Histogram
This is a form of representation like the bar graph, but it is used for
Let us represent the data given above graphically as follows:
| Weights (in kg) | Number of students |
|---|---|
| 30.5 - 35.5 | |
| 35.5 - 40.5 | |
| 40.5 - 45.5 | |
| 45.5 - 50.5 | |
| 50.5 - 55.5 | |
| 55.5 - 60.5 | |
| Total |
(i) We represent the weights on the horizontal axis on a suitable scale. We can choose the scale as 1 cm = 5 kg. Also, since the first class interval is starting from 30.5 and not zero, we show it on the graph by marking a kink or a break on the axis.
(ii) We represent the number of students (frequency) on the vertical axis on a suitable scale. Since the maximum frequency is 15, we need to choose the scale to accomodate this maximum frequency.
(iii) We now draw rectangles (or rectangular bars) of width equal to the class-size and lengths according to the frequencies of the corresponding class intervals. For example, the rectangle for the class interval 30.5 - 35.5 will be of width 1 cm and length 4.5 cm.
Observe that since there are no gaps in between consecutive rectangles, the resultant graph appears like a
Here, in fact, areas of the rectangles erected are proportional to the corresponding frequencies. However, since the widths of the rectangles are all equal, the lengths of the rectangles are proportional to the frequencies. That is why, we draw the lengths according to (iii) above.
Now, consider a situation different from the one above.
Example 3 : A teacher wanted to analyse the performance of two sections of students in a mathematics test of 100 marks. Looking at their performances, she found that a few students got under 20 marks and a few got 70 marks or above. So she decided to group them into intervals of varying sizes as follows: 0 - 20, 20 - 30, . . ., 60 - 70, 70 - 100. Then she formed the following table:
| Marks | Number of students |
|---|---|
| 10 - 20 | |
| 20 - 30 | |
| 30 - 40 | |
| 40 - 50 | |
| 50 - 60 | |
| 60 - 70 | |
| 70 - 80 | |
| 80 - 90 |
A histogram for this table was prepared by a student as shown in Fig
Carefully examine this graphical representation. Do you think that it correctly represents the data?
No, the graph is giving us a misleading picture. As we have mentioned earlier, the areas of the rectangles are proportional to the frequencies in a histogram. Earlier this problem did not arise, because the widths of all the rectangles were equal. But here, since the widths of the rectangles are varying, the histogram above does not give a correct picture. For example, it shows a greater frequency in the interval 70 - 100, than in 60 - 70, which is not the case.
So, we need to make certain modifications in the lengths of the rectangles so that the areas are again proportional to the frequencies. The steps to be followed are as given below:
- Select a class interval with the minimum class size. In the example above, the minimum class-size is
.
2. The lengths of the rectangles are then modified to be proportionate to the class-size 10.
For instance, when the class-size is 20, the length of the rectangle is 7. So when the class-size is 10, the length of the rectangle will be
Similarly, proceeding in this manner, we get the following table:
| Marks | Frequency | Width of the class | Length of the rectangle |
|---|---|---|---|
| 0 - 20 | 7 | 20 | |
| 20 - 30 | 10 | 10 | |
| 30 - 40 | 10 | 10 | |
| 40 - 50 | 20 | 10 | |
| 50 - 60 | 20 | 10 | |
| 60 - 70 | 15 | 10 | |
| 70 - 100 | 8 | 30 |
Since we have calculated these lengths for an interval of 10 marks in each case, we may call these lengths as “proportion of students per 10 marks interval”. So, the correct histogram with varying width is given in Fig
A histogram for this table was prepared by a student as shown in Fig