data types in statistics

data types in statistics

Data drives statistics. In traditional statistical analysis, data can usually be visualized by a matrix. Each column in the matrix

represents

a data

variable

(slightly different from the

mathematical definition

of a variable), and each row respesents an observation or outcome, in which case only one data variable is involved, or a vector of observations or outcomes where several data variables are involved.

The types of data that are being distinguished have to do with the data variables. Before going into the details, let’s begin with an example as a setting. A statistical analysis is conducted based on an observational study of autombile insurance data during a particular calendar year

Y

⁢

r

. A matrix of data is formed with the following data variables being observed:

whether a policy has been involved in an accident during

Y

⁢

r

,

NumAcc

number of accidents have a policy been involved in an accident during

Y

⁢

r

,

Cost

the total amount of money a policy cost the insurance company during

Y

⁢

r

,

Gen

gender of driver,

Mar

marital status of driver,

Age

age of driver,

Hist

number of accidents a driver had prior to year

Y

⁢

r

,

DrvZIP

zip code location where the driver lives,

AccZIP

zip code location where the accident happened,

AccSt

a numerical code corresponding to the state or province where the accident took place (for example, 0=Alabama, 1=Alaska, etc…, 50=Wyoming),

Inj

the extent of the injury sustained during an accident,

VehType

the type of vehicle in the policy, and finally,

VehWgt

the weight of the vehicle in the policy.

Now, we are ready to breakdown the data variables. First, the data variables can be broken down in terms of their uses:

1.

response variable

or

predicted variable

. From the above example, ,

NumAcc

,

Cost

can all be response variables. These are variables that we are trying to study, and predict.

2.

explanatory variable

or

predictor variable

or

control variable

. In the example above, given the response variable is , the explanatory variables can be any of the other variables except

NumAcc

,

Cost

, and

Inj

. Although possibly highly correlated with ,

NumAcc

,

Cost

, and

Inj

do not “explain” why an accident occurs. In particular,

Inj

is only valid when there was an accident.

Usually, the response variable(s)

𝒚

and the explanatory variable(s)

𝒙

can be related functionally as

𝒚

=

f

⁢

(

𝒙

)

.

A breakdown of data variables in terms of the natures of the variables is as follows:

1.

categorical variable

or

discrete variable

. These are data variables whose ranges are

countable

, often finite. Any value of a categorical variable is called a

level

, or a

category

. For example, is a categorical variable whose values are “Yes” (to mean that at least an accident occurred during year

Y

⁢

r

) and “No” (to mean otherwise). A categorical variable whose number of values is two is often called a

binary variable

or a

dichotomous variable

. A categorical variable that has more than two values is called a

multinomial variable

or a

polychotomous variable

.

DrvZip

,

Inj

(no injury, light, medium, serious injuries, or death),

VehType

(family sedan, sports coupe, etc…) and

NumAcc

are examples of a multinomial variable.

2.

continuous variable

. Any data variable that is not a categorical variable is a continuous variable.

Age

and

VehWgt

are both examples of continuous variables. In real situations, these continuous variables usually lie within a certain bounded interval or ball (in higher

dimensions

). For example, it is safe to say that the range of the variable

Age

is

[

0

,

140

]

.

In many statistical modeling situations, it is often convenient, sometimes even desirable to change continuous variables to

categorical

ones, and vice versa. Discretization is a way to turn a continuous variable into a categorical one. For example, the continuous variable

Age

can be turned into a dichotomous variable by the grouping: “Young” = Age

∈

[

0

,

25

]

and “Not Young” = Age

∈

(

25

,

140

]

. Another possible grouping rule may be “Young” = Age

∈

[

0

,

25

]

, “Mature” = Age

∈

(

25

,

55

]

Age and “Old” = Age

∈

(

55

,

140

]

.

Conversely

, to turn a categorical variable into a

continuous

one, either the method of

extension

or

transformation

, or both, are used. For example,

Hist

, the number of prior accidents is a discrete variable taking on non-negative integer values, can be extended to a continuous variable taking on all non-negative real values to suit a certain modeling

function

f

, even though non-integral values do not make sense and are not used in actual predictions.

AccZIP

can be transformed into a two-dimensional real-valued vector (longitude, latitude), since each (U.S.) zip code corresponds to an area with a unique

centroid

whose coordinate is measured in longitude and latitude.

Next, data variables can be grouped as whether they are:

1.

quantitative

. All variables such as

Age

,

NumAcc

,

Hist

, and

VehWgt

are quantitative variables since they take on numerical values. Variable

AccSt

is not a quantitative variable even though it is numeric in nature, since its values have no intrinsic numerical meanings. Another possible non-quantitative variable may be

DrvZIP

.

2.

qualitative

. Variables like

Gen

,

Mar

,

Inj

, as well as

AccSt

and

DrvZIP

are all qualitative variables.

Finally, data variables can be classified in terms of whether they can be ordered or not:

1.

nominal

variables have no intrinsic ordering

structure

.

Gen

and

Mar

are such examples, as are

AccSt

,

DrvZIP

and

VehType

.

2.

The meaning of

ordinal

variables is self-explanatory. Usually, numerical variables are ordinal, except when they are multi-dimensional or vectorial.

AccZIP

, when transformed into longitude,latitude, is not ordinal. However, fixing any one of the two coordinates turns the other coordinate into an ordinal variable. An example of a non-numerical ordinal variable is

Inj

. Since the levels of

Inj

can be ranked by their severity, from “no injury” to “death”, it is ordinal.

The data variables in the above example is summarized in the following table:

Title

data types in statistics

Canonical

name

DataTypesInStatistics

Date of creation

2013-03-22 14:44:27

Last modified on

2013-03-22 14:44:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Topic

Classification

msc 62-07

Defines

response variable

Defines

explanatory variable

Defines

continuous variable

Defines

discrete variable

Defines

categorical variable

Defines

nominal variable

Defines

ordinal variable

Defines

predictor

Defines

control variable

Defines

observation

Defines

qualitative variable

Defines

quantitative variable

Defines

dichotomous

Defines

polychotomous

\@unrecurse

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