Data can be represented in following manners:
(A) Bar graphs
(B) Histograms of uniform width, and of varying widths
(C) Frequency polygons
(A)Bar graphs
A bar graph is a pictorial representation of data in which usually bars of uniform width are drawn with equal spacing between them on one axis (say, the x-axis), depicting the variable. The values of the variable are shown on the other axis (say, the y-axis) and the heights of the bars depend on the values of the variable. (B) Histogram
This is a form of representation like the bar graph, but it is used for continuous class intervals. For instance, consider the frequency distribution Table 14.6, representing the weights of 36 students of a class:
| Weight (in kg) | Number of Students |
|---|---|
| 30.5-35.5 | 9 |
| 35.5-40.5 | 6 |
| 40.5-45.5 | 15 |
| 45.5-50.5 | 3 |
| 50.5-55.5 | 1 |
| 55.5-60.5 | 2 | Total | 36 |
Let us represent the data given above graphically as follows:
(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 $cm$.
(iv) In this way, we obtain the graph as shown in Fig. 14.3:
Observe that since there are no gaps in between consecutive rectangles, the resultant graph appears like a solid figure. This is called a histogram, which is a graphical representation of a grouped frequency distribution with continuous classes. Also, unlike a bar graph, the width of the bar plays a significant role in its construction.
C.) Frequency Polygon
There is yet another visual way of representing quantitative data and its frequencies. This is a polygon. To see what we mean, consider the histogram represented by Fig. 14.3. Let us join the mid-points of the upper sides of the adjacent rectangles of this histogram by means of line segments. Let us call these mid-points B, C, D, E, F and G. When joined by line segments, we obtain the figure BCDEFG (see Fig. 14.6). To complete the polygon, we assume that there is a class interval with frequency zero before 30.5 - 35.5, and one after 55.5 - 60.5, and their mid-points are A and H, respectively. ABCDEFGH is the frequency polygon corresponding to the data shown in Fig. 14.3. We have shown this in Fig. 14.6.
Although, there exists no class preceding the lowest class and no class succeeding the highest class, addition of the two class intervals with zero frequency enables us to make the area of the frequency polygon the same as the area of the histogram. Why is this so? (Hint : Use the properties of congruent triangles.)
Now, the question arises: how do we complete the polygon when there is no class preceding the first class? Let us consider such a situation.
Example: Consider the marks, out of 100, obtained by 51 students of a class in a test, given in Table 14.9.
| Marks | No. Of Students(frequency) |
| 0-10 | 5 |
| 10-20 | 10 |
| 20-30 | 4 |
| 30-40 | 6 |
| 40-50 | 7 |
| 50-60 | 3 |
| 60-70 | 2 |
| 70-80 | 2 |
| 80-90 | 3 |
| 90-100 | 9 |
| TOTAL | 51 |
Draw a frequency polygon corresponding to this frequency distribution table.