Frequency distribution table


A frequency distribution table is one of the methods of tabulation. In statistics, tabulation means the systematic presentation of information in the data in the form of a table with respect to their characteristics.

Frequency distribution table can also be viewed as a type of classification by constructing different classes or class intervals. Let us understand how to construct a frequency distribution table with the help of an example as follows.

 

Example

The weights (in pounds) of 30 newborn babies are listed below:

5.4, 5.5, 5.7, 5.8, 5.9, 6.1, 6.1, 6.4, 6.4, 6.5, 6.6, 6.7, 6.7, 6.7, 7.0, 7.0, 7.0, 7.1, 7.2, 7.2, 7.4, 7.5, 7.7, 7.7, 7.8, 8.0, 8.1, 8.1, 8.3, 8.7.

Construct the frequency distribution table for the above data. Use the appropriate number of classes.


Solution

Follow the steps given below to construct the frequency distribution table for the given data.

Step 1: Choosing the number of classes

Figure out how many classes do you need form. There is no such hard and fast rule to decide the number of classes, but the Sturges’ rule given by Prof. Sturges approximates the number of classes (k) based on the number of observations (N) in the data.

k = 1 + 3.322*log10N

Here, the number of observations, N = 30. Using the Sturges’ rule, we get the number of classes (k) for the given data as follows.

k = 1 + 3.322*log1030 = 5.9070

Round this value to the nearest integer to get, the number of classes k = 6.

 

Step 2: Find the range of the observations

Find the range of the given data using the formula, range = Max – Min.

Here Max = Largest observation in the data, and Min = Smallest observation in the data.

Here Max = Largest weight = 8.7, Min = Smallest weight = 5.4.

This gives, range = 8.7 – 5.4 = 3.3.

 

Step 3: Find the class width

The class with is nothing but the length of each class interval is given the following formula.


Observe that, all the observations in the given data are rounded up to one decimal place. Therefore, round the class-width to one decimal place to get 0.6.

 

Step 4: Find the lower limits of classes

The lower limit of the first class is the Min value (i.e., smallest observation in the data). To find the lower limits of all other classes, add class-width to the previous lower limit as follows.

Lower limit of Class 1 is, 5.4.

Lower limit of Class 2 is, 5.4 + 0.6 = 6.0.

Lower limit of Class 3 is, 6.0 + 0.6 = 6.6

Lower limit of Class 4 is, 6.6 + 0.6 = 7.2.

Lower limit of Class 5 is, 7.2 + 0.6 = 7.8.

Lower limit of Class 6 is, 7.8 + 0.6 = 8.4.

 

Step 5: Find the upper limits of classes

Observe that the least difference among all the observations is, 0.1. Therefore, the upper limit of first-class is given by the following formula.


The upper limit of Class 1 = Lower limit of Class 1 + Class width – Least difference


To find the upper limits of all other classes, add class-width to the previous upper limit as follows.

Upper limit of Class 1 is, 5.4 + 0.6 – 0.1 = 5.9.

Upper limit of Class 2 is, 5.9 + 0.6 = 6.5.

Upper limit of Class 3 is, 6.5 + 0.6 = 7.1.

Upper limit of Class 4 is, 7.1 + 0.6 = 7.7.

Upper limit of Class 5 is, 7.7 + 0.6 = 8.3.

Upper limit of Class 6 is, 8.3 + 0.6 = 8.9.

 

Step 6: Find the class intervals

The classes of a frequency distribution are written as, (Lower Limit, Upper Limit). Use the lower limits and upper limits found in steps 4 and 5 to write the classes as follows.

Class 1: (5.4, 5.9)

Class 2: (6.0, 6.5)

Class 3: (6.6, 7.1)

Class 4: (7.2, 7.7)

Class 5: (7.8, 8.3)

Class 6: (8.4, 8.9)

 

 Step 7: Find the frequencies of each class

The frequency of a class is the number of observations in the original data that lies within the class. Consider the given data on the weights (in pounds) of 30 newborn babies as follows.

5.4, 5.5, 5.7, 5.8, 5.9, 6.1, 6.1, 6.4, 6.4, 6.5, 6.6, 6.7, 6.7, 6.7, 7.0, 7.0, 7.0, 7.1, 7.2, 7.2, 7.4, 7.5, 7.7, 7.7, 7.8, 8.0, 8.1, 8.1, 8.3, 8.7.

Observe that, 5.4, 5.5, 5.7, 5.8, 5.9 are the only observations that lie within the class interval (5.4, 5.9). Therefore, the frequency of Class 1 is 5.

After finding the frequencies of all the classes in the same way we get the frequency distribution table as follows.


Class

Frequency

5.4 – 5.9

5

6.0 - 6.5

5

6.6 - 7.1

8

7.2 - 7.7

6

7.8 - 8.3

5

8.4 – 8.9

1

Total

30