the 2 distributions. Conversely, accepting 

it means that the 2 distributions are essentially the same 

with differences due to sampling or random variations.

Figure 5-5 summarizes these steps for the die example. 

The “difference” column shows the difference between the 

expected and actual numbers. The “squared difference” is the 

square of the difference obtained by multiplying the number 

by itself. The right-most column is the squared difference 

divided by the expected number; for example, the first figure 

is 0.067, obtained from 1 divided by 15. The chi-square value is 

1.34, which is the sum of the values in the last column.

Figure 5-5. Actual and expected results — die-tossing 

experiment

Dots 

visible

Actual 

number

Expected 

number Difference Squared difference

Divided

by 

expected 

number

1 16 15 1 1 0.067

2 17 15 2 4 0.267

3 12 15 -3 9 0.600

4 14 15 -1 1 0.067

5 17 15 2 4 0.267

6 14 15 -1 1 0.067

Total 90 90 1.340

Chi-square 

value

1.34

Degrees of 

freedom

5.00

Critical 

value

11.07

From the Appendix, the critical chi-square value for 5 degrees 

of freedom at the 0.05 level of significance is 11.07. Since the 

calculated chi-square value of 1.34 is less, the null hypothesis 

is accepted. Therefore, the results from the 90 throws do not 

provide evidence that the die is unfair.
Chapter 5: Analyses of Tables 51

Degrees of freedom have been defined as the number of 

categories minus 1. The rationale for determining degrees 

of freedom is that each category may be considered as 

contributing 1 piece of data to the chi-square statistic. These 

data are free to vary except for the last category, since it is 

determined already by what is left. It is, therefore, not free to 

vary. Thus, the values in all categories except 1 are free to vary. 

An illustration of this may be more helpful than an explanation. 

Suppose you were asked to name any 5 numbers. In response, 

you chose 25, 44, 62, 82 and 2. In this case, there were no 

restrictions on the choices. There were 5 choices and 5 

degrees of freedom. Now suppose you were asked to name 

any 5 numbers again. This time you chose 1, 2, 3 and 4, but 

were stopped at that point and told that the mean of the 5 

numbers must be equal to 4. Now you have no choice for the 

last number, because it must be 10 (1 + 2 + 3 + 4 + 10 = 20, and 

20 divided by 5 equals 4). The restriction caused you to lose 1 

degree of freedom in your choice. Instead of having 5 degrees 

of freedom as in the first example, you now have 5 minus 1, or 

4, degrees of freedom. Each statistical test of significance has 

its own built-in degrees of freedom based on the number and 

type of restrictions it makes. The chi-square has 1.

At this time, the question on whether the distribution of fires 

in Lincoln differs from the nationwide distribution of fires can 

be dealt with. It was noted that there were differences in some 

categories; for example, Figure 5-1 shows that vehicle fires 

account for 26.9% of the fires in Lincoln compared to 20.2% 

nationwide. Similarly, other fires account for 2.3% of the fires 

in Lincoln compared to 7.8% nationwide.

However, these are individual comparisons. The chi-square 

test allows all categories to be tested simultaneously. The 

null hypothesis is that “The percentage distribution of fires 

in Lincoln does not differ significantly from the nationwide 

picture.” If the calculated chi-square value is larger than 

the appropriate critical value in the Appendix, then the null 

hypothesis is rejected, which indicates that there is a significant 

difference. Otherwise, the null hypothesis is accepted, 

indicating no significant difference in the 2 distributions.
52 Fire Data Analysis Handbook

Figure 5-6 shows the calculations using the information in 

Figures 5-1 and 5-2. The “actual number” column comes 

directly from Figure 5-1. To obtain the expected number, the 

percentages from Figure 5-2 are applied to the 650 Lincoln 

fires. For example, 28.7% of the nationwide fires were 

structure fires, which means we expect 28.7% of the 650 fires 

in Lincoln to be structure fires. This calculation yields 186.6 

fires (28.7% times 650 fires).

The “difference” column gives the difference between the 

actual and expected numbers, and the next column is the 

squared difference (the difference multiplied by itself). The 

last column is the squared difference divided by the expected 

value. The calculated chi-square value is the sum of the 

column, which is 63.8.

