Formula for Median with Examples

Here you will learn what is the formula for median of grouped and ungrouped data and how to find median with examples.

Let’s begin –

What is Median ?

Median is defined as the measure of central term when they are arranged in ascending or descending order of magnitude.

Formula for Median :

(i) For ungrouped distribution : Let n be the number of variate in a series then

Median = \(({n + 1\over 2})^{th}\) term, (when n is odd)

Median = Mean of \(({n\over 2})^{th}\) and \(({n\over 2} + 1)^{th}\) terms, (where n is even)

Example : Find the median of 6, 8, 9, 10, 11, 12 and 13.

Solution : Total number of terms = 7

Here, n is odd.

The middle term = \(1\over 2\)(7 + 1) = 4th

Median = Value of the 4th term =10

Hence, the median of the given series is 10.

(ii) For ungrouped frequency distribution : First we prepare the cumulative frequency(c.f.) column and Find value of N then

Median = \(({N + 1\over 2})^{th}\) term, (when N is odd)

Median = Mean of \(({N\over 2})^{th}\) and \(({N\over 2} + 1)^{th}\) terms, (where n is even)

(iii) For grouped frequency distribution : Prepare c.f. column and find value of \(N\over 2\) then find the class which contain value of c.f. is equal or just greater to N/2, this is median class

Median = \(l\) + \(({N\over 2} – F)\over f\)\(\times\)h

where  \(l\) – lower limit of median class

f – frequency of median class

F – c.f. of the class preceding median class

h – class interval of median class

Example : Find the median of the following frequency distribution.

class 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
\(f_i\) 8 30 40 12 10

Solution :

class 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
\(f_i\) 8 30 40 12 10
c.f. 8 38 78 90 100

Here \(N\over 2\) = \(100\over 2\) = 50 which lies in the value of 78 of c.f. hence corresponding class of this c.f. is 20 – 30 is the median class, so

\(l\) = 20, f = 40, f = 38, h = 10

\(\therefore\) Median = \(l\) + \(({N\over 2} – F)\over f\)\(\times\)h = 20 + \((50 – 38)\over 40\)\(\times\)10 = 23

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