MATH

Median

Noun. (Statistics.)

The median of a list of 

Example 1: The median of the numbers  ,  ,  ,  ,  is . To see why this is so, arrange the numbers in order:  ,  ,  ,  , . Then choose the middle number in the list, which is .

Example 2: The median of the numbers  ,  ,  ,  ,  ,  is . To see why this is so, arrange the numbers in order:  ,  ,  ,  ,  , . Then choose the "middle" number, which is defined to be the mean of the values in the middle of the list, that is, the mean of  and .

Definition:

Given a list of  numbers (not necessarily distinct), label the numbers  so that .

1. If  is odd, then the median of the list is the middle value in the ordered list, namely,

.

(This is to say that the median is the number in position  in the list.)

2. If  is even, then the median of the list is the mean of the two middle values in the ordered list, namely,

.

(This is to say that the median is the mean of the numbers in the positions just above and just below position  in the list.)

Remarks:

· The median is the 50th percentile.

· Because the middle number of an ordered set of numbers (i.e., the median) will be the same whether the numbers are arranged in increasing or decreasing order, we may simply say "ordered" in the definition above without stating increasing or decreasing. However, in practice, it is more common to arrange the numbers in increasing order, as in calculating percentiles.

· The median is a measure of center of a set of numbers, like the mean and the mode. Unlike the mean, though, the median is resistant to outliers. This means that, in a sense, extreme values tend not to affect the value of median. For instance, the list of numbers  ,  ,  ,  ,  is just like the list of numbers in Example 1 above, except that the extreme value  in Example 1 has been replaced by the (much more extreme) value  here. The median is the same in both cases, but the means are very different ( in Example 1 and  here.)