In this example, there are 6 categories of fires, which means 

there are 5 degrees of freedom. From the Appendix, the critical 

chi-square value at the 0.05 level of significance is 11.07. Since 

the calculated chi-square value of 63.8 is greater than the 

critical value, the null hypothesis is rejected. The conclusion is 

that the distribution of fires in Lincoln differs significantly from 

those nationwide. As stated before, the table in the Appendix 

lists the critical values for chi-square at various levels. For 

the purposes of this type, the 0.05 level is sufficient, which 

means that the difference will be significant 95% of the time 

or at the 95% confidence level. In this particular example, the 

obtained chi-square value far exceeds the critical value for 

even the 0.001 level of significance, which is 20.51. This means 

that it is significant 99.9% of the time with a chance of error of 

only one-tenth of a percent! In most comparisons, this level of 

confidence is rarely obtained.
Chapter 5: Analyses of Tables 53

Figure 5-6. Actual and expected results — Lincoln fires

Type of

 fire

Actual 

number

Expected 

number Difference Squared difference

Divided

by 

expected 

number

Structure 162 186.6 -24.6 605.16 3.2

Outside 44 22.8 21.2 449.44 19.7

Vehicle 175 131.3 43.7 1,909.69 14.5

Grass 166 177.4 -11.4 129.96 0.7

Refuse 88 81.3 6.7 44.89 0.6

Other 15 50.7 -35.7 1,274.49 25.1

Total 650 650.0 63.8

Chi-square 

value

63.8

Degrees of 

freedom

5.0

Critical 

value

11.07

Some of the rationale behind the chi-square statistic may be 

helpful in understanding what it is actually reporting. The 

dynamics of what contributes to the chi-square value are 

evident in Figure 5-6. For example, the largest difference 

(regardless of sign) between the actual and expected numbers 

is 43.7 for vehicle fires. Squaring the difference and dividing 

by the expected number gives 14.5, as shown in the last 

column. As can be seen, vehicle fires are only the third largest 

contributor to the chi-square value, even though this type 

has the largest difference between the actual and expected 

number of fires. The reason for this is that larger categories 

have greater leeway for numerical variations, since it requires 

more to account for the same amount of actual change than 

smaller categories with fewer numbers to begin with. This 

can readily be seen by looking at the top 2 categories in 

contribution weight to the chi-square value. Outside fires with 

a difference of 21.2 and other fires with a difference of -35.7 

contribute 19.7 and 25.1, respectively, for a total of 44.8 toward 

the 63.8 chi-square value. That is, 70% of the chi-square value 

is made up of the 2 smallest categories! While the numerical 

difference is less than that of vehicle fires, the actual amount
54 Fire Data Analysis Handbook

of change in those categories is greater, because the numerical 

difference is greater proportionally to the number of fires in 

those categories. This is why it was stated earlier that “although 

the chi-square test is conducted in terms of frequencies, it is 

best viewed conceptually as a test about proportions.”

2-way contingency tables

Up to this point, chi-square has been applied in cases with 

only 1 variable. It also has important application to the analysis 

of bivariate frequency distributions. By studying bivariate 

distributions with 2 categorical variables, the statistical 

association between the 2 variables can be measured. 

Association allows the gaining of information about 1 

variable by knowing the value of the other. The strength of 

the association may run from no association to weak to quite 

strong. The chi-square measures its existence and strength.

Figure 5-7 is used as the star ting point to introduce 

contingency tables, statistical variable association and the 

chi-square statistic’s role in measuring it. The NFPA’s “Survey 

of Fire Departments for U.S. Fire Experience” for 1 year was 

used to develop the figure. To facilitate the example, 5 of the 

10 categories under “nature of injuries” were eliminated. The 

“type of duty” category is as it appears in the original table.

There are 5 categories for location or “type of duty.” The first is 

“responding to or returning from incident.” The next category, 

“fireground,” covers injuries while on site at a fire. Similarly, 

the third category, “nonfire emergency,” covers injuries while 

on site at all nonfire incidents. The “training” category includes 

any injuries sustained while the firefighter was training. The 

last category covers all injuries not under the other categories 

but while still on duty.

The nature of the injury also is divided into 5 categories. As 

mentioned above, there were originally 10 categories of injuries, 

but for simplicity’s sake, only the top 5 were used. They are:

1. Burns.

2. Smoke inhalation.

3. Wounds/cuts.

4. Strains/sprains.

5. Other.
Chapter 5: Analyses of Tables 55

Figure 5-7. Firefighter injuries — type of duty and nature 

of injuries

Type of 

duty

Nature of injuries

Burns Smoke 

inhalation

Wounds/

cuts

Strains/

sprains Other Row totals

Responding 

to or 

returning 

from 

incident

65 115 960 2,250 710 4,100

Fireground 3,255 2,580 9,210 16,410 3,635 35,090

Nonfire 

emergency

185 185 2,440 8,025 2,725 13,560

Training 345 40 1,380 3,860 625 6,250

Other on 

duty

245 105 2,780 8,185 2,495 13,810

Column 

totals

4,095 3,025 16,770 38,730 10,190 72,810

Figure 5-7 shows that there were a total of 72,810 injured 

firefighters. The top left number means there were 65 

firefighters who were burned either responding to or returning 

from an incident. Similarly, the number in the second row and 

fourth column indicates that there were 16,410 firefighters who 

suffered strains or sprains while on a fire incident. Further, this 

number is the mode of the contingency table.

Outside of identifying the mode and showing the relative 

position of each category within its variable, the numbers 

in the table do not relay much information. Next, various 

percentages are calculated from the table to provide more 

insight. Finally, a chi-square value is calculated to measure the 

strength of the relationship between the 2 variables.

Percentages for 2-way contingency tables

There are 3 ways to calculate percentages for 2-way 

contingency tables of frequencies. Each way highlights a 

different feature of the table. More importantly, each provides 

a different interpretation of the data and leads to different 
56 Fire Data Analysis Handbook

conclusions about the relationship between the 2 variables. 

The 3 ways of calculating percentages are:

1. Joint percentages.

2. Row percentages.

3. Column percentages.

The type of percentage used depends upon where the 

emphasis needs to be placed. Joint percentages allow the 

direct comparison of table entries with each other. Row 

percentages concentrate on the individual rows of the table 

with percentages along the row totaling 100%. Similarly, 

column percentages deal with the individual columns of the 

table with column totals equaling 100%.

