Chapter 37.7 - Ogive curves - Solved examples

In the previous section we saw ogive curves. We saw some examples too. In this section we will see some solved examples.
Solved example 37.15
The following distribution gives the daily income of 50 workers of a factory.

Table.37.55

Convert the distribution above to a less than type cumulative frequency distribution, and draw its ogive.
Solution:
1. To draw the less than ogive:
• We want the the upper limit of each class interval to use as the x coordinates
• We want the corresponding values in the 'less than' cumulative frequency column to be used as the y coordinates
• So x coordinates are already available in the question. The y coordinates are tabulated in the third column of the table 37.56 below:

Table.37.56

2. So the coordinates are:
(120,12), (140,26), (160,34), (180,40), (200,50)
• The required ogive is shown in fig.37.8 below:

Fig.37.8

• Note that a kink is given in the x axis between O and 110. This is because, the values between (0,0) and (110,0) are not required for this problem

Solved example 37.16
During the medical check-up of 35 students of a class, their weights were recorded as follows:

Table.37.57

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:
1. To draw the less than ogive:
• We want the the upper limit of each class interval to use as the x coordinates
• We want the corresponding values in the 'less than' cumulative frequency column to be used as the y coordinates
• So y coordinates are already available in the question. They are shown again in the third column of table 37.58 below.
The x coordinates can be obtained from the class intervals tabulated in the first column of the table 37.58 below:

Table.37.58

2. So the coordinates are:
(38,0), (40,3), (42,5), (44,9), (46,14), (48,28), (50,32), (52,35)
• The required ogive is shown in fig.37.9 below:

Fig.37.9

2. Calculation of median from the ogive:
(i) Mark P on the y axis such that, the y coordinate of P is (ⁿ⁄₂) = (³⁵⁄₂) = 17.5
(ii) Draw a horizontal dashed line through P. Let it meet the ogive at Q
(iii) Draw a vertical dashed line through Q. Let it meet the x axis at R
• The coordinates of R are: (46.5,0)
• So median = 46.5 kg
3. Now we will check the median using the formula:
• In this problem, the total number of observations 'n' = 35. So ⁿ⁄₂ = ³⁵⁄₂ = 17.5
• The cumulative frequency closest to and just greater than 17.5 is 28
• The class corresponding to the cumulative frequency 28 is 46-48
• So 46-48 is the median class. It's frequency = 14
■ The median is calculated using the following formula:

Where:
l = lower limit of the median class = 46
n = number of observations = 35
cf = cumulative frequency of the class preceding the median class = 14
f = frequency of the median class = 14
h = width of the class interval (assuming all classes are of the same width) = 2
• Substituting the values, we get:
• Numerator = (ⁿ⁄₂ - cf) = (³⁵⁄₂ - 14)  = (17.5-14) = 3.5
• Thus median = 46 + (^3.5⁄₁₄)×2 = 46 + 0.5 = 46.5 (Same as before)

Solved example 37.17
The following table gives production yield per hectare of wheat of 100 farms of a village.

Table.37.59

Change the distribution to a more than type distribution, and draw its ogive.
Solution:
1. To draw the more than ogive:
• We want the the lower limit of each class interval to use as the x coordinates
• We want the corresponding values in the 'more than' cumulative frequency column to be used as the y coordinates
• So x coordinates are already available in the question.

The y coordinates can be obtained from the cumulative frequency tabulated in the third column of the table 37.60 below:

Table.37.60

2. So the coordinates are:
(50,100), (55, 98), (60,90), (65,78), (70,54), (75,16)
• The required ogive is shown in fig.37.10 below:

Fig.37.10

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We have completed this discussion on topics in statistics. Next, we will be discussing class 11 math topics.

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