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Exercise 14.4 Page: 293

1. The following distribution gives the daily income of 50 workers if a factory. Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive.

Daily income in Rupees 100-120 120-140 140-160 160-180 180-200
Number of workers 12 14 8 6 10

Solution

Convert the given distribution table to a less than type cumulative frequency distribution, and we get

Daily income Frequency Cumulative Frequency
Less than 120 12 12
Less than 140 14 26
Less than 160 8 34
Less than 180 6 40
Less than 200 10 50

From the table plot the points corresponding to the ordered pairs such as (120, 12), (140, 26), (160, 34), (180, 40) and (200, 50) on graph paper and the plotted points are joined to get a smooth curve and the obtained curve is known as less than type ogive curve

2.During the medical check-up of 35 students of a class, their weights were recorded as follows:

Weight in kg Number of students
Less than 38 0
Less than 40 3
Less than 42 5
Less than 44 9
Less than 46 14
Less than 48 28
Less than 50 32
Less than 52 35

Draw a less than type ogive for the given data. Hence obtain the median weight from the graph and verify the result by using the formula.

Solution:

From the given data, to represent the table in the form of graph, choose the upper limits of the class intervals are in x-axis and frequencies on y-axis by choosing the convenient scale. Now plot the points corresponding to the ordered pairs given by (38, 0), (40, 3), (42, 5), (44, 9),(46, 14), (48, 28), (50, 32) and (52, 35) on a graph paper an join them to get a smooth curve. The curve obtained is known as less than type ogive.

Locate the point 17.5 on the y-axis and draw a line parallel to the x-axis cutting the curve at a point. From the point, draw a perpendicular line to the x-axis. The intersection point perpendicular to x-axis is the median of the given data. Now, to find the mode by making a table.

Class interval Number of students(Frequency) Cumulative Frequency
Less than 38 0 0
Less than 40 3-0=3 3
Less than 42 5-3=2 8
Less than 44 9-5=4 9
Less than 46 14-9=5 14
Less than 48 28-14=14 28
Less than 50 32-28=4 32
Less than 52 35-22=3 35

The class 46 – 48 has the maximum frequency, therefore, this is modal class

Here, = 46, h = 2, f1= 14, f0= 5 and f2 = 4

The mode formula is given as:

Now, Mode =

= 46 + 0.95 = 46.95

Thus, mode is verified.

3. The following tables gives production yield per hectare of wheat of 100 farms of a village.

Production Yield 50-55 55-60 60-65 65-70 70-75 75-80
Number of farms 2 8 12 24 38 16

Change the distribution to a more than type distribution and draw its ogive.

Solution:

Converting the given distribution to a more than type distribution, we get

Production Yield (kg/ha) Number of farms
More than or equal to 50 100
More than or equal to 55 100-2 = 98
More than or equal to 60 98-8= 90
More than or equal to 65 90-12=78
More than or equal to 70 78-24=54
More than or equal to 75 54-38 =16

From the table obtained draw the ogive by plotting the corresponding points where the upper limits in x-axis and the frequencies obtained in the y-axis are (50, 100), (55, 98), (60, 90), (65, 78), (70, 54) and (75, 16) on

this graph paper. The graph obtained is known as more than type ogive curve.

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