Mean, median, and mode

Statistics begins with a simple question: if you had to describe a whole dataset with a single number, what would it be? There are three honest answers — and knowing when each one lies is the real skill.

The three measures of center

• Mean — the arithmetic average: add everything up and divide by how many.
• Median — the middle value when the data is sorted (or the average of the two middle ones).
• Mode — the value that appears most often.

The mean has a clean formula:

$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{1}{n}\sum_{i=1}^{n} x_i$

The Σ (sigma) just means "add them all up" — you'll see it everywhere in statistics.

When the mean and median disagree

On symmetric data the mean and median sit together. But add a long tail — a few very large values — and they split apart: the mean gets dragged toward the tail, while the median barely moves.

That's why you hear "median household income," not mean. A handful of billionaires pull the mean income far above what a typical household earns; the median stays honest.

Why this matters

The center is the first number anyone quotes about data — and the first place statistics gets manipulated. Knowing that a skewed dataset has a mean and median that disagree lets you ask the right question: "Is that an average, or a median?" The gap between them is itself a clue about the data's shape.