# Frequency Distributions

A frequency distribution is a tool for organizing data. We use it to group data into categories and show the number of observations in each category. Here are some test scores from a math class.

 65 91 85 76 85 87 79 93 82 75 100 70 88 78 83 59 87 69 89 54 74 89 83 80 94 67 77 92 82 70 94 84 96 98 46 70 90 96 88 72

It's hard to get a feel for this data in this format because it is unorganized. To construct a frequency distribution, you should first identify the lowest and highest values in the list. We do this because we want to be sure that each value in the list fits into one of our categories. The low value here is 46, and the high is 100. A set of categories that would work here is 41-50, 51-60, 61-70, 71-80, 81-90, and 91-100. Here's a finished product :

 Class Frequency 41-50 1 51-60 2 61-70 6 71-80 8 81-90 14 91-100 9

We can now see that the biggest number of tests were between 81 and 90, and most of the tests were between 71 and 100.

The low number in each category (or class) is called the lower class limit, and the high number is called the upper class limit.

Now for some guidelines for constructing a frequency distribution.

• Each value should fit into a category. The classes should be mutually exhaustive.
• No value should fit into more than 1 category. The classes should be mutually exclusive, there should be no overlapping of classes.
• Make the classes of equal size if possible. This makes it easier to compare the frequency in one class to another.
• Avoid open-ended classes if possible such as "75 and over".
• Try to use between 5 and 20 classes if possible. If you have fewer than 5 classes, you're not really breaking up the data, and if you use more than 20 classes, this will probably be information overflow.
• It is usually convenient to use class sizes of 5 or 10, in other words, to have each class containing 5 or 10 possible values.
• It is usually convenient to make the lower limit of the first category a multiple of the class size.

After the first two rules above, the rest are merely suggestions. Each set of data may require you to violate some of these suggestions. The best advice is to try and follow them whenever possible.

One further extension to the frequency distribution is to look at the percentage of values that show up in each category. This is called a relative frequency distribution or percent frequency distribution. Here's how the above data would be presented in this way.

 Class Frequency Relative Frequency Percent 41-50 1 1/40 2.5% 51-60 2 2/40 5% 61-70 6 6/40 15% 71-80 8 8/40 20% 81-90 14 14/40 35% 91-100 9 9/40 22.5%

The final frequency distribution that we will discuss is the cumulative frequency distribution. Think about the word cumulative, it generally refers to some sort of total. A cumulative frequency distribution is a way to list how many values fit into the first class, the first 2 classes, the first 3 classes, etc., or the last class, the last 2 classes, etc. Here's a cumulative less than frequency distribution for the above set of data.

 Class Frequency Cumulative (Less Than) 41-50 1 1 51-60 2 3 61-70 6 9 71-80 8 17 81-90 14 31 91-100 9 40

The 1 means that there is 1 value that is 50 or less, the 3 means that there are 3 values that are 60 or less, the 9 means that there are 9 values that are 70 or less, and so on.

Now for a cumulative greater than frequency distribution.

 Class Frequency Cumulative (Greater Than) 41-50 1 40 51-60 2 39 61-70 6 37 71-80 8 31 81-90 14 23 91-100 9 9

The 40 means that there are 40 values that are 41 or more, the 39 means that there are 39 values that are 51 or more, the 37 means that there are 37 values that are 61 or more, and so on.

Practice Questions