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Categorical Data

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- Types of Data in Statistics
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Have you every filled out a customer satisfaction survey? How about one where you were asked about your income level? Then you have participated in **categorical data** collection!

First let's take a quick look at what categorical data is.

**Categorical data** is data which can be divided into different groups instead of being measured numerically.

So, some examples of categorical data would be hair color, type of pets someone has, and favorite foods. On the other hand things like height, weight, and the number of cups of coffee that someone drinks per day would be measured numerically, and so are not categorical data.

For a more thorough explanation of categorical data and what it is used for see the article Categorical Variables. To see the various types of data and how they are used you can take a look at One-Variable Data and Data Analysis.

In fact there are two types of categorical data, nominal and ordinal.

** Nominal categorical data** would be data that isn't assigned a number. An example would be, if you asked people if they lived in a rural area or a city area. "Rural" and "city" would be nominal categories.

** Ordinal categorical data** would be data that can be assigned numbers, but you couldn't really add the numbers together. For example if you did a customer satisfaction survey, and asked people to rate the service on a scale from \(1\) to \(5\), that would be ordinal categorical data. Notice that you can't add a satisfaction level of \(2\) and a satisfaction level of \(4\) together to get a satisfaction level of \(6\)!

Now you know what categorical data is, but how is that different from quantitative data? It helps to look at the definition first.

**Quantitative data **is data that is a count of how many things in a data set we have a particular quality.

**Quantitative data** usually answers questions like "how many" or "how much". For example quantitative data would be collected if you wanted to know how much people spent on buying a cell phone. Quantitative data is often used to compare multiple sets of data together. For a more complete discussion of quantitative data and what it is used for, take a look at Quantitative Variables.

Categorical data is qualitative, not quantitative!

All right, what about continuous data? Can that be categorical? Let's take a look at the definition of continuous data.

**Continuous data** is data that is measured on a scale of numbers, where the data could be any number in the scale.

A good example of continuous data is height. For any of the numbers between \(4 \, ft.\) and \(5 \, ft.\) there could be someone of that height. In general, categorical data is not continuous data.

Now that you have seen some comparisons between categorical data and other types, let's look at some more examples of categorical data.

Suppose you are having a party, and you want to make sure everyone has a dessert that they can eat. So you ask people to fill out a survey telling you their favorite dessert, and you gather up their data into a table like the one below.

Favorite Dessert | Frequency |

Ice cream | \(4\) |

Cake | \(2\) |

Fruit | \(17\) |

Pudding | \(5\) |

Cookies | \(10\) |

Is the data in the table categorical data?

**Solution**

Yes. Because the data is divided up into categories (favorite dessert) this is categorical data. In fact this would be considered ordinal categorical data.

Let's take a look at another example.

Suppose you were asked to give a survey to decide whether people liked a particular soft drink and got back the following information:

- 14 people liked the soft drink; and
- 50 people did not like it.

Is this categorical data?

**Solution **

Yes. You can divide up the answers into two categories, in this case "liked it" and "didn't like it". This would be an example of nominal categorical data.

Let's take a look at one more example.

Suppose you came across a survey someone had done which measured how far away from the center of a city someone lived and compared it to their income. Would this be categorical data?

**Solution**

It depends on the questions asked when gathering data. Let's take a look at a couple of surveys.

**Survey 1**

Question 1: How far do you live from the city center?

(a) I live in the city center

(b) I live within 1 mile of the city center, but not in it.

(c) I live between 1 and 5 miles of the city center.

(d) I live more than 5 miles from the city center.

Question 2: What is your income?

(a) Less than $10,000 per year.

(b) Between $10,000 and $20,000 per year.

(c) More than $20,000 per year.

**Survey 2**

Question 1: How many miles do you live from the city center?

Question 2: What is your income per year?

Then Survey 1 has the information divided up into categories. It is actually collecting two types of categorical data, and those can be compared together.

On the other hand, Survey 2 asks people for numbers. Their answer can be any number which is positive. So this survey is collecting continuous data. The data is not divided into categories, so it is not categorical data.

It is reasonable to ask how you analyze categorical data.

Two of the most frequent ways to look at categorical data is in bar charts and pie charts.

Let's go back to the example about soft drinks, where you discovered that 14 people liked the soft drink and 50 didn't. You could just look at the total number of responses, and make a bar chart showing the information.

You could also make a pie chart with the data.

Either one gives you a visual comparison of the data. For many more examples of how to construct a chart for categorical data, see Bar Graphs.

If you go back to the example about dessert, there was a table of data. It listed how many people liked each kind of dessert. This kind of table is also called a **frequency table**. You could change the heading "number of responses" to "frequency" (shorthand for frequency of response) and the table would give exactly the same information.

Favorite Dessert | Frequency |

Ice cream | \(4\) |

Cake | \(2\) |

fruit | \(17\) |

Pudding | \(5\) |

Cookies | \(10\) |

Let's take a look at the more formal definition.

A **categorical frequency distribution** is a table that organizes categorical data into frequencies.

So in fact the table above could be called a** categorical frequency distribution**!

Once you know that, it is normal to ask questions like "what percentage of the party goers like fruit for dessert?". That is asking for the relative frequency.

The **relative frequency** is the proportion of the number of times a category appears in the data set when compared to the total number in the data set.

In other words, the relative frequency is just the number in that category divided by the total number of responses. Because these are really percentages, if you add up all of the relative frequencies in a table you should get \(1\), or \(100 \%\). Let's do an example.

From the table of dessert choices, make a table of relative frequencies.