Joint percentages

To calculate joint percentages, each entry in the table is 

divided by the overall total. Figure 5-8 shows the percentage 

calculation for the counts from Figure 5-7. The lower-left entry 

is simply 4,095 divided by 72,810, which equals 5.6%. This 

means that 5.6% of the total firefighter injuries sustained were 

burns. The sum of all the entries in the table is 100.0%.

More logical comparisons can be made with joint percentages 

than with just the raw counts. For example, the table shows 

that 22.5% of all injuries were sprains or strains that occurred 

while the firefighter was on the fireground. In a similar manner, 

only 1.3% of all injuries were wounds or cuts suffered by 

firefighters responding to or returning from an incident.

Figure 5-8 also provides important information from the 

row and column totals. For example, from the second row 

it is apparent that nearly half (48.2%) of all the injuries were 

sustained at a fireground. There are 2 methods that can be 

used to derive this percent. The first is to add the 5 percentages 

across the row (4.5 + 3.5 + 12.6 + 22.5 + 5.0 = 48.1). The second 

is to divide the row total of 35,090 (from Figure 5-7) by 72,810 

to yield the 48.2%. (Note: Due to rounding, the percentages are 

not always exactly the same using these methods.)
Chapter 5: Analyses of Tables 57

Similarly, column percentages provide information about 

the nature of the injuries involved. For example, only 5.6% 

of firefighters injured suffered from burns, 4.2% from smoke 

inhalation, 23% from wounds or cuts, 14% from other injuries, 

and over half (53.2%) from strains or sprains.

Figure 5-8. Firefighter injuries — joint percentages

Type of 

duty

Nature of injuries

Burns Smoke 

inhalation

Wounds/

cuts

Strains/

sprains Other Row totals

Responding 

to or 

returning 

from 

incident

0.1 0.2 1.32 3.1 1.0 5.6

Fireground 4.5 3.5 12.6 22.5 5.0 48.2

Nonfire 

emergency

0.3 0.3 3.4 11.0 3.7 18.6

Training 0.5 0.1 1.9 5.3 0.9 8.6

Other on 

duty

0.3 0.1 3.8 11.2 3.4 19.0

Column 

totals

5.6 4.2 23.0 53.2 14.0 100.0

While Figure 5-8 provides more insight into these 2 variables, it 

does not directly address other questions. For example, direct 

comparisons between burns and smoke inhalation injuries 

for any particular type of duty cannot be made. Similarly, 

comparisons between types of duty for any particular injuries 

cannot be made. To make these types of comparisons, row and 

column percentage calculations must be made.

Row percentages

To convert table counts into row percentages, each entry in the 

table must be divided by its row total. Therefore, the top-right 

entry is calculated by dividing 710 by 4,100. This indicates that 

17.3% of the total firefighters responding to or returning from 

an incident sustained other types of injuries.
58 Fire Data Analysis Handbook

Figure 5-9. Firefighter injuries — row percentages

Type of 

duty

Nature of injuries

Burns Smoke 

inhalation

Wounds/

cuts

Strains/

sprains Other Row totals

Responding 

to or 

returning 

from 

incident

1.6 2.8 23.4 54.9 17.3 100.0

Fireground 9.3 7.4 26.2 46.8 10.4 100.0

Nonfire 

emergency

1.4 1.4 18.0 59.2 20.1 100.0

Training 5.5 0.6 22.1 61.8 10.0 100.0

Other on 

duty

1.8 0.8 20.1 59.3 18.1 100.0

A table of row percentages allows for comparisons among the 

categories represented by the rows. The total for each row is 

100%, and these figures appear on the right of the table as a 

reminder that row percentages are represented. (Note: Due 

to rounding, the row percentages may not add up to 100%.)

As indicated, 17.3% suffered other types of injuries when they 

were responding to or returning from an incident. A total of 

1.6% had burn injuries, 2.8% had smoke inhalation injuries, 

23.4% sustained wounds or cuts, and the vast majority, 54.9%, 

had sprains or strains.

Looking at the second row, which is for firefighters injured at 

the fireground, a somewhat different picture emerges. Burns 

and smoke inhalation injuries accounted for 9.3% and 7.4%, 

respectively. These are followed by wounds and cuts at 26.2%, 

sprains and strains at 46.8%, and 10.4% for the other category. 

Once again, these percentages total 100, accounting for all 

firefighters injured while at the fireground.

Column percentages

To convert table counts into column percentages, each entry in 

the table must be divided by the total for its column. The top-left 
Chapter 5: Analyses of Tables 59

entry would be calculated by dividing 65 by 4,095, yielding 1.6%. 

This indicates that only 1.6% of the firefighters who received 

burns were responding to or returning from an incident.

