What is standard deviation?
Standard deviation measures how spread out numerical values are around their mean. A small standard deviation means the observations tend to cluster relatively close to the mean; a larger standard deviation means they are more dispersed. Unlike variance, standard deviation is expressed in the same units as the original data, which often makes it easier to interpret.
OpenStax defines standard deviation as a measure of overall variation in a data set and distinguishes two formulas depending on whether the observations represent an entire population or a sample. For a population, squared deviations are averaged using N. For a sample used to estimate population variability, the familiar sample formula divides by n − 1. The calculator keeps that choice explicit instead of silently assuming one version.
That sample-versus-population distinction is one of the most common sources of disagreement between calculators, spreadsheets and hand calculations. Two people can enter the same values and receive slightly different answers simply because one used the population formula while the other used the sample formula. Our result identifies which denominator was used and exposes the sum of squared deviations so the arithmetic can be checked.
How to use the Standard Deviation Calculator
- Enter the data values. Paste or type numbers separated by commas, spaces, semicolons or new lines.
- Choose Sample or Population. Use Sample when the listed observations are a subset used to learn about a larger population. Use Population when the listed values are the complete group you intend to describe.
- Calculate. The blue card shows the standard deviation, which is the primary answer.
- Review the supporting statistics. Count, mean, median, variance and range help you understand the data before interpreting spread.
- Inspect the calculation table. Each value is shown with its deviation from the mean and squared deviation. The calculator then totals those squared deviations before applying the selected denominator and square root.
Sample and population standard deviation formulas
The population formula is:
The sample formula is:
OpenStax presents these same formulas and notation: lowercase s for sample standard deviation and Greek sigma σ for population standard deviation. In both cases, the core idea is identical: measure each observation's deviation from the mean, square those deviations so positive and negative values do not cancel, average the squared deviations using the appropriate denominator, and then take the square root.
Why are deviations squared?
If you simply added deviations from the mean, positive and negative differences would cancel. Squaring makes every contribution nonnegative and gives more weight to observations farther from the mean. The sum of squared deviations therefore captures dispersion without losing it to cancellation.
The square root at the end returns the measure to the original units. If the data are measured in centimeters, variance is in square centimeters, while standard deviation returns to centimeters.
Why does sample standard deviation divide by n − 1?
When the sample mean x̄ is estimated from the same sample, the deviations are constrained to sum to zero. Once n − 1 deviations are known, the final deviation is determined, leaving n − 1 degrees of freedom. Dividing the sum of squared deviations by n − 1 produces the usual unbiased estimator of the population variance under standard random-sampling assumptions.
This adjustment is often called Bessel's correction. It does not mean the population formula is wrong; the formulas answer different questions. If your data list is literally the complete population of interest, divide by N. If it is a sample used to estimate variability in a larger population, the conventional sample variance uses n − 1.
NIST's statistical handbook likewise presents the sample variance estimate as Σ(Xᵢ − X̄)²/(n − 1) and takes its square root to estimate population standard deviation. This is why statistical software often labels the two functions differently rather than offering one universal “standard deviation” command.
Worked standard deviation examples
Data: 2, 4, 4, 4, 5, 5, 7, 9. Mean = 5. Sum of squared deviations = 32. Population variance = 32/8 = 4, so population standard deviation = √4 = 2.
Data: 10, 12, 14. Mean = 12. Squared deviations are 4, 0 and 4, totaling 8. Sample variance = 8/(3 − 1) = 4, so sample standard deviation = 2.
Data: 6, 6, 6, 6. Every deviation from the mean is 0. Both population and sample standard deviation are 0 because every observation is identical.
Same data, different denominator
Take 3, 5 and 7. The mean is 5 and the squared deviations are 4, 0 and 4, for a total of 8. Treating those three values as the entire population gives variance 8/3 ≈ 2.6667 and standard deviation ≈ 1.633. Treating them as a sample gives variance 8/2 = 4 and standard deviation 2. The raw observations did not change—the statistical role of the observations did.
How to interpret standard deviation
Standard deviation has no universal “high” or “low” threshold. Its interpretation depends on the units, scale and subject matter. A standard deviation of 5 might be tiny for annual income measured in thousands, large for a tightly controlled manufacturing dimension, or ordinary for an exam scored from 0 to 100.
Comparisons are most straightforward when two data sets use the same units and have similar scales. OpenStax gives the intuition that, when two groups have the same mean, the group with the larger standard deviation shows greater variation. If the means or measurement scales differ substantially, a relative measure such as the coefficient of variation may sometimes be more informative.
Standard deviation keeps the original unit
This is an important practical advantage over variance. If delivery times are measured in minutes, their standard deviation is also measured in minutes. You can say that observations typically vary around the mean on a scale of several minutes, whereas variance would be expressed in minutes squared.
