Not just "what's typical": how spread out is it?

Two datasets can share the exact same mean and median and still look completely different: one tightly bunched, one scattered wide. This page is about measuring that spread, and why the measure you pick matters just as much here as it did for mean vs. median.

Range, variance, and standard deviation

The range is the simplest possible answer, just the biggest value minus the smallest. It's also the most fragile: it's computed from exactly two points and ignores everyone else entirely.

Variance uses every point instead: take each point's distance from the mean, square it (so distance in either direction counts the same, and big misses count far more), then average all of those squares. Its square root, the standard deviation, undoes the squaring so the result is back in the data's original units, "how far from the mean, typically."

11 points below. Drag any of them and watch how differently the range and the standard deviation react.

Computing variance, step by step

1
Find the mean

The reference point every deviation is measured from.
2
Subtract it from every value

xᵢ − x̄: each point's deviation. These always sum to exactly zero, which is exactly why the next step is needed.
3
Square each deviation, then average

Squaring makes every term positive (so they don't cancel out) and penalizes large deviations disproportionately. That average is the variance.
4
Take the square root

Variance is in squared units (points²), which nothing in the real world is measured in. The square root (the SD) converts back to the original units.

Drag any point along the line; it snaps to the nearest whole number.

a point deviation line, point above the mean deviation line, point below the mean

Each line's length is that point's |deviation| from the mean. Square every length, average them all: that average is the variance.

Drag a point to see how each measure reacts.

Mean: —
Range (max − min): —
Variance: —
Standard deviation: —

Numbers = [ ]

Variance = Σ(x − x̄)² / n

= =

SD = √Variance

= √ =

Try it: send a point far out, then compare how far each metric moved. The range changes by exactly the size of the jump; it's blind to everything else. The standard deviation moves too, but by much less proportionally, because it's an average spread across every point, not just the two extremes.

The box plot and the IQR: a robust alternative

Just like the median ignores how extreme the extremes are, the interquartile range (IQR) measures spread using only the middle 50% of the data, the gap between the first quartile (Q1, the 25th percentile) and the third (Q3, the 75th percentile). Points beyond that middle chunk simply don't affect it.

A box plot draws this directly: a box from Q1 to Q3 with a line at the median, whiskers reaching out to the most extreme point that's still within 1.5×IQR of the box, and any point beyond that plotted separately as a flagged outlier.

Pick a preset and read the spread straight off the shape.

Reading a box plot

1
The box: Q1 to Q3

Contains the middle 50% of the data. Its width is the IQR.
2
The line inside: the median

Off-center inside the box is a visible sign of skew, the same way it was on the histogram presets on the central-tendency page.
3
The whiskers

Extend to the furthest point still within 1.5×IQR of the box, not necessarily all the way to the true min/max.
4
Dots beyond the whiskers: flagged outliers

Anything past 1.5×IQR from the box gets plotted on its own: exactly the points that would otherwise inflate the range and SD the most.

Pick a spread and read Q1, median, Q3, and any outliers straight off the plot.

a raw data point Q1–Q3 box flagged outlier

—

Q1, median, Q3: —
IQR: —
Range: —
Standard deviation: —

The jargons

Every measure below answers the same question — how spread out is the data — but they disagree about how much a single extreme value should be allowed to sway the answer.

Range

Maximum minus minimum. The simplest measure of spread, and the one most easily distorted by a single extreme value.

Variance

The average of every point's squared deviation from the mean. In squared units, which is exactly why the standard deviation usually gets reported instead.

Standard deviation SD

The square root of variance, back in the data's original units. Roughly, "how far from the mean a typical point sits."

Quartile Q1, Q3

Q1 is the median of the lower half of the data, Q3 the median of the upper half: the 25th and 75th percentiles.

IQR interquartile range

Q3 − Q1: the spread of just the middle 50% of the data. Unaffected by anything happening in the tails: the range's robust counterpart.

1.5×IQR rule

The standard convention for flagging outliers on a box plot: anything more than 1.5×IQR beyond Q1 or Q3 gets plotted separately instead of extending the whisker.