Standard Deviation

Standard deviation is the square root of variance and describes typical distance from the mean in the data's original units.

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Centre and spread answer different questions about the same distribution. Two dot plots share the same dashed centre line but have visibly different horizontal widths. The wider lower plot shows why equal centres can coexist with unequal spread.

Standard deviation translates squared spread back into the same units as the observations.

Definition

Population standard deviation is

\sigma=\sqrt{\frac{\sum(x_i-\mu)^2}{N}},

while sample standard deviation is

s=\sqrt{\frac{\sum(x_i-\bar x)^2}{n-1}}.

It is always nonnegative and equals zero only when every value is identical.

Interpretation

Standard deviation measures how far observations typically lie from the mean, with squared deviations giving greater influence to distant values. A small standard deviation indicates clustering; a large one indicates greater spread.

It does not describe the direction of deviation and does not state that every value lies within one standard deviation.

Worked example

Unlike variance, the answer uses the original data units.

Normal-distribution interpretation

For approximately normal data, about $68\%$ lie within one standard deviation of the mean, $95\%$ within two, and $99.7\%$ within three.

These percentages rely on normal shape. Chebyshev-type guarantees are broader but weaker for arbitrary distributions.

Effect of transformations

Adding a constant to every observation leaves standard deviation unchanged. Multiplying every value by $a$ multiplies standard deviation by $|a|$:

\operatorname{SD}(aX+b)=|a|\operatorname{SD}(X).

Changing units from metres to centimetres therefore multiplies the standard deviation by $100$.

Common mistakes

Using variance as though it has original units. Take the square root for standard deviation.

Selecting $\sigma_x$ instead of $s_x$ without considering design. Match population or sample.

Applying the 68–95–99.7 rule to strongly non-normal data. Shape matters.

Calling small standard deviation accurate. It measures precision/spread, not bias from truth.

Ignoring outliers. They can dominate the measure.