Standard deviation vs variance, range and mean absolute deviation
| Measure | What it describes | Main limitation / feature |
|---|---|---|
| Standard deviation | Spread around the mean | Same units as data; sensitive to extremes |
| Variance | Average squared spread around mean | Squared units |
| Range | Maximum minus minimum | Uses only two observations |
| Mean absolute deviation | Average absolute distance from a center | Different weighting from squared deviations |
| IQR | Spread of middle 50% | More resistant to extreme values |
Variance and standard deviation contain the same underlying squared-deviation information; one is simply the square of the other. Range is much easier to calculate but ignores every value except the minimum and maximum. The interquartile range is often useful when the data are skewed or contain extreme observations because it focuses on the middle half of the data.
Standard deviation and the 68–95–99.7 rule
If a distribution is approximately normal, standard deviation has a familiar probability interpretation: about 68% of observations fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. This empirical rule is a property of the normal distribution, not a definition of standard deviation.
Do not apply those percentages automatically to strongly skewed, multimodal or heavy-tailed data. Standard deviation can still be calculated for such data, but the “within one or two standard deviations” proportions may differ substantially from the normal benchmark.
Standard deviation and z-scores
A z-score uses standard deviation to express an individual observation's position relative to the mean: z = (x − mean)/standard deviation. A z-score of 1.5 means the observation is 1.5 standard deviations above the mean; −2 means two standard deviations below. Standard deviation therefore describes the spread of the entire data set, while a z-score describes one observation's standardized position within that spread.
Rounding and numerical accuracy
For hand calculations, keep extra decimal places through the mean, deviations and squared deviations, then round the final result. NIST warns that excessive intermediate rounding in variance calculations can cause significant loss of precision because subtraction is involved. The calculator therefore performs its arithmetic using JavaScript numeric precision and rounds only for display.
For extremely large values combined with extremely tiny differences, or for specialist high-precision scientific work, dedicated statistical software using numerically stable algorithms may be preferable. This browser calculator is designed for ordinary educational and practical data sets rather than arbitrary-precision computation.
Limitations, outliers and robust measures of spread
Standard deviation is sensitive to extreme observations because deviations are squared. One very distant value can substantially increase both the mean and the standard deviation. That sensitivity is not necessarily a flaw—sometimes an extreme value is real and should influence the description—but it means standard deviation should not be interpreted without looking at the data distribution.
For skewed data or data with influential outliers, the median and interquartile range can provide a more resistant summary of center and spread. This calculator includes the median alongside standard deviation for that reason. If mean and median are far apart, or if a few values sit far beyond the rest, inspect the data rather than relying on one summary statistic.
Can standard deviation be negative?
No. Variance is built from squared deviations and is therefore nonnegative, and standard deviation is the nonnegative square root of variance. The smallest possible standard deviation is 0, which occurs when all observations are equal.
Common standard deviation mistakes
- Using the wrong denominator. Population variance divides by N; conventional sample variance divides by n − 1.
- Forgetting to square deviations. Raw deviations around the mean cancel to zero.
- Stopping at variance. Standard deviation is the square root of variance.
- Mixing units. Values in one data set should represent the same variable on a compatible scale.
- Assuming a larger number is automatically bad. “Large” or “small” depends on units and context.
- Applying the 68–95–99.7 rule to every data set. Those percentages rely on an approximately normal distribution.
- Ignoring outliers or skewness. Standard deviation can be strongly affected by extreme observations.
- Rounding the mean too early. Premature rounding can change squared deviations and the final result.
How to calculate standard deviation by hand
For a data set such as 4, 7, 9, first find the mean: (4 + 7 + 9)/3 = 20/3 ≈ 6.6667. Subtract that mean from every observation, square each deviation, and add the squared deviations. Then divide by N for a population or n − 1 for a sample. Finally, take the square root.
The table displayed by this calculator mirrors exactly those core steps. It is especially useful for checking homework because you can see whether an error came from the mean, a deviation, the squaring step, the denominator or the final square root.
Frequently asked questions
What does standard deviation tell you?
It describes the spread of data around the mean in the same units as the original observations. Larger values indicate more dispersion, while 0 means all observations are identical.
Should I choose sample or population?
Choose Population if your list is the complete group you intend to describe. Choose Sample when the values are a subset used to estimate or describe variability in a larger population.
Why does sample standard deviation use n − 1?
Estimating the sample mean uses one degree of freedom. Dividing by n − 1 is the conventional correction used when estimating population variance from sample data.
Can standard deviation equal zero?
Yes. It equals 0 when every value is the same because every deviation from the mean is zero.
Can standard deviation be negative?
No. Variance is nonnegative and standard deviation is its nonnegative square root.
What is the difference between variance and standard deviation?
Variance is the average squared spread around the mean; standard deviation is the square root of variance and returns to the original data units.
Does a high standard deviation mean the data are inaccurate?
No. Standard deviation describes variability, not accuracy. Whether the observed spread is acceptable depends on the application.
Does standard deviation require normally distributed data?
No. Standard deviation can be calculated for non-normal data. However, normal-distribution rules such as 68–95–99.7 should not be assumed unless a normal model is appropriate.