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Frequency Histograms


We will start by looking at a graphical method for studying the variation known as the 'Frequency Histogram'.

To create a frequency histogram, group the data into ‘bins’, each bin containing a range of values. The vertical axis represents the number of observations in each range, known as the 'frequency'.

The data below show the test results for 25 students:

Results

 

Bin

Midpoint

Frequency

38

10

60

90

88

 

  >0-20

10

7

96

1

41

86

14

 

  >20-40

30

8

25

5

3

16

22

 

  >40-60

50

5

2

29

34

55

36

 

  >60-80

70

0

37

36

91

47

43

 

  >80-100

90

5

There are various ways of calculating the number of bins. I find that using the square root of the number of data values gives as good a result as the more complicated methods. The value is usually on the low side, but you can adjust it upwards to get convenient bin boundaries. Treat the calculated number of bins as a starting point, and adjust it as necessary to give the result you prefer.

In the example there are 25 data values. The square root rule gives 5 bins. The smallest data value is 1, the largest is 96. A scale stretching from 0 through 100 will contain all the values; this conveniently gives 5 bins of span 20.

Now we group the data into the bins. You must decide on how to handle values that fall on the boundary between two bins. Various conventions are used, I include values greater than the lower boundary up to and including the upper boundary; this is consistent with the convention used by Excel.

For example, there are 8 data values greater than 20 and up to and including 40. I’ve highlighted them in the table to make them easy to count. After calculating all the frequencies we can create the histogram:

There are two conventions for showing the bin values on the histogram:

  1. show the midpoint of the bin range
  2. show the upper limit of each bin range, the 'cutpoint'

The histogram above shows the midpoint convention. Pass your cursor over the image to see the alternative 'cutpoint' convention.


 
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