Creating Frequency Distributions and Histograms Example: It is often quite useful to represent data graphically. One popular way to represent data is the histogram. The following set of 16 values can be organized into a frequency distribution (table) and then the frequency distribution is used to create a histogram.
18, 16, 14, 12, 13, 19, 17, 14, 10, 9, 11, 14, 13, 19, 10, 16
The sorted data set is: 9, 10, 10, 11, 12, 13, 13, 14, 14, 14, 16, 16, 17, 18, 19, 19
To create a histogram from this set of data we start by constructing a frequency distribution or table. A frequency distribution shows how data is split up into categories or classes by listing
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the classes along with the number of data values in each class. The first step in creating a frequency distribution is deciding how many classes we wish to use. For this data set let’s use four classes. Once that decision has been made there is a step-by-step process we can follow.
Step 1: To calculate the class width, find the following value.
h = −
If this value is a decimal, round up to the nearest integer (illustrated in this example). If this value is an integer, you may have to add one in order to have enough classes to accommodate the data.
For the data set listed above, the width is:
width = 19−9 = 2.5 4
This value is rounded up to give a width of 3.
Step 2: Choose a value for the first lower class limit. Typically, this is the minimum value but it could also be a conveniently chosen value.
For the data set listed above, the first lower class limit will be 9.
Step 3: Use the first lower class limit and class width to list the other lower class limits. Add the class width to each lower class limit to determine the next lower class limit.
First lower class limit = 9
Second lower class limit = 9+3 = 12 Third lower class limit = 12+3 = 15 Fourth lower class limit = 15+3 = 18
This creates the start of a frequency distribution table as shown below:
Class Frequency 9-
12 -
15 -
18 -
Step 4: Determine the upper class limits. The upper class limit in the FIRST class will be the value just below the lower class limit in the SECOND class. Then add the width to get the remaining upper class limits.
First upper class limit = 11, because 11 is the value right before 12 Second upper class limit = 11+3 = 14
Third upper class limit = 14+3 = 17
Fourth upper class limit = 17+3 = 20
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Now the classes in the table are complete, as shown below.
Class Frequency 9 - 11
12 - 14
15 - 17
18 - 20
Step 5: Use the sorted list to determine how many values belong in each class and enter the value into the Frequency column. Be sure to check that all values are included by adding the frequencies to get the sample size.
9, 10, 10, 11, 12, 13, 13, 14, 14, 14, 16, 16, 17, 18, 19, 19
The first class has a frequency of 4, because 9, 10, 10, 11 are in the first class.
The second class has a frequency of 6, because 12, 13, 13, 14, 14, 14 are in the second class. The third class has a frequency of 3, because 16, 16, 17 are in the third class.
The fourth class has a frequency of 3, because 18, 19, 19 are in the fourth class.
Class Frequency 9 - 11 4
12-14 6
15-17 3
18-20 3
For the frequency distribution above, the sum of the frequencies is equal to the sample size: 4+6+3+3=16
Step 6: Follow the directions from the link on how to “Create a Histogram From a Frequency Table” (included below) to create a histogram similar to the one shown here.
7 6 5 4 3 2 1 0
Histogram
9 - 11
12 - 14
Data Values
18 - 20
15 - 17
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