median() — The middle value

median(x)

Key idea

• The median is the middle value once the data is sorted from smallest to largest. (e.g. median of 1,3,100 = 3)
• With an even count of numbers, average the two middle values. (e.g. median of 2,4,6,8 = (4+6)/2 = 5)
• It is robust: extreme outliers (far-off values) barely move the median. (e.g. median of 1,3,100 = 3)
• Half the data falls below the median and half above it.
• Use the median for skewed (lop-sided) data like incomes or house prices.

How to find the median in three steps

1. Line your numbers up in order, from smallest to largest.
2. If you have an ODD number of values, the median is the one in the middle. With n numbers, the middle one sits at position (n + 1) / 2 — e.g. 5 numbers → position 3.
3. If you have an EVEN number of values, add the two middle numbers and divide by 2. The midpoint (n + 1) / 2 falls between them — e.g. 4 numbers → position 2.5, so average the 2nd and 3rd.

Definition

$$\tilde{x}=\{\begin{array}{ll}{x}_{(\frac{n+1}{2})}& n\text{ odd}\\ \frac{x_{(n/2)}+x_{(n/2+1)}}{2}& n\text{ even}\end{array}$$

$x_{(k)}$

the k-th value once sorted

$n$

how many values

$\tilde{x}$

the median (middle value)

Where you'd use it

• Five houses on a street sell for $200k, $220k, $240k, $260k and one mansion at $2M. The mean is dragged up past $580k, but median picks the middle value, $240k, which describes a typical home far better.
• A company pays $30k, $35k, $40k, $45k and a boss on $500k. The median salary is $40k, ignoring the one huge pay that would inflate the average.
• Clinic waits are 10, 12, 15, 18 minutes with one rare 90-minute delay. The median wait stays a realistic ~15 minutes, unmoved by that single long wait.

Frequently asked questions

What if there’s an even number of values?

The median is the average of the two middle values, so it may not be an actual data point.