Histograms and cumulative frequency

Histograms
Histograms are similar to bar charts apart from the consideration of areas. In a bar chart, all of the bars are the same width and the only thing that matters is the height of the bar. In a histogram, the area is the important thing.

Example: Draw a histogram for the following information.

Frequency:

Height (feet):         (Number of pupils)        Relative frequency:

0-2                        0                                0

2-4                        1                                1

4-5                        4                                8

5-6                        8                                16

6-8                        2                                2

(Ignore relative frequency for now). It is difficult to draw a bar chart for this information, because the class divisions for the height are not the same. If you do not understand, try drawing a bar chart.

When drawing a histogram, the y-axis is labelled 'relative frequency'. You must work out the relative frequency before you can draw a histogram. To do this, first work out the standard width of the groups. Some of the heights are grouped into 2s (0-2, 2-4, 6-8) and some into 1s (4-5, 5-6). Most are 2s, so we shall call the standard width 2. To make the areas match, we must double the values for frequency which have a class division of 1 (since 1 is half of 2). Therefore the figures in the 4-5 and the 5-6 columns must be doubled. If any of the class divisions were 4 (for example if there was a 8-12 group), these figures would be halved. This is because the area of this 'bar' will be twice the standard width of 2 unless we half the frequency.

Cumulative Frequency
This is the running total of the frequencies. On a graph, it can be represented by a cumulative frequency polygon, where straight lines join up the points, or a cumulative frequency curve.

Example:

Height (cm) Frequency: Cumulative frequency:
0 - 100                     4                     4
100-120                    6                    10             (4 + 6)
120-140                    3                    13             (4 + 6 + 3)
140-160                    2                    15             (4 + 6 + 3 + 2)
160-180                    6                    21             (4 + 6 + 3 + 2 + 6)
180-220                    4                    25             (4 + 6 + 3 + 2 + 6 + 4)

These data are used to draw a cumulative frequency polygon by plotting the cumulative frequencies against the upper class boundaries.