Median (middle value)

The median is the value that lies right in the middle of a sorted list.

Idea

To find the median:

1. Sort the list from smallest to largest (or largest to smallest).
2. If the count $n$ is odd, the median is the value at position $\tfrac{n+1}{2}$.
3. If $n$ is even, the median is the mean of the two middle values at positions $\tfrac{n}{2}$ and $\tfrac{n}{2}+1$.

Example: $5, 8, 3, 7, 6$. Sort: $3, 5, 6, 7, 8$. Odd count ($5$), middle is the $3$rd value: median $= 6$.

Example: $10, 6, 8, 4$. Sort: $4, 6, 8, 10$. Even count ($4$), middle two are $6$ and $8$. Median $= \tfrac{6+8}{2} = 7$.

Why median? The median is the "$50\%$ value" , half the data is below it, half is above. Unlike the mean, it is not pulled by outliers. Adding a value of $100$ to $1, 2, 3, 4$ changes the mean drastically but the median only slightly.

Median is good for income data, house prices, anything skewed.

Worked examples

Example 1. Find the median of $4, 9, 6, 1, 5$.

Sort: $1, 4, 5, 6, 9$. Middle value (3rd): $5$.

Example 2. Find the median of $10, 15, 20, 25, 30, 35$.

Sort (already): $10, 15, 20, 25, 30, 35$. Middle two: $20, 25$. Median $= 22.5$.

Example 3. Five exam scores: $80, 90, 70, 60, 85$. Median?

Sort: $60, 70, 80, 85, 90$. Median: $80$.

Example 4. Add the value $1000$ to the list in Example 3. How do the mean and median change?

Before: mean was $\tfrac{60+70+80+85+90}{5} = 77$, median was $80$. After adding $1000$ (six values): mean $= \tfrac{385+1000}{6} \approx 230.83$. Median: sort $60, 70, 80, 85, 90, 1000$; middle two are $80, 85$; median $= 82.5$. Mean shot up; median changed only a little.

Try it yourself

1. Median of $7, 3, 9, 1, 5$.
2. Median of $8, 4, 2, 6, 10, 12$.
3. Find median of $11, 13, 14, 17, 20, 25$.
4. Marks: $35, 50, 60, 75, 100, 100$. Median?
5. Median of $1$ to $9$.
6. List $2, 4, 6, 8, 10, 12, 14$ , median?
7. Median vs mean for $1, 2, 3, 4, 5, 100$.
8. List $3, 5, 5, 8, 9, 11$. Median?

Activity

Collect heights (in cm) of $10$ classmates. Sort the list. Find the median. Compare with the mean.