Formula for median

CFA Level I
#1

The CFA curriculum says the following:

The median is the value of the middle item of a set of items that has been sorted into ascending or descending order. In an odd-numbered sample of n items, the median occupies the (n + 1)/2 position. In an even-numbered sample, we define the median as the mean of the values of items occupying the n/2 and (n + 2)/2 positions (the two middle items).

(Institute 220)

Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.

Call me crazy, but does anyone else see the equivalence between (n + 1)/2 and the mean of n/2 and (n + 2)/2?

Let’s do the math real quick:

Mean of n/2 and (n + 2)/2 = [n/2 + (n + 2)/2]/2 = [n/2 + n/2 + 2/2]/2 = [n + 2/2]/2 = (n + 1)/2

I guess I understand why they have “defined” the median the way they have, but it seems odd that they don’t highlight that the formula/procedure for finding the median is the same for a set w/ an even number of items as it is for one w/ an odd number of items.

Am I missing something?

#2

Your calculation is absolutely right and (n+1)/2 is eligible for both odd and even number of observations. However, whenever you have even number of observations it will end with 0.5 as it won’t be completely divisible by 2.

So, anyway you have to find the value for 0.5 (which would be average between the two observations). Such as in the following example.

0, 2 ,2, 4, 11, 17, 60, 100, 200, 4000

Now if I know the direct formula for even obervations i will take 5th and 6th observation and average (11+17)/2=14

Going by (n+1)/2 rule it is 5.5th observation. 5th observation is 11 and 0.5 of difference between 5th and 6th observation which is 3 (0.5x6). Median= 11+3 =14

It is just easier to rember the direct formula and do the average of observations right away.