Grouped frequency distribution

When the range of a data is large, the data must be grouped into classes that are more than one unit in width, in what is called a grouped frequency distribution. Then also when a data is grouped into classes together with corresponding frequencies and then put into a table. It is referred to as grouped frequency distribution table.

For example, the table below shows the distribution of marks obtained by 50 students in an examination.

Processing a grouped frequency distribution:

Depending on questions asked, the following entries can be prepared from a grouped frequency distribution.

NB the following:

Class interval:

The groups 0 – 9, 10 – 19, 20 – 29, etc are called classes and the range of values 0 – 9, 10 – 19, 20 – 29, etc are called class interval. Students must note that the class interval may be given in an exam.

Class limits:

For a class 0 – 9 for example, 0 and 9 are the class limits. 0 is the lower class limit and 9 is the upper class limit. Also for the class 20 – 29, 20 is the lower class limit and 29 is the upper class limit.

Class boundaries:

The numbers in the second column of the table above are called class boundaries. These numbers are used to separate the classes so that there are no gaps in the frequency distribution. The gaps are due to the limits; for example there is a gap between 29 and 30.

Students sometimes have the difficulty finding class boundaries when giving the class limits. The basic rule of thumb is that the class limits should have the same decimal place as the data, but the class boundaries should have one additional place value and ends in a 5.

The accurate way of obtaining the class boundaries are demonstrated below:

v  If the values of the class in the data set are whole numbers such as 10 – 19, 20 – 29 etc. Then boundaries are found by subtracting 0.5 from 10 (the lower class limit) and adding 0.5 to 19 (the upper class limit).

Lower limit – 0.5 = 10 – 0.5 = 9.5 = lower boundary

Upper limit + 0.5 = 19 + 0.5 = 19.5 = upper boundary

v  If the data are in tenths, such as 7.8 – 8.8, 8.8 – 9.8 etc. Then boundaries are found by subtracting 0.05 from 7.8 (the lower class limit) and adding 0.05 to 8.8 (the upper class limit).

Lower limit – 0.05 = 7.8 – 0.05 = 7.75 = lower boundary

Upper limit + 0.05 = 8.8 + 0.05 = 8.85 = upper boundary

Class Width:

The class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the lower (or upper) class limit of the next class. For example, the class width in the preceding distribution (i.e. fig.2) is 10, found from 10 – 0 = 10.

The class width can also be found as the differences between the lower and the upper class boundaries of a class interval. That is from fig.2, CLASS WIDTH = 9.5 – (-0.5) = 10.

NB: Do not subtract the limits of a single class. It will result in an incorrect answer.

Class Midpoint or Class mark (x):

The class midpoint or class mark is found by adding the lower class limit to the upper class limit and then divide the resulting by 2. It can also be found by adding the lower class boundary to the upper class boundary and then divide the answer by 2. Thus for the class 0 – 9. Class midpoint = (0 + 9) ÷ 2 = 9 ÷ 2 = 4.5 or class midpoint = (-0.5 + 9.5) ÷ 2 = 9.0 ÷ 2 = 4.5.

NOTE: To complete a table for class midpoints just get the first one and then add the class width to each.

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