Variance and standard deviation: dispersion in 2 formulas

The formulas

For data $x_1, \ldots, x_n$ with mean $\bar{x}$:

Variance: $V = \dfrac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$

Standard deviation: $\sigma = \sqrt{V}$

In words

• Variance is the mean squared deviation from the mean.
• Standard deviation is the square root of variance.

Std deviation is easier to read: same units as the data.

Step-by-step

Data: 2, 4, 6, 8, 10.

1. Mean: 30/5 = 6.
2. Deviations: −4, −2, 0, 2, 4.
3. Squared: 16, 4, 0, 4, 16.
4. Variance: 40/5 = 8.
5. Std dev: $\sqrt{8} \approx 2.83$.

Interpretation

• Small std: data close to mean, low spread.
• Large std: data spread out.

Alternative formula (often faster)

$V = \dfrac{1}{n} \sum x_i^2 - \bar{x}^2$

“Mean of squares minus square of mean.”

Common mistakes

1. Confusing variance and std dev: variance is in units², std dev in units.
2. Forgetting the square root at the end.
3. Dividing by $n-1$ instead of $n$: that’s the “sample variance” (inferential statistics).

?À toi de jouer

Data: 1, 3, 5, 7, 9. Mean is 5. Variance?

A.8B.4C.10D.25

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