Linear Interpolation

The following table shows the marks obtained for mathematics in a certain group:

ClassFrequency
11 - 2010
21 - 3018
31 - 4030
41 - 5010
51 - 6014
61 - 7016
71 - 802

Now, let's make the corresponding cumulative frequency table for the same data. It is as follows:

ClassFrequencyCumulative Frequency
11 - 201010
21 - 301828
31 - 403058
41 - 501068
51 - 601482
61 - 701698
71 - 802100

In order to plot a cumulative frequency graph, we have to plot cumulative frequency against the upper-class-boundary of each class. The curve should look like the following:

interpolation

 

 

Finding the median

The median is the n/2 th value.
100/2 = 50
50th value.

50th value lies in the 31 - 40 class - i.e. anywhere between 30.5 and 40.5. We use linear interpolation to find it.

Since we treat the segment of the curve as a straight line in this class - shown in brown colour - the process is called linear interpolation. Let's consider the gradient of the line segment as follows:

q1

Gradient = (58-28)/(40.5-30.5)
Gradient = (50-28)/(m-30.5)
So, 30/10 = 22/(m-30.5)
3 = 22/(m-30.5)
3m - 91.5 = 22
3m = 113.5
m = 37.8
median = 37.8

 

 

Finding the first quartile - Q1

The first quartile is the n/4 th value.
100/4 = 25
25th value.

25th value lies in the 21 - 30 class - i.e. anywhere between 20.5 and 30.5. We use linear interpolation to find it.

Let's consider the gradient of the red line segment as follows:

median

Gradient = (28-10)/(30.5-20.5)
Gradient = (25-10)/(Q1-20.5)
So, 18/10 = 15/(Q1-20.5)
1.8 = 15/(Q1-20.5)
1.8Q1 - 36.9 = 15
1.8Q1 = 51.9
Q1 = 28.8
First Quartile = 28.8

 

 

Finding the third quartile - Q3

The third quartile is the 3n/4 th value.
100 X (3/4) = 75
75th value.

75th value lies in the 51 - 60 class - i.e. anywhere between 50.5 and 60.5. We use linear interpolation to find it.

Let's consider the gradient of the yellow line segment as follows:

median

Gradient = (82-68)/(60.5-50.5)
Gradient = (75-68)/(Q3-50.5)
So, 14/10 = 7/(Q3-50.5)
1.4 = 7/(Q3-50.5)
1.4Q3 - 70.7 = 7
1.4Q3 = 77.7
Q3 = 55.5
Third Quartile = 55.5

 

 



 

 

 

 

Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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