Figure 5-10. Firefighter injuries — column percentages

Type of 

duty

Nature of injuries

Burns Smoke 

inhalation

Wounds/

cuts

Strains/

sprains Other

Responding 

to or 

returning 

from 

incident

1.6 3.8 5.7 5.8 7.0

Fireground 79.5 85.3 54.9 42.4 35.7

Nonfire 

emergency

4.5 6.1 14.5 20.7 26.7

Training 8.4 1.3 8.2 10.0 6.1

Other on 

duty

6.0 3.5 16.6 21.1 24.5

Total 100.0 100.0 100.0 100.0 100.0

The table of column percentages looks at a particular nature 

of injury across the 5 types of duty. With burn injuries, you 

can see that most (79.5%) occurred at the fireground, 8.4% 

during training, 6% on other types of duty, 4.5% at nonfire 

emergencies, and only 1.6% while responding to or returning 

from an incident. The “other” injury category shows a very 

different breakdown. A total of 7% of the injuries occurred 

while responding to or returning from an incident, while 

35.7% were sustained at the fireground. Nonfire emergencies 

accounted for 26.7%, followed by 24.5% for other on duties, 

and lastly 6.1% during training.

Selecting a percentage table

The choice of a percentage table depends on the uses of the 

data. Joint percentage tables are beneficial when the emphasis 

is on the interrelationship between the 2 variables in the table. 

For example, Figure 5-8 reveals that the combination of burns 
60 Fire Data Analysis Handbook

at the fireground accounted for 4.5% of the total. This figure 

can be compared to other combinations in the table.

The row percentage table provides a way of emphasizing the 

nature of injury for each type of duty. When a firefighter was 

responding to or returning from an incident, Figure 5-9 shows 

54.9% of the injuries were from strains or sprains, 23.4% from 

wounds or cuts, 17.3% from other types of injuries, 2.8% from 

smoke inhalation, and 1.6% from burns. These are useful 

results by themselves and can be compared to distributions 

in other rows.

The column percentage table emphasizes the type of duty 

for each nature of injury. For burns only, Figure 5-10 shows 

that 79.5% were sustained at the fireground, 8.4% occurred 

during training, 6% were other on duty, 4.5% were on a nonfire 

emergency, and 1.6% were responding to or returning from an 

incident. Interestingly, the percentage of those burned while 

responding to or returning from an incident is the same for 

both the row and the column percentages.

Testing for independence in a 2-way contingency 

table

This section uses the chi-square test to determine whether the 

2 variables in a 2-way contingency table are independent of 

each other. As before, a step-by-step procedure for calculating 

the chi-square value is provided. It should be noted that, with 

the chi-square calculations, as with the other calculations 

that have been performed, virtually all statistical packages 

automatically calculate the values. You can see that manual 

calculation is arduous and time-consuming. Additionally, 

manual calculations are more subject to error. Therefore, 

a statistical software package should be used whenever 

possible. However, the details of the computations are 

shown here to enhance the understanding of the underlying 

principles that are involved.

Before calculating the chi-square, however, a discussion of what 

is meant by independence is needed. 2 variables are said to be 

independent if knowledge about 1 variable cannot be used in 

predicting the outcome of the other variable. In general, the 

null hypothesis of independence for a 2-way contingency 
Chapter 5: Analyses of Tables 61

table is equivalent to hypothesizing that in the population, 

the relative frequencies for any row (across the categories 

of the column variable) are the same for all rows, or that the 

population of the relative frequencies for any column (across 

the categories of the row variable) are the same for all columns. 

So once again, the hypothesis to be tested by chi-square can 

be seen as one concerning proportions. For example, there are 

almost 9 times as many injuries sustained on the fireground as 

there are responding to or returning from an incident, but if the 

type of duty is unrelated to the number of injuries sustained, 

then on a proportional basis, the number of injuries should 

be the same for each type of duty.

Constructing a table of expected values

To calculate the chi-square value, the expected values for each 

cell must be determined. The expected values are the counts 

that would occur if the 2 variables were independent. The first 

step in developing a table of expected values is to calculate the 

proportion of cases in each cell. This can be done by column 

or row. 

Using the column, divide each column total by the grand 

total. The proportion for the first column, “burns,” would be 

calculated as follows: 4,095 divided by 72,810 equals 0.056. 

Subsequent column proportions would be “smoke inhalation” 

(0.042), “wounds/cuts” (0.23), “strains/sprains” (0.532), and 

“other” (0.14). Note that the proportions are the same as the 

column total percentages calculated for the joint percentages 

in Figure 5-8.

It is easy to calculate the expected cell frequencies from the 

expected cell proportions. For each cell, multiply the expected 

column proportion for that cell by the row total for that cell. 

For example, the cell representing firefighters responding to 

or returning from an incident who sustained burns would be: 

0.05624 (column proportion) times 4,100 (row total) equals 230.6. 

Figure 5-11 shows the results of the remaining expected values.
62 Fire Data Analysis Handbook

Figure 5-11. Firefighter injuries — table of expected values

Type of 

duty

Nature of injuries

Burns Smoke 

inhalation

Wounds/

cuts

Strains/

sprains Other Row totals

Responding 

to or 

returning 

from 

incident

230.6 170.3 944.3 2,180.9 573.8 4,100

Fireground 1,973.5 1,457.9 8,082.1 18,665.5 4,910.9 35,090

Nonfire 

emergency

762.6 563.4 3,123.2 7,213.0 1,897.8 13,560

Training 351.5 259.7 1,439.5 3,324.6 874.7 6,250

Other on 

duty

776.7 573.8 3,180.8 7,346.0 1,932.8 13,810

Column 

totals

4,095 3,025 16,770 38,730 10,190 72,810

The table of expected values is the distribution of proportions in 

each row (or column) that would be expected in the absence of 

a dependent relationship between the 2 variables. In this case, 

it would mean that the expected values are those that reflect 

no relationship between the nature of injuries sustained and 

the type of duty performed. As stated before, the same results 

could have been obtained by calculating the row proportions 

and multiplying them by the column totals. It should also be 

noted that the row and column totals are exactly the same as 

the original table of counts. That is, the development of the 

expected value table preserves these totals. However, slight 

discrepancies may exist due to rounding of decimals.

