Standard Deviation

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

Definition

Population standard deviation is

while sample standard deviation is

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.

Population or sample

Use $\sigma$ and denominator $N$ when the data include the entire population of interest. Use $s$ and $n-1$ when a sample estimates population variability.

Calculator menus often report both $\sigma_x$ and $s_x$. Choosing the wrong one changes the result, especially for small samples.

Z-scores

Standard deviation creates a common scale:

A z-score of $2$ means the value lies two standard deviations above the mean; $-1.5$ means $1.5$ standard deviations below.

Standardized scores allow comparison across distributions with different units or scales, provided the underlying comparison is meaningful.

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 $|a|$:

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

Outliers and shape

Standard deviation is sensitive to outliers because it is based on squared deviations from the mean. A single extreme observation can increase both mean and standard deviation.

For skewed or outlier-heavy data, pair the median with interquartile range. For roughly symmetric data without strong outliers, mean and standard deviation work well together.

Comparing consistency

If two groups measure the same quantity on the same scale, smaller standard deviation indicates more consistency around each group's mean. If means or units differ substantially, compare standardized measures or coefficient of variation with care.

A low standard deviation does not imply high accuracy; measurements can be consistently biased.

Reporting

Report the centre, spread, sample size, units, and enough context to identify the variable. Avoid more decimal places than the measurements justify.

Software output is not self-explanatory; state whether the statistic is sample or population standard deviation.

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.

Quick self-check

• Is population or sample standard deviation appropriate?
• Was variance square-rooted?
• Are units the same as the original variable?
• Does the magnitude match the visible spread?
• Is a normal-rule interpretation justified by shape?
• Should resistant measures or bias information also be reported?

Related topics

• Variance
• Z-Scores
• Normal Distribution
• Mean