Ugrás a tartalomhoz

## SOCIAL STATISTICS

Renáta Németh, Dávid Simon

ELTE

The index of qualitative variation (IQV)

## The index of qualitative variation (IQV)

Supplemental topic not covered in the exam. We discuss it in order to assign a measure of variability to each level of measurement.

With nominal or ordinal variables.

Takes values from 0 to 1.

• If all the cases are in the same category, there is no variability and IQV is 0.

• In contrast, when the cases are distributed uniformly across the categories, there is maximum variability and IQV is 1.

Example (ISSP, 1998, Hungary). Education by employment status.

 Education Below high school High school College or university Total Self employed 27 35.5% 32 42.1% 17 22.4% 76 100.0% Employee 516 62.6% 195 23.7% 113 13.7% 824 100.0%

Level of education is more homogeneous within employees: two third of them do not have a high school degree. Compute the IQV within the two groups.

IQV = number of observed differences / maximum possible differences

Calculating the number of observed differences

Consider the sample below:

János without high school degree         István university degree         Károly university degree         Ildikó high school degree

the pairs below differ in their level of education:

János-István János-Károly János-Ildikó István-Ildikó Károly-Ildikó

So there are 5 “differences”. A simpler way is to follow the formula:

without high school degree                1 person university degree                        2 persons high school degree                        1 person

Different pairs: without high school degree vs. university degree – 2 pairs, without high school degree vs. high school degree – 1 pair, university degree vs. high school degree – 2 pairs, a total of 5 pairs.

If we have K categories, and fi denotes the frequency in the ith category, the following formula gives the result:

Σi=1..K, j=(i+1)..Kfifj

Following the formula for self-employeds, the number of observed differences is:

27*32+27*17+32*17=1,867

For employees:

516*195+516*113+195*113=180,963

 Education Below high school High school College/university Total Self employed 27 32 17 76 Employee 516 195 113 824

Calculating the maximum number of possible differences

(K(K-1)/2)*(N/K)2

where K is the number of categories of the variable, and N is the sample size.

For self-employeds:

(3*2/2)*(76/3)2  = 1,925

For employees:

(3*2/2)*(824/3)2  = 226,325

Calculating IQV

IQV = number of observed differences / maximum possible differences

For self-employeds: 1867/1925 = 0.97

For employees: 180,963/226,325 = 0.8

That is, the values of IQV support our earlier observation: level of education has a lower level of variability within employees.

In the previous example IQV was calculated for an ordinal variable. However, the IQV is not sensitive to the ordering of the categories. Its application with ordinal variables causes some information deficit.

Remark:

IQV can be calculated from percentages as well.

In the previous example, IQV for self-employeds is (35.5*42.1+35.5*22.4+42.1*22.4)/((3*2/2)*(100/3)2) = 0.97

Example

Racial diversity in eight states of the USA in 2006 (categories are white/black/Asian/Latino/other (Native American etc)). Interpret the data.

 State IQV Hawaii 0.89 California 0.84 New York 0.63 Alaska 0.54 Washington 0.33 Florida 0.48 Maine 0.05 Vermont 0.05

Source: Frankfort-Nachmias