Calculation of chi-square for a 2-way contingency 

table

The chi-square value for a 2-way contingency table is calculated 

similarly to the one done for a single categorical variable.

1. Develop the table of expected values, as shown in Figure 

5-11, using the method discussed in the previous section.
Chapter 5: Analyses of Tables 63

2. For each table entry, subtract the expected value from the 

corresponding entry in the original table of counts and then 

square the result. This difference measures the discrepancy 

between the actual counts and what would be expected if 

the variables were independent.

3. Divide the results from step 2 by the expected value. This 

adjustment allows for the larger expected numbers which 

are usually associated with larger deviations.

4. Sum the results from step 3. This is the chi-square statistic. 

The larger the chi-square statistic, the more likely that 

there is a significant statistical association between the 2 

variables. However, the chi-square statistic also depends 

on the number of categories, which must be taken into 

account in the following steps.

5. Find the degrees of freedom, which are calculated for a 

2-way contingency table by multiplying the number of rows 

minus 1 times the number of columns minus 1. In the current 

example, there are 5 rows and 5 columns. Therefore, the 

number of degrees of freedom is (5-1) x (5-1) = 16.

6. Compare the computed chi-square statistic from step 4 to 

the value in the chi-square table in the Appendix using the 

appropriate degrees of freedom. The table value is called 

the critical chi-square value.

7. If the computed chi-square statistic is greater than the 

critical value in the table, then the null hypothesis of 

independence is rejected and the variables are related. If 

the computed chi-square statistic is less than the critical 

value, the null hypothesis of independence is accepted and 

the variables are not related.

It is important to keep in mind that in a 2-way contingency 

table, the 2 variables are independent. If the null hypothesis is 

accepted, it means that knowing the value of 1 of the variables 

does not help in predicting the value of the other variable. In 

the current example, the null hypothesis is that the type of 

duty engaged in is independent of the nature of the injuries 

sustained.
64 Fire Data Analysis Handbook

Figure 5-12. Firefighter injuries — table of chi-square 

entries

Type of duty

Nature of injuries

Burns Smoke 

inhalation

Wounds/

cuts

Strains/

sprains Other

Responding to 

or returning 

from incident

118.92 17.96 .26 2.19 32.33

Fireground 832.15 863.65 157.40 272.55 331.54

Nonfire 

emergency

437.48 254.15 149.45 91.41 360.55

Training 0.12 185.86 2.46 86.22 71.28

Other on duty 363.98 383.01 50.50 95.82 163.53

Total 

chi-square 

value = 

5,324.77

Critical 

value = 

26.3

Figure 5-12 shows the chi-square entries for the 2-way 

contingency table. These entries are the results after step 3 

above. The top-left entry was calculated as follows: 

1. Figure 5-7 gave an actual count of 65 for this entry, and 

Figure 5-11 gave an expected value of 230.6. 

2. Subtracting the expected value from the actual count yields 

-165.6 (65 minus 230.6), and squaring that figure results in 

27,423.36. 

3. Dividing this number by the expected value, 230.6, provides 

the chi-square value of 118.92. 

4. This value is then entered in Figure 5-12, and the procedure 

is repeated for each of the other entries. 

5. When all of the entries are calculated, they are all totaled. 

This total is the total chi-square value. 

6. In Figure 5-12, this total is 5,324.77. It is entered at the 

bottom of the table.
Chapter 5: Analyses of Tables 65

Now, to test the hypothesis about the independence of the 

2 variables, type of duty and nature of injury, compare the 

total chi-square value to the critical chi-square value from 

the Appendix. The critical chi-square value for 16 degrees of 

freedom at the 0.05 level of significance is 26.3. Since the total 

chi-square value greatly exceeds this value, the null hypothesis 

is rejected. Therefore, there is a statistical association between 

type of duty and nature of injury.

As a cautionary note, remember that a significant outcome of 

the chi-square test is directly applicable only to the data taken 

as a whole. The chi-square obtained is inseparably a function 

of the, in this case, 25 contributions composing it — 1 from each 

cell. Therefore, it cannot be said whether 1 group is responsible 

for the finding of significance or whether all are involved.
66 Fire Data Analysis Handbook
Chapter 6: Correlation 67

Chapter 6: Correlation

Introduction

This chapter deals with the concept of correlation for 

continuous (quantitative) data. Correlation is a statistical 

measure which indicates the degree to which 1 variable 

changes with another variable. For example, calls for EMS 

generally increase with population growth. That is, as 

population increases, more medical service calls would be 

expected. This would indicate a positive correlation between 

population and EMS calls. The correlation measures the 

strength of the association between the 2 variables.

