Median and its stability

If five friends report their pocket money , $\mathrm{Rs.}\,50, 60, 70, 80, 1000$ , what's the "typical" amount? The mean comes out to $\mathrm{Rs.}\,252$, which feels misleading. Almost everyone has under $\mathrm{Rs.}\,100$. The median, by contrast, gives $\mathrm{Rs.}\,70$. That single rich friend doesn't drag it.

Concept

The median is the middle value of a list of numbers after sorting them from smallest to largest. It splits the data into two halves with equal count.

Finding the median.

• Sort the data.
• If the count $n$ is odd, the median is the value at position $\dfrac{n+1}{2}$.
• If the count $n$ is even, the median is the average of the two middle values at positions $\dfrac{n}{2}$ and $\dfrac{n}{2}+1$.

Example (odd $n$). Sort: $4, 7, 9, 12, 15$. Middle (3rd) is $9$.

Example (even $n$). Sort: $4, 7, 9, 12, 15, 18$. Middle two are $9$ and $12$. Median $= \dfrac{9+12}{2} = 10.5$.

Mean vs median , when do they differ?

• For a symmetric, evenly spread dataset, they are nearly equal.
• For data with one or more extreme values (outliers), the mean is pulled towards the outlier while the median stays put.
• That makes the median the safer "typical" measure for skewed data like incomes, house prices, or test scores with a few very-low or very-high marks.

Stability of the median. Adding values has a surprisingly mild effect on the median.

• Adding a value equal to the median doesn't change the median (it just adds another copy to the middle).
• Adding a value larger than the current median can push the median up by at most one position.
• Adding a value smaller than the current median can push it down by at most one position.
• Removing a value works similarly , only the position of the middle shifts, not the values themselves much.

A useful fact. If the median of a list is $M$, then adding a value larger than $M$ and a value smaller than $M$ together leaves the median unchanged in most cases , the new pair lands one on each side, and the middle stays where it was.

Worked examples

Example 1. Median of $3, 9, 12, 15, 6, 11, 20$.

• Sort: $3, 6, 9, 11, 12, 15, 20$.
• $n = 7$ (odd). Middle position $4$. Median $= 11$.

Example 2. Median of $8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92$.

• Already sorted; $n = 16$ (even). Middle two are at positions $8, 9$: values $41$ and $41$. Median $= 41$.

Example 3. Compare the mean and median of $\{2, 3, 4, 5, 100\}$.

• Mean $= 114/5 = 22.8$. Median $= 4$.
• The median better represents the "typical" value here; the mean is yanked up by the outlier $100$.

Example 4. Consider data $12, 47, 8, 73, 18, 35, 39, 8, 29, 25, p$. For which values of $p$ is the median equal to $29$?

• Without $p$, the sorted ten values are $8, 8, 12, 18, 25, 29, 35, 39, 47, 73$.
• With $p$, count becomes $11$ (odd) , the median is the $6$th value.
• If $p \le 25$, the $6$th value is $25$. If $p \ge 29$, the $6$th value is $29$. If $25 < p < 29$, the $6$th is $p$.
• So the median is $29$ when $p \ge 29$. That covers $p \in \{29, 30, 35, 40, 47, 100, \dots\}$ but not $p = 10$ or $25$.

Try it yourself

1. Median of $5, 9, 1, 12, 7$.
2. Median of $14, 22, 18, 30, 25, 28$.
3. The mean of $7$ numbers is $20$. Their median is $14$. Are the numbers symmetric? Why or why not?
4. From the list $4, 6, 9, 12, 14, 17$ remove $4$. New median?
5. To $\{10, 20, 30\}$, what value should you add so the median becomes $25$?
6. True/false: "Including a value less than the median always decreases the median." Justify.
7. The median of a class's marks is $50$. A topper scoring $98$ joins. Does the median move much? Does the mean?
8. Find the median of the first $50$ natural numbers.

Activity

Median sticker line. Get $11$ classmates to each write any positive number on a small sticker. Stand in a line ordered by the number. The person in the middle is the median holder. Now invite a $12$th person. The median is now the average of the two people in the centre. Try adding a person with a huge number ($1000$): notice the median person barely changes, but if you computed the mean, it would jump dramatically.