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This is an example of a frequency distribution for which the data consists of whole numbers:

The first class contains the numbers 1 through 10.  The lower class limit is 1 and the upper class limit is 10.  The frequency is the number of values in the class, so there are 5 values in the range 1 to 10. 

The class width is the difference between any two successive lower (or upper) limits.  All class widths must be the same.  In this case the class width is 11 - 1 = 10. Explanation

The classes must not overlap and there must be no gaps between classes.  Using whole numbers each lower limit is one more than the preceding upper limit.  What if the data are not whole numbers?

The first and last frequencies must not be zero.  Intermediate frequencies can be zero.

How to construct a frequency distribution

Construct a frequency distribution with five classes for the following data:

22, 38, 11, 40, 10, 32, 26, 12, 47, 39, 28, 40, 17, 34.

Note you have whole number data, so the limits will be whole numbers. 

First determine the class width.  Figure

and then round up to the next whole number, even if the calculation gives a whole number.  Explanation

The class width is 8.  To construct the classes make the first lower limit the lowest value and add the class width going down to get other lower limits.

Remember upper limits have to one less than the next lower limit (for whole number data)

Make a tally by going through the data one by one and placing bars in the appropriate classes.  Then count up and replace tallies by numbers. 

Do I have to do it this way?

Class       Tally             Class       Frequency

10 - 17      ||||                  10 - 17      4

18 - 25      |                    18 - 25      1

26 - 33      |||                   26 - 33      3

34 - 41      ||||                  34 - 41      5

42 - 49      |                    42 - 49      1

Class Boundaries

Class boundaries are found by adding 0.5 to upper limits and subtracting 0.5 from lower limits for whole number data.  If the data had one decimal place you would use 0.05 and so on.

Here is the above distribution using boundaries

Class       Frequency

9.5 - 17.5        4

17.5 - 25.5      1

25.5 - 33.5      3

33.5 - 41.5      5

41.5 - 49.5      1

Class Midpoints

The class midpoint is the average of the two limits or boundaries. 

For example the midpoint of the third class using limits is .

Using boundaries it is

Relative Frequencies

If you add the frequencies you get the total number of data, usually denoted n. 

In the last example n = 4 + 1 + 3 + 5 + 1 = 14

Class       Relative Frequency

9.5 - 17.5        0.286

17.5 - 25.5      0.071

25.5 - 33.5      0.214

33.5 - 41.5      0.357

41.5 - 49.5      0.071

Note that the sum of relative frequencies is one.  If you round decimals to get relative frequencies your sum may differ slightly from 1.  Above you get 0.999

Cumulative Frequencies

The idea of a cumulative frequency is to give the number of data less than a given value. You add frequencies going down to get cumulative frequencies.  Note on Concepts and Ideas

For the last example you get:

Class        Cumulative Frequency

9.5 - 17.5        4

17.5 - 25.5      5

25.5 - 33.5      8

33.5 - 41.5      13

41.5 - 49.5      14                              Note the last cumulative frequency is n, the total number of data

Cumulative Relative Frequencies

To get a cumulative relative frequency you divide the cumulative frequencies by n, the total number of data.  Do not add relative frequencies because if they are rounded errors might creep in.

For the last example you get:

Class        Cumulative Relative Frequency

9.5 - 17.5        0.286

17.5 - 25.5      0.357

25.5 - 33.5      0.571

33.5 - 41.5      0.929

41.5 - 49.5      1                           Note the last cumulative relative frequency is 1

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