The short answer
Standard deviation measures how spread out data are around the mean, in the same units as the original data. Compute it by finding each value's squared deviation from the mean, averaging those squared deviations (variance), then taking the square root. A tight cluster of values gives a small SD; widely scattered values give a large SD.
Key takeaways
- Sample SD divides by (n−1), not n — Bessel's correction — to avoid systematically underestimating the true population spread.
- Standard deviation shares the same units as the original data; variance is in squared units, which is why SD, not variance, is usually reported.
- The empirical rule (68-95-99.7) only applies to roughly bell-shaped (normal) data — it doesn't hold for skewed or multimodal distributions.
- Squaring deviations before averaging (rather than just averaging absolute deviations) penalizes larger deviations more heavily, which is why variance and SD became the standard measure of spread.
The standard deviation formula, step by step
| Step | Action |
|---|---|
| 1 | Find the mean of the data set |
| 2 | Subtract the mean from each value (the deviation) |
| 3 | Square each deviation |
| 4 | Average the squared deviations (divide by n or n−1) — this is the variance |
| 5 | Take the square root of the variance — this is the standard deviation |
Sample vs. population: the n−1 correction
For the data set {2, 4, 6, 8}, mean = 5, and the squared deviations are 9, 1, 1, 9 (sum = 20):
Population variance = 20 ÷ 4 = 5 → population SD = √5 ≈ 2.236
Sample variance = 20 ÷ 3 ≈ 6.667 → sample SD = √6.667 ≈ 2.582
The sample version is always equal to or larger than the population version for the same data — dividing by a smaller number (n−1 instead of n) inflates the estimate slightly, correcting for the fact that a sample tends to underestimate the true spread of the full population.
The empirical rule for normal data
| Range | % of data (normal distribution) |
|---|---|
| Mean ± 1 SD | 68% |
| Mean ± 2 SD | 95% |
| Mean ± 3 SD | 99.7% |
This rule is a quick sanity check for bell-shaped data, not a universal law — it breaks down for skewed, multimodal, or heavy-tailed distributions.
Common mistakes to avoid
- Forgetting the final square root step — variance and standard deviation are frequently confused, but they aren't interchangeable, since they're in different units.
- Using the population formula (÷n) when trying to estimate a population's spread from a sample — this understates true variability.
- Applying the empirical rule to data that isn't roughly bell-shaped — skewed data won't follow the 68-95-99.7 pattern.
- Computing deviations from the median instead of the mean — standard deviation is specifically defined around the mean.
Related calculators
- Average Calculator — compute the mean this calculator measures spread around.
- Mean, Median, Mode & Range Calculator — get a fuller statistical picture alongside SD.
- Confidence Interval Calculator — use SD to build a range estimate for the population mean.
- Z-Score Calculator — see how many standard deviations a value sits from the mean.