If there is a correlation between 2 variables, then predictions 

better than chance can be made from an individual score (or 

whatever is being measured) on 1 variable to its predicted 

score on the correlate variable. Any problem in correlation 

requires 2 pairs of corresponding scores, 1 for each variable. 

Generally, the greater the association (correlation) between 2 

variables, the more accurately a prediction can be made on the 

standing in 1 variable from the standing in the other.

This chapter starts with the scatter diagram illustrated in 

Chapter 3 and proceeds with a discussion of the correlation 

coefficient. Next, a typical calculation of a correlation is 

presented for demonstration purposes. The chapter concludes 

with a discussion of the applicability and uses of a correlation 

and mention of other types of correlation.

Scatter diagram

Figure 6-1 shows a scatter diagram presented in Chapter 3 on 

the number of fire deaths and the number of other residential 

fires for a 10-year period. The horizontal axis gives the number 

of fires (in thousands), and the vertical axis gives the number 

of deaths. You can see from the figure that deaths are higher 

with greater numbers of fires. The general trend is clear even 

though the pattern is not perfect. The term “not perfect” refers 

to the fact that the points do not fall on a straight line.
68 Fire Data Analysis Handbook

Figure 6-1. Fires and deaths — other residential

Fires (thousands)

Deaths

10 11 12 13 14 15 16

20

40

60

80

100

120

With relationships depicted in this manner, the usual 

terminology is to label 1 variable as the independent variable, 

and the other as the dependent variable. In the case of Figure 

6-1, “fires” serves as the independent variable and “deaths” 

as the dependent variable. Obviously, the number of fires 

influences the number of fire-related deaths. Generally 

speaking, the more fires there are, the greater number of 

fire deaths. This represents a positive correlation since the 

increase in the independent variable is accompanied by an 

increase in the dependent variable.

It is important to emphasize 2 points about correlations. First, 

correlations assume an underlying linear relationship; that is, 

a relationship that can be best represented by a straight line. It 

should be noted, however, that not all relationships are linear. 

There are, for example, curvilinear relationships where the 

points on a scatter diagram cluster about a curved line.

Secondly, while correlation can be used for prediction, it does 

not imply causation. The fact that 2 variables vary together is 

a necessary — but not a sufficient — condition to conclude 

that there is a cause-and-effect connection between them. A 

strong correlation between variables is often the starting point 

for further research.
Chapter 6: Correlation 69

Correlation coefficient

The correlation coefficient measures the strength of 

association between 2 variables. The term “correlation 

coefficient” is used by most statisticians but is the same as 

the more commonly used correlation. The correlation is 

always between -1 and +1. A correlation of exactly -1 or +1 is 

called a perfect correlation and means that all the points fall 

on a straight line. A correlation of 0 indicates no relationship 

between the variables and would be represented on a scatter 

diagram as random points with no discernible direction. As a 

correlation coefficient moves from 0 in either direction, the 

strength of the association between the 2 variables increases.

As stated before, a positive correlation means that, as the 

independent variable increases, so does the dependent variable. 

In a negative correlation, as the independent variable increases, 

the dependent variable decreases. The sign of the correlation 

indicates direction, not magnitude. Magnitude is indicated 

by the size of the number regardless of the sign. Therefore, a 

correlation of -0.82 is greater than a correlation of +0.63.

To summarize the relationship between a scatter diagram 

and the correlation coefficient, the correlation coefficient is 

a number that indicates how well the data points in a scatter 

diagram “hug” the straight line of best fit. With perfect 

correlations, all the data points fall exactly on a straight 

line that summarizes the relationship, and the value of the 

coefficient is +1 or -1. When the association between the 

2 variables is less than perfect, the data points show some 

scatter about the straight line that summarizes the relationship 

as in Figure 6-1, and the absolute value (regardless of sign) 

of the correlation coefficient is less than 1. The weaker the 

relationship, the more scatter and the lower the absolute value 

of the correlation coefficient.

Another impor tant point is that correlations are not 

arithmetically related to each other. For example, a correlation 

of 0.6 is not twice as strong as a correlation of 0.3. Although it is 

obvious that a correlation of 0.6 reflects a stronger association 

than a correlation of 0.3, there is no exact specification of the 

difference. Subsequently, there is no relationship between 

correlations and percentages.
70 Fire Data Analysis Handbook

To make direct comparisons between correlations, the 

correlation coefficient must be converted to a coefficient 

of determination. The coefficient of determination is the 

square of the correlation multiplied by 100. This yields the 

percentage of association between the 2 variables. For 

example, a correlation of 0.50 would indicate a 25% (0.50 times 

0.50 equals 0.25 times 100 equals 25) association between 

variables. A perfect correlation of 1.00 would be equal to a 

100% coefficient of determination. So a correlation of 1.00 is 

4 times as strong as a correlation of 0.50, not twice as strong, 

as might appear from comparing the correlations directly. 

Additionally, the differences between successive correlation 

coefficient values do not represent equal differences in 

degree of relationship. For example, the difference between 

a correlation of 0.40 and 0.50 does not represent the same 

difference as that between correlations of 0.90 and 1.00. 

This can be seen more clearly by examining the coefficients 

of determination and their corresponding correlations in 

Figure 6-2. There is more than double the difference between 

correlations of 0.90 and 1.00 than between 0.40 and 0.50 when 

the corresponding coefficients of determination are compared.

