A Complete Guide to Descriptive Statistics  Central Tendency and Dispersion
Source: tai_blog

In a world filled with data  statistics is the compass guiding us through the huge seas of numbers. Statistics play an important role in predicting the weather  analyzing market trends  or assessing public health.   In this blog  well understand the essence of statistics  diving into one of its main branches: descriptive statistics.   But before starting with descriptive statistics  lets take a step back and understand what exactly statistics is. And why is it so crucial?   What is Statistics?According to Wikipedia:   Statistics is the discipline that concerns the collection  organization  analysis  interpretation  and presentation of data.   In simple terms  statistics means collecting information  summarizing  and determining what it means. Statistics helps us understand the patterns and trends within the data.   A world without Statistics?   Without statistics  we will never be able to understand how data behaves  what happened  or what may happen in the future.   All these things require a fundamental understanding of statistics.   With that context  lets get started with the topic for today i.e.  descriptive statistics.   Descriptive Statistics: Painting a Picture of DataDescriptive statistics help us summarize and describe the main features of a data set.   Imagine you have a dataset of students test scores. Descriptive statistics will tell you the average score  the range of scores  and how scores are distributed. It provides a snapshot and a concise overview of the data at hand.   Key Concepts in Descriptive Statistics1. Measures of Central TendencyA single number/statistic that quantifies the central behavior of the dataset. The central tendency can be measured using the following statistics:   i) Mean: Mean or arithmetic mean is the average of a set of numbers. For a dataset with n values  the mean is calculated using the following formula:   Mean uses all data points  providing a comprehensive measure of the numerical columns.   ii) Median: The middle value of a set of numbers arranged in ascending order.   For an even set of numbers  the median is the average of the middle 2 numbers while for the odd set of numbers  its the middle number.Just like the mean  the median can be applied to numeric data only.Median does not use all data points  potentially losing some information.If there are outliers in a numerical column  the preferred way to measure central tendency is the median (as outliers influence mean value but not median).iii) Mode: The most frequently occurring score. A dataset can have one mode (unimodal)  more than one mode (bimodal or multimodal)  or no mode at all if no number repeats.   How to Find the Mode:   Identify the Frequency: Count how many times each value appears in the dataset.Determine the Most Frequent Value: The value with the highest frequency is the mode.When to use Mode:   When analyzing categorical data where you want to know the most common category.When dealing with discrete data and interested in the most frequent value.When examining data distributions that are not symmetrical.But Central Tendency is not sufficient?   While Central Tendency measures provide important information about where the data is centered  they do not provide a complete picture of the datas distribution. Relying solely on measures of central tendency can be misleading  as they do not capture the variability or spread of the data.   Datasets with the same mean  median  or mode can have very different distributions. For example:   Example 1: Consider two datasets with the same mean:   Dataset A: [50  50  50  50  50]Dataset B: [10  30  50  70  90]Both have a mean of 50  but Dataset A has no variability  while Dataset B has a wide range of values. The mean alone does not capture this difference.   2. Measures of DispersionMeasures of dispersion quantify the spread or variability of the data points around the central value.   i) Range: The range is the difference between the maximum and minimum values in a column. It tells us the span of the data and gives a basic indication of the spread.   How to Calculate the Range:   Identify the maximum value in the dataset.Identify the minimum value in the dataset.Subtract the minimum value from the maximum value.Range=Maximum ValueMinimum Value   When to Use the Range:   When you need a quick and simple measure of dispersion.When comparing the variability of two or more datasets.In preliminary data analysis to get an overview of the data spread.ii) Variance: Variance measures the average squared deviation of each data point from the mean. In simpler terms  it tells us how spread out the data points are around the mean.   A higher variance indicates that the data points are more spread out  while a lower variance indicates that they are closer to the mean.   The formula for Variance: The formula for variance differs slightly depending on whether we are dealing with a population or a sample.   When to use:   Variance provides a good understanding of the distribution of values within the dataset (around the mean).iii) Standard Deviation: The square root of the variance  indicating how spread out the scores are around the mean.   Formula for Standard Deviation: Just like Variance  the formula for standard deviation depends on whether you are dealing with a population or a sample.   Variance vs Standard Deviation  Which one to use when?Mathematical Properties and Interpretation:   Variance: Provides a measure that is useful for mathematical and statistical calculations. It is often used in theoretical contexts where the squaring of deviations simplifies further mathematical manipulation.Standard Deviation: Offers a more intuitive measure of spread  as it is in the same units as the data  making it easier to interpret.Analytical Convenience(more on this in future blogs):   Variance: In many statistical formulas and tests (e.g.  ANOVA  regression analysis)  working with variance is more convenient because of its additive properties.Standard Deviation: When communicating results to a non-technical audience or comparing the spread of different datasets  standard deviation is preferred due to its direct interpretability.SummaryStatistics is a powerful tool that helps us make sense of data  uncover patterns and making informed decisions. Descriptive statistics provide a summary of the data  giving us insights into its central tendencies and variability.   So next time you come across a data set  remember to use the power of statistics in order to turn those numbers into meaningful insights.