Statistics

Median: position, not average

Sort the data. For $n$ values: odd $n$ → the $\left(\frac{n+1}{2}\right)$-th value; even $n$ → the mean of the $\frac{n}{2}$-th and $\left(\frac{n}{2}+1\right)$-th values. The median is a POSITION — no arithmetic on the values themselves happens except the final halving. Unsorted data is the single biggest source of error; sort first, always.

Mode: the most frequent value

The mode is the value appearing most often. A list may have one mode, several (bimodal), or none (all values equally frequent). In small exam datasets the mode is found by counting — 30 seconds, zero formulas.

The 3M empirical relation

$\text{mode} = 3\,\text{median} - 2\,\text{mean} \quad\Leftrightarrow\quad \text{median} = \frac{\text{mode} + 2\,\text{mean}}{3} \quad\Leftrightarrow\quad \text{mean} = \frac{3\,\text{median} - \text{mode}}{2}$ Treated as exact in competitive exams (it is only approximate in real statistics). Given any two of the three, the third is a substitution. Expect mean/median/mode triplets like (27, 33, 45) that satisfy the relation exactly.

Additive shifts and scales

Add $k$ to every value: mean, median and mode each shift by $k$ (order is preserved, frequency peaks unchanged). Multiply every value by $a$: all three multiply by $a$. These transformation rules appear as "the mean and median of the new data are..." — answer both in one stroke, and remember the empirical relation is preserved because all three scale identically.

Median vs mean: which moves?

The median is robust: changing one extreme value barely moves it (only its position matters). The mean absorbs every change fully. A question where "one value 100 is replaced by 1" usually asks which measure changes more — the mean, by $\frac{99}{n}$; the median may not move at all.

Worked example: adding values around the median

Data 3, 7, 9, 12, 15 has median 9. Add 1 and 20 → 1, 3, 7, 9, 12, 15, 20: the median stays 9 because one value landed on each side. Add 10 and 20 instead → 3, 7, 9, 10, 12, 15, 20: the median slides to 10. Before recomputing, count how many new values fall on each side of the old median — equal counts mean no change.

Even-count trap

For 4, 8, 10, 14, 18, 22 the median is $\frac{10 + 14}{2} = 12$ — a value that does not appear in the data. Options often include 10 and 14 to catch anyone who picks a single middle value.

Quick revision

• Sort first. Odd $n$: middle term. Even $n$: average of the two middles.
• Mode = most frequent; may be multiple or none.
• $\text{mode} = 3\,\text{median} - 2\,\text{mean}$ (any two give the third).
• $+k$ shifts all three; $\times a$ scales all three.
• Median resists outliers; the mean does not.