FINDING MEDIAN FOR GROUPED DATA

Example 1 :

The following are the marks scored by the students in the Summative Assessment exam. 

Calculate the median. 

Solution :

Median class  =  (N/2)^th value

=  (50/2)^th value

=  25^th value

Median class  =  30 to 40

l = 30, N//2  =  25, m = 24, f = 10 and c = 10

Substitute.

Median  =  30 + ([25 - 24]/10) x 10

=  30 + 1

≈  31

Example 2 :

The following table gives the weekly expenditure of 200 families. Find the median of the weekly expenditure.

Solution : 

Median class  =  (N/2)^th value

=  (200/2)^th value

=  100^th value

Median class  =  2000 - 3000

l = 2000, N//2  =  100, m = 74, f = 54 and c = 1000

Substitute.

Median  =  2000 + ([100 - 74]/54) x 1000

=  2000 + (26/54) x 1000

=  2000 + 481.5

=  2481.5

Example 3 :

The Median of the following data is 24. Find the value of x.

Solution : 

Since the median is 24 and median class is 20 – 30.

l = 20 N = 55 + x, m = 30, c = 10, f = x

Substitute. 

24  =  20 + {[(55 + x)/2 - 30] / x} ⋅ 10

4  =  {[(x - 5)/2] / x} ⋅ 10

4  =  {(x - 5) / 2x} ⋅ 10

4  =  {(x - 5) / x} ⋅ 5

4  =  (5x - 25) / x

4x  =  5x - 25

25  =  5x - 4x

25  =  x

Example 4 :

The table below gives data on the heights in cm, of 51 children.

a) Estimate the mean height

b) Estimate the median height

c) Find the modal class

Solution :

a) Finding mean :

Mean = Sum of all values / total number of values

= 8215 / 51

= 161.07

b)  Finding median :

N - number of terms is even, then median = (N + 1)/2

= (51 + 1)/2^th value

= 52/2^th value

26^th value

To get 26th value, we need to get 4^th value. Approximately 162 is the median.

c) Finding the modal class :

The modal class is 160 ≤ h < 170.

Example 5 :

A door to door salesman keeps a record of the number of homes he visits each day.

a) Estimate the mean number of homes visited.

b)  Estimate the median

c)  What is the modal class.

Solution :

Since the given interval is not continuous, to make it as continuous.

a) Finding mean :

Mean = Sum of all values / total number of values

= 3723.5 / 116

= 32.09

b)  Finding median :

N - number of terms is even, then median = N/2

= 116/2^th value

= 58^th value

The median will lie in between the interval 20 to 29. 

= L + [(N/2 - m)/f] x c

L = 29.5, f = 60, N/2 = 58, c = 10, m = 11

= 29.5 + [(58 - 35)/60] x 10

= 29.5 + 0.383 x 10

= 29.5 + 3.83

= 33.33

c) Modal class is 30 to 39.

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