What is Median? — Mathematics

Definition 8.5 High School Mathematics · Chapter 8 — Descriptive Statistics

The median is a value that splits the sorted series in two halves: at least half the values are $\leq$ the median, and at least half are $\geq$ it. In practice, sort the $N$ values;

if $N$ is odd, the median is the middle value, in position $\frac{N+1}{2}$ ;
if $N$ is even, take the midpoint of the two middle values, in positions $\frac N2$ and $\frac N2 + 1$ .

Examples

Example 8.6

Sorted series of $7$ house prices (in thousands): $120$ , $150$ , $160$ , $180$ , $210$ , $240$ , $900$ . The median is the $4$ th value, $180$ . The mean is $\frac{1960}{7} = 280$ — larger than $6$ of the $7$ prices! One extreme value ( $900$ ) pulls the mean far more than the median: the median is robust, the mean is not.

Example 8.9

Take the $11$ sorted values

2,\ 3,\ 5,\ 5,\ 6,\ 7,\ 8,\ 9,\ 9,\ 10,\ 12 .

Median: position $\frac{11+1}{2} = 6$ , so the median is $7$ . $Q_1$ : $\frac{11}{4} = 2.75$ , round up to $3$ ; the $3$ rd value is $5$ . $Q_3$ : $\frac{33}{4} = 8.25$ , round up to $9$ ; the $9$ th value is $9$ . Range: $12 - 2 = 10$ ; interquartile range: $9 - 5 = 4$ .

Example 8.10 (Comparing two groups)

Two classes take the same test (scores out of $20$ ):

	median	$Q_1$	$Q_3$
class A	$12$	$10$	$14$
class B	$12$	$6$	$17$

Same median, very different spreads: class A is homogeneous (half its scores in $\intcc{10}{14}$ ), class B is heterogeneous (its central half spans $\intcc{6}{17}$ ). Indicators of center alone never tell the whole story.

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Definition 17.2 High School Mathematics · Chapter 17 — Descriptive Statistics

A median of a sorted dataset is a value that splits it in half: at least half the values are $\leq$ it, and at least half are $\geq$ it. In practice, sort the $n$ values; if $n$ is odd, the median is the middle value; if $n$ is even, take the midpoint of the two middle values.

Examples

Example 17.3

For the running dataset, the sum is $144$ , so $\bar x = \frac{144}{12} = 12$ . The two middle values ( $6$ th and $7$ th of the sorted list) are both $12$ , so the median is $12$ too. Mean and median need not coincide: the median ignores how far the extreme values are, the mean does not. Replacing the top mark $17$ by $20$ moves the mean to $12.25$ but leaves the median at $12$ .

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