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