Understanding your standard deviation
For approximately normal (bell-shaped) data, the empirical rule describes how much of the data falls within a given number of standard deviations of the mean.
| Interval around the mean | Approximate share of data (normal distribution) |
|---|---|
| Within 1 standard deviation | About 68% |
| Within 2 standard deviations | About 95% |
| Within 3 standard deviations | About 99.7% |
- The 68-95-99.7 empirical rule applies to approximately normal distributions; heavily skewed or heavy-tailed data can deviate substantially from these percentages.
- Use the sample form (n - 1) when the data are a sample used to infer about a larger population; use the population form (n) only when the data set is the entire population of interest.
- Standard deviation is sensitive to outliers because deviations are squared; a single extreme value can inflate it markedly.
- This calculator requires at least two values, since spread cannot be estimated from a single observation.
What is standard deviation?
Standard deviation quantifies the typical distance of data values from their mean. A small standard deviation means the values cluster tightly around the mean; a large one means they are widely scattered. It is expressed in the same units as the data, which makes it easier to interpret than the variance (its square).
The core teaching point is the distinction between the sample and population forms. The population standard deviation (sigma) divides the sum of squared deviations by n, and is correct only when the data set is the entire population of interest — for example, the exam scores of every student in one specific class. The sample standard deviation (s) divides by n - 1 instead, and is correct when the data are a sample used to estimate the spread of a larger population.
Dividing by n - 1 is known as Bessel's correction. Because the sample mean is calculated from the same data, the deviations around it are systematically slightly smaller than the deviations around the true (unknown) population mean; dividing by n - 1 rather than n compensates for this and makes the sample variance an unbiased estimator of the population variance. The correction matters most for small samples and becomes negligible as n grows large.
How to use this standard deviation calculator
- Enter at least two numbers separated by commas, for example: 10, 12, 23, 23, 16, 23, 21, 16.
- Read the sample standard deviation (s) if your data are a sample from a larger population — the usual case in surveys, experiments and quality checks.
- Read the population standard deviation (sigma) if your data are the complete population of interest.
- Use the mean, sample variance and count to check the calculation or carry it into further analysis such as confidence intervals and z-scores.
Standard deviation formulas
Both forms start the same way: compute the mean, subtract it from each value, square each deviation, and sum the squares. The population form divides that sum by n and takes the square root; the sample form divides by n - 1 and takes the square root.
Worked example: the values 2, 4, 4, 4, 5, 5, 7, 9. Step 1 — mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5. Step 2 — squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Step 3 (population) — 32 / 8 = 4, and sqrt(4) = 2. Step 4 (sample) — 32 / 7 = 4.571429 (the sample variance), and sqrt(4.571429) = 2.13809 (rounded).
The sample value is always slightly larger than the population value for the same data, because it divides by a smaller number. The gap shrinks as the sample grows.
Common mistakes
- Using the population formula (dividing by n) on sample data — this systematically understates the spread of the population the sample came from; divide by n - 1 for samples.
- Confusing variance with standard deviation: variance is the square of the standard deviation and is expressed in squared units.
- Applying the 68-95-99.7 rule to data that are clearly skewed or heavy-tailed, where it does not hold.
- Comparing standard deviations of data sets measured in different units or on very different scales without standardizing (for relative spread, the coefficient of variation divides by the mean).
Questions fréquentes
What is the difference between sample and population standard deviation?
The population standard deviation divides the sum of squared deviations by n and is correct when your data are the entire population. The sample standard deviation divides by n - 1 (Bessel's correction) and is correct when your data are a sample used to estimate the spread of a larger population. For the same data the sample value is always slightly larger; the difference fades as the sample size grows.
Why divide by n - 1 instead of n?
Because the sample mean is computed from the same data, the observed deviations around it are systematically a little smaller than the deviations around the true population mean. Dividing by n - 1 instead of n compensates for this bias, making the sample variance an unbiased estimator of the population variance. This adjustment is called Bessel's correction.
How do I calculate standard deviation by hand?
First find the mean. Then subtract the mean from each value, square each result, and add the squares. For a sample, divide this sum by n - 1; for a population, divide by n. Finally take the square root. Example: for 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squared deviations sum to 32, and the sample standard deviation is sqrt(32 / 7) = 2.138 (rounded).
What does a standard deviation of 0 mean?
It means every value in the data set is identical — there is no spread at all. The larger the standard deviation, the more the values differ from the mean on a typical basis. Standard deviation can never be negative.
What is the 68-95-99.7 rule?
For approximately normal (bell-shaped) data, about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. The rule is a quick way to judge whether an observation is unusual, but it is only reliable when the distribution is roughly normal.
Is variance the same as standard deviation?
No. Variance is the average of the squared deviations from the mean, and standard deviation is its square root. Variance is expressed in squared units (for example, cm^2 for data measured in cm), which is why the standard deviation — in the original units — is usually quoted when describing spread.
Références
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.6: Measures of Scale. nist.gov.
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. W. H. Freeman (sample vs population standard deviation).
- Weisstein, Eric W. "Standard Deviation" and "Bessel's Correction." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.