Favorite Dessert | Frequency |

Ice cream | \(4\) |

Cake | \(2\) |

Fruit | \(17\) |

Pudding | \(5\) |

Cookies | \(10\) |

**Solution**

First you need to know how many responses there were to the survey. You can find that by adding up the frequency column of the table, so

\[ \mbox{total responses } = 4+2+17+5+10 = 38.\]

Then you can find the relative frequency of each category by dividing the frequency by the total number of responses. For example the relative frequency of ice cream is

\[ \mbox{relative frequency of ice cream } = \frac{4}{38} = 0.105 \]

to three decimal places.

You can fill in the rest of the table in exactly the same way.

Favorite Dessert | Frequency | Relative Frequency |

Ice cream | \(4\) | \(0.105\) |

Cake | \(2\) | \(0.053\) |

Fruit | \(17\) | \(0.447\) |

Pudding | \(5\) | \(0.132\) |

Cookies | \(10\) | \(0.263\) |

Notice that if you add up all of the relative frequencies, you get \(1\), so you know these are more than likely correct. It is a good check to do to see if you are on the right track.

You can also look at a table that includes the **cumulative relative frequency**. That is just a fancy way of saying that the table includes the sum of all the relative frequencies before it.

Let's go back to the dessert table (which sounds like we should all be getting a piece of cake instead of more math). The cumulative relative frequency of the first row is just the relative frequency of the first row. The cumulative relative frequency of the second row is given by the sum of the relative frequency of the first row PLUS the relative frequency of the second row. Here is the table with cumulative relative frequency.

Favorite Dessert | Frequency | Relative Frequency | Cumulative Relative Frequency |

Ice cream | \(4\) | \(0.105\) | \(0.105\) |

Cake | \(2\) | \(0.053\) | \(0.105 + 0.053 = 0.158\) |

Fruit | \(17\) | \(0.447\) | \(0.447 + 0.158 = 0.605\) |

Pudding | \(5\) | \(0.132\) | \(0.132 + 0.605 = 0.737\) |

Cookies | \(10\) | \(0.263\) | \(0.263 + 0.737 = 1\) |

But what happens if you have two kinds of categorical data and want to compare them?

Two-way tables are a way to compare types of categorical data. This is easiest to understand with an example. Let's go back to the survey

Question 1: How far do you live from the city center?

(a) I live in the city center.

(b) I live within 1 mile of the city center, but not in it.

(c) I live between 1 and 5 miles of the city center.

(d) I live more than 5 miles from the city center.

Question 2: What is your income?

(a) Less than $10,000 per year.

(b) Between $10,000 and $20,000 per year.

(c) More than $20,000 per year.

There are two questions, and each one is a type of categorical data. Suppose you got the following responses from the survey:

Person Number | Question 1 | Question 2 | Person Number | Question 1 | Question 2 |

1 | a | a | 7 | b | c |

2 | a | b | 8 | c | a |

3 | d | a | 9 | a | b |

4 | b | c | 10 | d | c |

5 | c | c | 11 | d | b |

6 | d | a | 12 | b | c |

It is kind of hard to see if there might be any relationships between distance from the city center and income this way! So instead you can make a two-way table. It has the columns as one of the first question responses, the rows are the second question responses. The empty two-way table would be:

| City Center | Within 1 mile | 1 to 5 miles | More than 5 miles |

Less than $10,000 | ||||

$10,000 - $20,000 | ||||

More than $20,000 |

The entry in each part of the table is the total number of responses given that has both the row answer and the column answer.

For example, in the table above \(2\) people answered (a) (city center) for Question 1 and (b) (between $10,000 and $20,000) for Question 2. So at the intersection of "City Center" and "$10,000 - $20,000" there should be a \(2\).

| City Center | Within 1 mile | 1 to 5 miles | More than 5 miles |

Less than $10,000 | ||||

$10,000 - $20,000 | 2 | |||

More than $20,000 |

You can fill in the rest of the table the same way.

| City Center | Within 1 mile | 1 to 5 miles | More than 5 miles |

Less than $10,000 | 1 | 0 | 1 | 2 |

$10,000 - $20,000 | 2 | 0 | 0 | 1 |

More than $20,000 | 0 | 3 | 1 | 1 |

Now it is much easier to see any connections between distance from the city center and income. Notice that if you add up all the entries in the table you get \(12\), which is exactly the same as the number of survey responses. You can graph these in a bar chart just like you normally would.

To look at the data in a two-way table graphically you can make a **segmented bar graph**. In a segmented bar graph, each bar of the graph is divided up into percentages based on the number of answers of that type. Sometimes a segmented bar graph is called a **stacked bar chart**. A segmented bar graph makes it easier to see what percentage of the total falls into each category.

Using the graph above, you can quickly see that more than half of the people making more than $20,000 per year live within 1 mile of the city center!

- Categorical data is data which can be divided into different groups instead of being measured numerically.
- Nominal categorical data would be data that isn't assigned a number.
- Ordinal categorical data would be data that can be assigned numbers, but you couldn't really add the numbers together.
- A categorical frequency distribution is a table that organizes categorical data into frequencies.
- The relative frequency is the proportion of the number of times a category appears in the data set when compared to the total number in the data set.

- Categorical data is qualitative, not quantitative.
- Continuous data is data that is measured on a scale of numbers, where the data could be any number in the scale.
- Two of the most frequent ways to look at categorical data is in bar charts and pie charts.
- Two-way tables are a way to compare types of categorical data.
- To look at the data in a two-way table graphically you can use a segmented bar graph, also called a stacked bar chart.

**Categorical data** is data which can be divided into different groups instead of being measured numerically.

Bar charts or pie graphs.

Examples of categorical data include hair color, type of pets someone has, and favorite foods.

Yes, for example income divided into ranges.

Bar charts.

More about Categorical Data

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