Figure 6-2. Relationship between correlations and 

coefficients of determination

Correlation coefficient Coefficient of determination

1.00 100%

0.90 81%

0.80 64%

0.70 49%

0.60 36%

0.50 25%

0.40 16%

0.30 9%

0.20 4%

0.10 1%

0.00 0%
Chapter 6: Correlation 71

Calculating the correlation

Today, most calculators include a program to calculate the 

correlation coefficient. Additionally, virtually all statistical 

software packages calculate the various types of correlations. 

However, for those who must calculate a correlation by 

hand and show what factors make the coefficient positive or 

negative and what factors result in a high or low value, the 

deviation-score method is used.

Figure 6-3 shows the number of fires and civilian fire deaths 

for a 10-year period. The correlation between these 2 variables 

will be computed using the deviation-score method. The most 

widely used correlation formula is the Pearson. Its full name is 

the Pearson product-moment correlation coefficient. There 

are other types of correlations suited for special situations, 

but the Pearson is by far the most common. In fact, when 

researchers speak of a correlation coefficient without being 

specific about which one they mean, it may safely be assumed 

they are referring to the Pearson product-moment correlation 

coefficient. The term moment is borrowed from physics and 

refers to a function of the distance of an object from the 

center of gravity. With a frequency distribution, the mean is 

the center of gravity and, therefore, deviation scores are the 

moments. As shown, the Pearson correlation is calculated by 

taking the products of the paired moments.

Figure 6-3. Total United States fires and civilian fire deaths

Year Fires (thousands) Deaths

1 2,041.5 4,465

2 1,964.5 4,730

3 1,952.5 4,635

4 2,054.5 4,275

5 1,965.5 4,585

6 1,975.0 4,990

7 1,795.0 4,050

8 1,755.0 4,035

9 1,823.0 3,570

10 1,708.0 4,045

Sum 19,034.5 43,380
72 Fire Data Analysis Handbook

As shown in Figure 6-3, fires tended to decrease over the 10-year 

period, while civilian fire deaths seem to have no obvious pattern 

overall (though the last 4 years have an apparent decrease). 

From this, it would seem that there is little association between 

the variables that should result in a low correlation.

The computation of the Pearson correlation using the deviationscore method is illustrated in Figure 6-4 and summarized in the 

following steps:

1. List the pairs of scores in 2 columns. The order in which 

the pairs are listed makes no difference in the value of the 

correlation. However, if 1 raw score is shifted, the one it is 

paired with must be shifted as well.

2. Find the mean for the raw scores of each variable.

3. Convert each score in both variables to a deviation score 

by subtracting the respective mean from each.

4. Calculate the standard deviation (S.D.) for both variables. 

Since the deviation scores are already done, they need only 

to be squared and summed. Divide each of these totals by 

the number of pairs (in this case 10) and take the square 

root of each.

5. Multiply each pair of deviation scores, known as the crossproduct, and total the results.

6. Next, multiply the 2 standard deviations by each other and 

multiply that result by the number of pairs (10).

7. Divide the results of step 5 by the results of step 6. The result 

is the Pearson product-moment correlation coefficient.

8. Square this for the coefficient of determination.
Chapter 6: Correlation 73

Figure 6-4. Deviation score calculation for Pearson 

correlation coefficient

Year Fires — 

mean

Deaths — 

mean

Fires — 

mean 

squared

Deaths — 

mean 

squared

Cross 

product

1 +138.05 +127 19,057.8 16,129 +17,532.35

2 +61.05 +392 3,727.1 153,664 +23,931.60

3 +49.05 +297 2,405.9 88,209 +14,567.85

4 +151.05 -63 22,816.1 3,969 -9,516.15

5 +62.05 +247 3,850.2 61,009 +15,326.35

6 +71.55 +652 5,119.4 425,104 +46,650.60

7 -108.45 -288 11,761.4 82,944 +31,233.60

8 -148.45 -303 22,037.4 91,809 +44,980.35

9 -80.45 -768 6,472.2 589,824 +61,785.60

10 -195.45 -293 38,200.7 85,849 +57,266.85

Sum 0 0 135,448.2 1,598,510 +303,759.00

Mean Fires 

1,903.45

Deaths 

4,338

Correlation 

coefficient

Coefficient of 

determination

S.D. 116.382 399.814 +.653 42.6%

The correlation obtained in Figure 6-4 is relatively high, as 

demonstrated by the coefficient of determination of 42.6%. 

This indicates a measure of relationship between the variables. 

It does not mean that the relationship is necessarily causal. For 

example, a high positive correlation probably exists between 

the amount of beer consumed and the amount of automobile 

accidents over each year from 1900 to the present. Rather 

than believe that beer consumption and the number of auto 

accidents are causally related, however, it is more reasonable 

to suggest that some condition such as an increase in 

population accounts for the increase in both beer consumption 

and automobile accidents.

