♦ Graphical Representation of Cumulative Frequency Distribution

As we all know, pictures speak better than words. A graphical representation helps us in understanding given data at a glance.

As we've represented the data through bar graphs, histograms and frequency polygons. Let us now represent a cumulative frequency distribution graphically.

For example, let us consider the cumulative frequency distribution given in Table 14.13.

Recall that the values `10, 20, 30, . . ., 100` are the upper limits of the respective class intervals. To represent the data in the table graphically, we mark the upper limits of the class intervals on the horizontal axis (x-axis) and their corresponding cumulative frequencies on the vertical axis ( y-axis), choosing a convenient scale.

The scale may not be the same on both the axis. Let us now plot the points corresponding to the ordered pairs given by (upper limit, corresponding cumulative frequency), i.e., `(10, 5), (20, 8), (30, 12), (40, 15), (50, 18), (60, 22), (70, 29), (80, 38), (90, 45), (100, 53) `on a graph paper and join them by a free hand smooth curve.

The curve we get is called a cumulative frequency curve, or an ogive (of the less than type). (See Fig. 14.1)

Next, again we consider the cumulative frequency distribution given in Table 14.14 and draw its ogive (of the more than type).

Recall that, here` 0, 10, 20, . . ., 90` are the lower limits of the respective class intervals `0 - 10, 10 - 20, . . ., 90 - 100`. To represent ‘the more than type’ graphically, we plot the lower limits on the x-axis and the corresponding cumulative frequencies on the y-axis.

Then we plot the points (lower limit, corresponding cumulative frequency), i.e., `(0, 53), (10, 48), (20, 45), (30, 41), (40, 38), (50, 35), (60, 31), (70, 24), (80, 15), (90, 8)`, on a graph paper, and join them by a free hand smooth curve.

The curve we get is a cumulative frequency curve, or an ogive (of the more than type). (See Fig. 14.2)

`"Remark"`

Note that both the ogives (in Fig. 14.1 and Fig. 14.2) correspond to the same data, which is given in Table 14.12.

Now, are the ogives related to the median in any way? Is it possible to obtain the median from these two cumulative frequency curves corresponding to the data in Table 14.12? Let us see.

One obvious way is to locate `n/2 = 53/2 = 26.5` on the y-axis (see Fig. 14.3). From this point, draw a line parallel to the x-axis cutting the curve at a point. From this point, draw a perpendicular to the x-axis.

The point of intersection of this perpendicular with the x-axis determines the median of the data (see Fig. 14.3).

Another way of obtaining the median is the following :

Draw both ogives (i.e., of the less than type and of the more than type) on the same axis. The two ogives will intersect each other at a point. From this point, if we draw a perpendicular on the x-axis, the point at which it cuts the x-axis gives us the median (see Fig. 14.4).

As we've represented the data through bar graphs, histograms and frequency polygons. Let us now represent a cumulative frequency distribution graphically.

For example, let us consider the cumulative frequency distribution given in Table 14.13.

Recall that the values `10, 20, 30, . . ., 100` are the upper limits of the respective class intervals. To represent the data in the table graphically, we mark the upper limits of the class intervals on the horizontal axis (x-axis) and their corresponding cumulative frequencies on the vertical axis ( y-axis), choosing a convenient scale.

The scale may not be the same on both the axis. Let us now plot the points corresponding to the ordered pairs given by (upper limit, corresponding cumulative frequency), i.e., `(10, 5), (20, 8), (30, 12), (40, 15), (50, 18), (60, 22), (70, 29), (80, 38), (90, 45), (100, 53) `on a graph paper and join them by a free hand smooth curve.

The curve we get is called a cumulative frequency curve, or an ogive (of the less than type). (See Fig. 14.1)

The term ‘ogive’ is pronounced as ‘ojeev’ and is derived from the word ogee. An ogee is a shape consisting of a concave arc flowing into a convex arc, so forming an S-shaped curve with vertical ends.

In architecture, the ogee shape is one of the characteristics of the 14th and 15th century Gothic styles .

Next, again we consider the cumulative frequency distribution given in Table 14.14 and draw its ogive (of the more than type).

Recall that, here` 0, 10, 20, . . ., 90` are the lower limits of the respective class intervals `0 - 10, 10 - 20, . . ., 90 - 100`. To represent ‘the more than type’ graphically, we plot the lower limits on the x-axis and the corresponding cumulative frequencies on the y-axis.

Then we plot the points (lower limit, corresponding cumulative frequency), i.e., `(0, 53), (10, 48), (20, 45), (30, 41), (40, 38), (50, 35), (60, 31), (70, 24), (80, 15), (90, 8)`, on a graph paper, and join them by a free hand smooth curve.

The curve we get is a cumulative frequency curve, or an ogive (of the more than type). (See Fig. 14.2)

`"Remark"`

Note that both the ogives (in Fig. 14.1 and Fig. 14.2) correspond to the same data, which is given in Table 14.12.

Now, are the ogives related to the median in any way? Is it possible to obtain the median from these two cumulative frequency curves corresponding to the data in Table 14.12? Let us see.

One obvious way is to locate `n/2 = 53/2 = 26.5` on the y-axis (see Fig. 14.3). From this point, draw a line parallel to the x-axis cutting the curve at a point. From this point, draw a perpendicular to the x-axis.

The point of intersection of this perpendicular with the x-axis determines the median of the data (see Fig. 14.3).

Another way of obtaining the median is the following :

Draw both ogives (i.e., of the less than type and of the more than type) on the same axis. The two ogives will intersect each other at a point. From this point, if we draw a perpendicular on the x-axis, the point at which it cuts the x-axis gives us the median (see Fig. 14.4).

Q 3280245117

The annual profits earned by 30 shops of a shopping complex in a

locality give rise to the following distribution :

Draw both ogives for the data above. Hence obtain the median profit.

Class 10 Chapter 14 Example 9

locality give rise to the following distribution :

Draw both ogives for the data above. Hence obtain the median profit.

Class 10 Chapter 14 Example 9

We first draw the coordinate axes, with lower limits of the profit along the horizontal axis, and the cumulative

frequency along the vertical axes. Then, we plot the points (5, 30), (10, 28), (15, 16), (20, 14), (25, 10), (30, 7) and (35, 3). We

join these points with a smooth curve to get the ‘more than’ ogive, as shown in Fig. 14.5.

Now, let us obtain the classes, their frequencies and the cumulative frequency from the table above.

Using these values, we plot the points `(10, 2), (15, 14), (20, 16), (25, 20), (30, 23), (35, 27), (40, 30)` on the same axes as in

Fig. 14.5 to get the ‘less than’ ogive, as shown in Fig. 14.6.

The abcissa of their point of intersection is nearly 17.5, which is the median. This can

also be verified by using the formula. Hence, the median profit (in lakhs) is Rs 17.5.

`text (Remark : ) ` In the above examples, it may be noted that the class intervals were

continuous. For drawing ogives, it should be ensured that the class intervals are

continuous. (Also see constructions of histograms in Class IX)