Median-Discrete Series

For calculating Median-Discrete Series, the steps are as follows:

1. At first arrange the data in ascending or descending order of magnitude.
2. Then find out the cumulative frequencies.
3. After that apply the formula median= size of (N + 1)/2
4. Now look at the cumulative frequency column and find the total which is either equal to (N + 1)/2 or next higher to that and determine the value of the variable corresponding to it. That gives the value of the median.

Illustration:

From the following data find the value of median.

Solution: 

Calculation of median

Median = size of {N + (1/2)} th item = 122 + (1/2) = 61.5th item.

Size of 61.5th item = $ 6,800, hence the median income is $ 6,800.

Calculation of median - continuous series

The Steps which determine the particular class in which the value of median lies use N/2 as the rank of the median instead of N + (1/2). Some writers have suggested that while calculating the median in continuous series I should be added to total frequency if it is odd (say, 100) however 1 is to be added in case of individual and discrete series as specific totals and individual values are involved, in a continuous frequency distribution all the frequencies lose their individuality the effort now is not to find the value of one specific item but to find a particular point on a curve- the one value which will have 50 percent of frequencies on one side of it and 50 per cent of the frequencies on the other side. It will be totally wrong to use the above rule. Hence it is N/2 which will divide the area of curve into two equal parts and as such we must use N/2 instead of N + (1/2), in a continuous series. After ascertaining the class in which the median lies, the following formula is used for determining the exact value of the median.

Median = L + (N/2) - c.f / f xi

L = lower limit of the median class, in which the middle item of the distribution lies.

c.f = the cumulative frequency of the class preceding the median class or sum of the frequencies of all classes lower than the median class.

F= simple frequency of the median class.

I = the class interval of the median class.

It should be keep in mind that while interpolating the median value in a frequency distribution it is assumed that the variable is continuous and that there is an orderly and even distribution of items within each class.

Illustration:

Calculate the median for the following frequency distribution

Solution:

First arrange the data in ascending order and then find out median.

Calculation of median

Med = size of N/2 item = 200/2 = 100th item

Median lies in the class 25-30

Med = L + N/2 - c.f. /f x i

L = 25, N/2 = 100, v. f. = 77 f = 42. I = 5 = 25 + 100 - 72 / 42 x 5 = 25 + 2.74 = 27. 74