Since the correlation is positive, it means that as the amount 

of fires increase/decrease, the number of deaths increases/

decreases as well. While on the surface this would seem intuitive, 

as with the beer/accident example, there can be other conditions 
74 Fire Data Analysis Handbook

that would account for the common variance. For example, 

an increase in fires would be expected as the population and 

buildings and residences increased. On the other hand, as 

knowledge and use of fire safety programs and procedures 

increased over time, the number of fire deaths would be 

expected to go down. The point is that there are usually many 

alternate and rational explanations for changes other than a 

causal one between 2 simultaneously changing variables.

The next step after obtaining a correlation that shows there 

is a relationship is to use it as a predictor. This is done by 

defining the straight line that the data points cluster around, 

known as the regression line. The regression line is defined 

algebraically, and the formula is used to make the predictions. 

The predictions become more reliable as the correlation 

increases. A discussion of the regression method is beyond 

the scope of this handbook but is mentioned here to give a 

fuller meaning to the correlation coefficient.

Other types of correlations

While the Pearson correlation is by far the most commonly 

used, there are other types of correlations derived directly 

or indirectly from the Pearson. These correlations are used 

with data that are not continuous and quantitative as with 

the Pearson. Several of them are presented here with a brief 

description of their use. Details of their computation and use 

can be found in some of the texts cited earlier.

ĵ Rank-order correlation. Sometimes it is useful to categorize 

data by ranking. The largest gets a rank of 1, the second 

largest a rank of 2, and so on. When both variables consist of 

ranks, a rank-order correlation coefficient is calculated. It is 

sometimes called the Spearman rank-order correlation. It is 

found merely by applying Pearson’s procedure to the ranks.

ĵ Biserial correlation. The biserial correlation is suited to 

cases in which 1 variable is continuous and quantitative and 

the other would be, except that it has been reduced to just 

2 categories (for example, if the correlation between the 

number of fires and whether or not the number of civilian 

fire deaths was above or below the median). This would 

require the use of the biserial technique since 1 variable 

is continuous and the other is expressed dichotomously.
Chapter 6: Correlation 75

ĵ Point biserial correlation. This is used as in the biserial, 

except that the second variable is qualitative and 

dichotomous and cannot be expressed as continuous and 

quantitative. For example, a correlation between the number 

of fires and the number of male and female civilian deaths.

ĵ Phi coefficient. This is the Pearson correlation coefficient 

for 2 variables that are both qualitative and dichotomous.

ĵ Partial correlation. The partial correlation shows the 

Pearson correlation coefficient between 2 variables in the 

absence of 1 or more other variables. For example, with 

the correlation of fires to deaths, the relationship each 

has to the passage of time may account for the change in 

each rather than a relationship to each other. By doing a 

partial correlation between fires and deaths for each month 

within a given year, time can be held constant. The resulting 

correlations reflect a truer picture of the relationship 

between fires and deaths.

There are other variations of correlations used for determining 

variable relationships with different circumstances, but these 

cover most of what is likely required. As stated before, these 

tools along with the ones discussed in the previous chapters 

are readily available in various statistical packages. Most of 

them guide the user through the process with clear, concise 

directions. The purpose of manually calculating these statistics 

is to provide the user with a more complete understanding 

of how the data are analyzed. This should make it easier 

to interpret the results from using a statistical package. It 

also serves as a good foundation for any further study with 

statistical texts and course work.
76 Fire Data Analysis Handbook
Appendix: Critical Values of Chi-Square 77

Appendix: Critical Values of 

Chi-Square

Level of significance

Degrees of 

freedom 0.05 0.025 0.01 0.005 0.001

1 3.84 5.02 6.63 7.88 10.83

2 5.99 7.38 9.21 10.60 13.82

3 7.81 9.35 11.34 12.84 16.27

4 9.49 11.14 13.28 14.86 18.47

5 11.07 12.83 15.09 16.75 20.51

6 12.59 14.45 16.81 18.55 22.46

7 14.07 16.01 18.48 20.28 24.32

8 15.51 17.53 20.09 21.95 26.12

9 16.92 19.02 21.67 23.59 27.88

10 18.31 20.48 23.21 25.19 29.59

11 19.68 21.92 24.73 26.76 31.26

12 21.03 23.34 26.22 28.30 32.91

13 22.36 24.74 27.69 29.82 34.53

14 23.68 26.12 29.14 31.32 36.12

15 25.00 27.49 30.58 32.80 37.70

16 26.30 28.85 32.00 34.27 39.25

17 27.59 30.19 33.41 35.72 40.79

18 28.87 31.53 34.81 37.16 42.31

19 30.14 32.85 36.19 38.58 43.82

20 31.41 34.17 37.57 40.00 45.31

21 32.67 35.48 38.93 41.40 46.80

22 33.92 36.78 40.29 42.80 48.27

23 35.17 38.08 41.64 44.18 49.73

24 36.42 39.36 42.98 45.56 51.18

25 37.65 40.65 44.31 46.93 52.62

26 38.89 41.92 45.64 48.29 54.05

27 40.11 43.19 46.96 49.65 55.48

28 41.34 44.46 48.28 50.99 56.89

29 42.56 45.72 49.59 52.34 58.30

30 43.77 46.98 50.89 53.67 59.70
78 Fire Data Analysis Handbook
Reference 79

Reference

Mosteller, F., Fienberg, S. E., & Rourke, R. E. K. (1983). Beginning 

statistics with data analysis. Addison-Wesley Publishing 

Company.


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