Statistics

Mean of Grouped Data Using the Direct Method

A pictorial representation always gives a better understanding than a written statement. A graphical representation helps us in understanding of a given data in an easier and detailed manner. In order to understand the median of a grouped data properly, we have to draw an ogive.

Firstly, let us discuss what is an ogive?

“When the data is given as ‘less than’ or ‘more than’ type and a graph is plotted between either of the limits and the cumulative frequency, the smooth curve so obtained is known as ogive or cumulative frequency curve”.

Let us discuss with an example that how an ogive is helpful to find out the median of a grouped data.

Let us consider that 80 students of a class appeared in a Geography test. The marks obtained (out of 100) by them are given in the following frequency distribution table.

Table - 1

Marks obtained (out of 100)

Number of students

0 − 10

2

10 − 20

1

20 − 30

3

30 − 40

5

40 − 50

9

50 − 60

15

60 − 70

22

70 − 80

12

80 − 90

7

90 − 100

4

We can write the above table in following two ways.

Less than type More than type1. For less than type

We can write the given table in less than type as follows.

Marks obtained (out of 100)

Number of students (Cumulative frequency)

Less than 10

2

Less than 20

3

Less than 30

6

Less than 40

11

Less than 50

20

Less than 60

35

Less than 70

57

Less than 80

69

Less than 90

76

Less than 100

80

Construction of Ogive of less than type

The smooth curve drawn between the upper limits of class intervals and cumulative frequency is called cumulative frequency curve or ogive (of less than type). The upper limits of the intervals and cumulative frequency are shown in the above table.

The method of drawing an ogive of less than type is as follows.

(1) Firstly we draw two perpendicular lines, one is horizontal (x-axis) and the other is vertical (y-axis), on a graph.

(2) Now, we mark the upper limits on the horizontal line and the cumulative

frequencies on the vertical line by taking suitable scale.

(3) After this, we plot the points (10, 2), (20, 3), (30, 6), (40, 11), (50, 20), (60, 35), (70, 57), (80, 69), (90, 76), (100, 80). These are the points corresponding to the upper limit and the cumulative frequency.

(4) Now, we join these points to obtain a smooth curve.

After following these steps, we obtain the following graph.

The smooth curve obtained in this figure is the ogive of less than type of the given data.

To find the median with the help of ogive

An important application of ogive in statistics is to find the median.

Let us see the method of finding the median through the same example.

In the previous example, number of observations, n = 80

Mark the point 40 on the vertical line and then draw a horizontal line through this point. Let this horizontal line intersect the ogive at point A. Now, draw a vertical line through A. Median is the point at which this vertical line intersects the horizontal line.

In this example, the median is 62 (approximately).

2. For more than type

We can write the given table for more than type as follows.

Marks obtained

Number of students (Cumulative frequency)

More than or equal to 0

80

More than or equal to 10

78

More than or equal to 20

77

More than or equal to 30

74

More than or equal to 40

69

More than or equal to 50

60

More than or equal to 60

45

More than or equal to 70

23

More than or equal to 80

11

More than or equal to 90

4

Construction of Ogive of more than type

The smooth curve drawn between the lower limits of class intervals and cumulative frequencies is called cumulative frequency curve or ogive (for more than type).

The method of construction of more than type ogive is same as the construction of less than type. For more than type ogive, we take the lower limits on x-axis and the cumulative frequencies on the y-axis. In the above table, the lower limits and the cumulative frequencies have been represented.

The ogive of more than type is obtained by plotting the points (0, 80), (10, 78), (20, 77), (30, 74), (40, 69), (50, 60), (60, 45), (70, 23), (80, 11), (90, 4).

The ogive of more than type of the previous table has been shown below.

To find the median with the help of ogive

Number of observations, n = 80

Mark the point 40 on the vertical line and then draw a horizontal line from this point. Let this line intersect the ogive at point A. Now, draw a vertical line through A. Median is the point at which this vertical line intersects the horizontal line.

Hence, the median is 62(approximately).

Relation between the ogive of less than type and more than type

Let us draw the graph of both less than type and more than type on the same graph paper.

We observe from the above graph that,

“If we construct a line parallel to y-axis through the point of intersection of both the ogives i.e., of less than type and more than type, then the point at which this line intersects x-axis represents the median of the given data”.

Let us solve some more examples to understand the concept better.

Example 1:

The following distribution table gives the daily wages of 50 workers in a factory.

Wage (in Rs)

Number of workers

50 − 100

5

100 − 150

25

150 − 200

10

200 − 250

7

250 − 300

3

Convert it into a less than type distribution and draw its ogive.

Solution:

We can write the given distribution table as a less than type distribution as follows.

Wage (in Rs)

Number of workers (cumulative frequency)

Less than 100

5

Less than 150

5 + 25 = 30

Less than 200

30 + 10 = 40

Less than 250

40 + 7 = 47

Less than 300

47 + 3 = 50

To obtain the ogive, we have to plot the points (100, 5), (150, 30), (200, 40), (250, 47), (300, 50) on a graph paper taking the upper limits on x-axis and the cumulative frequencies on y-axis.

The ogive obtained has been represented in the following figure.

Here, n = 50

∴

First, we mark the point 25 on the vertical line and then draw a horizontal line through this point. Let this horizontal line intersect the ogive at point A. Now, draw a vertical line through point A. Med

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