Exploring One-Variable Data

Standard Deviation

Calculating and interpreting standard deviation as a measure of variability.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Measures of Center & Spread

A Measure of Spread That Uses Every Value

The 10 noon temperatures from the last lesson had a mean of 74°F. Before reading on: if you computed how far each individual day's temperature sat from that mean, then simply added all those differences together, what do you think that sum would come out to?

Definition — Standard Deviation

The sample standard deviation, s = √[Σ(x−x̄)²/(n−1)], measures the typical distance of a data value from the mean. Deviations are squared before averaging because the raw deviations from a mean always sum to exactly 0 — squaring removes the cancellation between positive and negative deviations, so the result actually measures spread instead of vanishing to nothing.

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

Worked Example — Confirming Raw Deviations Sum to Zero

For the 10 temperatures (mean 74), the deviations are −6, −4, −3, −2, −1, 0, 1, 2, 4, 9. Adding them directly: −6−4−3−2−1+0+1+2+4+9 = 0 exactly — confirming that a raw sum of deviations from the mean is always 0, no matter what the data looks like, which is exactly why standard deviation squares the deviations instead of just averaging them directly.

Worked Example — Computing the Standard Deviation

Squaring each deviation: 36, 16, 9, 4, 1, 0, 1, 4, 16, 81. These sum to 168. Dividing by n−1 = 9 gives a variance of 168/9 ≈ 18.67, and s = √18.67 ≈ 4.32°F. Interpretation: the noon temperature typically differed from the mean (74°F) by about 4.32 degrees.

Tip

A larger standard deviation means the data is more spread out around its mean; a smaller one means it's more tightly clustered — comparing two datasets' standard deviations (measuring the same kind of thing, in the same units) is a direct way to compare their variability.

Common Mistakes

Reporting standard deviation without its units, or confusing it with variance.
Standard deviation is in the same units as the original data (here, °F) and describes a typical distance from the mean directly — variance (in squared units, °F²) is an intermediate step in computing it, not the final interpretable quantity.
Dividing by n instead of n−1 when computing a sample's standard deviation.
The n−1 denominator is standard for a sample standard deviation — dividing by n instead describes the population standard deviation, appropriate only when the data represents an entire population rather than a sample from it.

Key Takeaways

Standard deviation measures a typical distance from the mean, using every value in the dataset.
Deviations from the mean are squared before averaging specifically because raw deviations always sum to exactly 0.
A larger standard deviation signals more spread-out data; a smaller one signals tighter clustering around the mean.

Summary

This closes the exploration of a single variable's distribution. The next unit turns to relationships between two variables at once, starting with how to picture and measure them.

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