## Statistics

### 12.1 Graphical Representation of Data

The representation of data by tables has already been discussed. Now let us turn our attention to another representation of data, i.e., the graphical representation. It is well said that one picture is better than a thousand words. Usually comparisons among the individual items are best shown by means of graphs. The representation then becomes easier to understand than the actual data. We shall study the following graphical representations in this section.

(A) Bar graphs

(B) Histograms of uniform width, and of varying widths

(C) Frequency polygons

#### (A) Bar Graphs

In earlier classes, you have already studied and constructed bar graphs. Here we shall discuss them through a more formal approach. Recall that 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.

**Activity 1 :** Continuing with the same four groups of Activity 1, represent the data by suitable bar graphs.

Let us now see how a frequency distribution table for continuous class intervals can be represented graphically.

#### (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 12.2, representing the weights of 36 students of a class:

Table 12.2

Weights (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 |

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 $1 \mathrm{~cm}=5 \mathrm{~kg}$. 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 $1 \mathrm{~cm}$ and length $4.5 \mathrm{~cm}$.

(iv) In this way, we obtain the graph as shown in Fig. 12.3:

Fig. 12.3

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.

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.

#### (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. 12.3. 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. 12.6). 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 $\mathrm{A}$ and $\mathrm{H}$, respectively. $\mathrm{ABCDEFGH}$ is the frequency polygon corresponding to the data shown in Fig. 12.3. We have shown this in Fig. 12.6.

Fig. 12.6

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.

### 12.2 Summary

In this chapter, you have studied the following points:

**1.** How data can be presented graphically in the form of bar graphs, histograms and frequency polygons.