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Describing Qualitative Data

Here we study ways of describing a variable that is qualitative.

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Say 50 people you know purchased a soft drink from a machine recently. A variable of interest might be the BRAND PURCHASED. Say the brands are made up of the 5 soft drinks Coke Classic, Diet Coke, Dr. Pepper, Pepsi-Cola, and Sprite (of course there are more varieties of soft drinks, but this is an illustrative example.)

Here each specific brand represents a different value on the variable brand purchased. Each specific brand represents an nonoverlapping class.

Here the variable brand purchased is a qualitative variable - values of the variable represent categories.

One thing that makes sense to do is ask each of the 50 people what they purchased. Then we could count the number of people who purchased Coke Classic and the others. The total number of people of the 50 who purchased Coke Classic would be the frequency.

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Soft Drink Frequency Relative Frequency

Percent Frequency

Coke Classic

19 0.38 38

Diet Coke 8 0.16 16

Dr. Pepper 5 0.1 10

Pepsi-Cola 13 0.26 26

Sprite 5 0.1 10

Total 50 1.00 100

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The first two columns on the previous screen, the Soft Drink and Frequency columns, make up what is called a frequency distribution. It is a tabular summary of data showing the number, or frequency, of items in each of several nonoverlapping classes.

The third column shows the relative frequency. We need the second column to create the third. To get the relative frequency in each row

take the frequency in that row and divide by the total frequency.

The fourth column shows the percent frequency. The fourth column equals the third column multiplied by 100.

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Do you know why we put information in columns? Because then we can call’um as we see’um. Sorry:)

So, the frequency, relative frequency and percent frequency distributions are different ways of summarizing information about a categorical variable.

Notes about our table.

1) The total, or sum, of the frequency column is equal to the number of observations, sometimes called n, in general.

2) The total, or sum, of the relative frequency column is equal to 1.

3) The total, or sum, of the percent frequency column is equal to 100 (sometimes it may be a little off due to rounding of decimal places).

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In our example here we had 50 people and we asked what soft drink they purchased. Studies occur that have thousands of people and they are asked several questions. Using a computer can help in the counting of responses. Later we will use Microsoft Excel to help in the counting.

Bar Graphs

Bar graphs just put the the frequency, relative frequency and percent frequency distributions into visual form. The form is a graph with certain properties.

The horizontal axis does not have numbers on it and the axis represents the categories. In our soft drink example we would put each brand in a different location on the axis.

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Imagine you have a piece of construction paper that is blue. Do you remember way back when in school you would cut strips of paper and then curl the paper with the scissors? Well, we will not need to curl the paper here!

I mention this silly example because I want you to think about cutting strips that are one inch wide. The height of each strip would then represent the frequency, relative frequency or percent frequency on the variable. You would tape each strip onto the graph above each category.

So the vertical axis, or height, in the bar graphs is either the frequency, relative frequency or percent frequency distributions.

In constructing the bar graph on a qualitative variable a space is left between each bar to help us remember we have a qualitative a variable.

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RuralSuburbanUrban

Percent

50

40

30

20

10

0

This is an example of what a percent frequency graph would look like. The variable is “what is the type of area in which you live” and the height of each bar is the percent frequency. (See how each bar is like a cut out from a piece of paper?)

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Pie Charts

Say we order a pizza pie and it is cut up into pieces. Below I show a pizza pie cut, and I wanted it to show it cut into slices that hits the middle. If you get a quarter of the pie, you get one of the sections shown. 0.25 of the pie is an example of the relative frequency. So, par charts show each category getting its relative share of the pie.

A pie chart could really be a the frequency, relative frequency and percent frequency pie, but the size of each piece of the pie is always the relative frequency.