When it is desired to calculate the average of series from the average of its component parts The method involves the calculation of weighted arithmetic average of the different means, using the number of items in each case, as the weights. Thus, if the average of a series is 20 and the number of items in it is 10 and the average of a series is 25 and the number of items in it is 15, the combined average of the two series would be equal to the weighted average of these two averages, the weights being 10 and 15 respectively (the number of items in each case). The weighted arithmetic average would be:—
(20×10)+(25×15)/(10+15) or 23
The simple arithmetic average of the two averages would be or (20+25)/2 22.5. This is an inaccurate average; as if it is multiplied by the total frequency (now 25) it would not give the correct aggregate. If, however, we multiply the weighted arithmetic average or 23, by the total frequency or 25, the product would be 575 which is the total of the aggregates of the two series (200+375).
Formula No. 1
M = 
where, M = The value of the median
l1 and l2 = The lower and the upper limit of the class in which the median lies
f1 = The frequency of the median class
m = The middle number whose value is median (N/2).
c = The cumulative frequency of the class preceding the median class.
The above formula has been written in a slightly different form by some authors according to their nomenclature.
Median = L +(N\2-c.f.)/f x i
where L is the lower limit of the median class, N/2 == the middle number, c.f. — the cumulative frequency of the class preceding the median class, f — the frequency of the median class and i = the magnitude of the median class-interval.
Formula No. 2
There is another formula for calculating median when instead of the lower limit of the median class we take its upper limit. The formula is as follows:
M= l- (l2-l1)/f1 (m-c)
This formula should be used only when the class-intervals are in descending order and the frequencies have been cumulated from top to bottom.