Evenly spaced values
Values: 4, 6, 8, 10, 12; mean = 8 and squared deviations total 40.
Population: variance 40 / 5 = 8, SD = 2.828. Sample: variance 40 / 4 = 10, SD = 3.162.
Free statistics calculator
Standard Deviation Calculator: Sample & Population
Paste a list of values to calculate spread around the mean. Choose a sample or a full population and see the variance, formula, and intermediate statistics.
Example result: For 4, 6, 8, 10, and 12, the population standard deviation is 2.828 and the sample standard deviation is 3.162.
Calculate standard deviation
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Population standard deviation
2.828
Example result
The five values average 8. Their population variance is 8, so the population standard deviation is the square root of 8. The sample result uses one fewer value in its denominator.
4681012
Values (n)
5
Mean
8.000
Selected variance
8.000
Range
8.000
Formula and calculation steps
Population variance = 40 / 5 = 8
Population standard deviation = square root of 8 = 2.828How this result was calculated
- Find the mean: (4 + 6 + 8 + 10 + 12) / 5 = 8.
- Square each distance from 8. The squared deviations total 40.
- Divide 40 by the population count, 5, to get variance 8.
- Take the square root of 8 and display 2.828.
For the same five values, sample variance is 10 and sample standard deviation is 3.162 because the sample formula divides by n - 1.
The selected mode controls the large result. The statistics below also show the count, mean, matching variance, and range so you can check the calculation.
How to use this standard deviation calculator
Choose the formula that matches your data, then review both the result and the values used to produce it.
1
Enter the observations
Paste numeric values from a worksheet or type them with commas, spaces, semicolons, or one value per line. Do not include labels, units, or a spreadsheet header in the list.
2
Choose the data set
Select population when your data includes every member of the group you are describing. Select sample when the observations are a subset used to estimate a larger group.
3
Calculate and inspect
Submit the list to see the selected standard deviation, mean, variance, count, and range. The formula steps show the denominator used for the selected mode.
4
Keep useful precision
Choose a display precision that fits your work. Rounding changes only the shown digits; calculations use the entered numeric values before formatting.
What does standard deviation measure?
Standard deviation describes how far values tend to fall from their arithmetic mean. Square each difference, average those squares with the population or sample denominator, then take the square root. The result uses the original units.
A small value indicates observations cluster near the mean; a larger value indicates more spread. There is no universal cutoff: a spread of 3 may be minor in miles but large on a scale measured in tenths.
Variance is the mean squared distance from the mean. Standard deviation is its square root, so it is easier to compare with the original measurements.
The calculator describes only the entered values. It cannot show whether a sample is representative, observations are independent, or a distribution model fits; those depend on data collection and your question.
Standard deviation is not the full story
A single spread value can hide skew, clusters, trends, or outliers. Inspect the observations or a chart when distribution shape matters.
Population vs. sample standard deviation
The population formula divides squared deviations by N. Use it when your list is the complete group you want to describe, such as all scores in a class.
The sample formula divides by n - 1, Bessel's correction. It is commonly used to estimate a larger population's variance under random-sampling assumptions; it cannot fix selection bias or make an unrepresentative sample reliable.
Use the population formula to describe the exact list. For an estimate of a wider group from a subset, sample standard deviation is the conventional choice; report which denominator you used.
| Case | Variance denominator | When to use it | What it describes |
|---|---|---|---|
| Population | N | The entered values are the complete group being summarized. | Spread within those exact observations. |
| Sample | n - 1 | The entered values are a subset used to estimate a wider population. | An adjusted estimate based on the sample. |
| One observation | Population only | A complete group contains one value. | Population spread is zero; sample spread cannot be estimated. |
| Repeated identical values | N or n - 1 | Every observation has the same value. | Both valid results are zero when the sample has at least two values. |
Worked standard deviation examples
These examples show the formulas and the effect of unusual values.
Identical readings
Four readings are all 7.5, equal to the mean.
Both formulas give zero spread (the sample requires at least two readings). Instrument uncertainty is not captured by identical observations.
A list with an outlier
Values 10, 11, 12, 13, 40 include one distant observation.
The outlier raises the mean and spread. Check the values or a plot before treating that spread as typical.
How to interpret a standard deviation result
Read the result with its units, mean, and distribution.
Compare with the same scale
Compare standard deviations only for variables with compatible units and collection methods; raw values from different scales are not directly rankable.
Look at the mean and the data
Extreme values can shift the mean. Lists with identical mean and spread may still have different shapes; inspect a plot, median, and range when clusters or outliers matter.
Do not confuse spread with error
Standard deviation describes variation among observations, not the standard error, confidence interval, measurement tolerance, or forecast uncertainty.
Consider the distribution
Within-one-or-two-standard-deviation rules rely on a distribution assumption, often approximate normality. Do not apply them automatically to skewed, bounded, or multimodal data.
Input format, assumptions, and limits
Enter plain numeric observations; each value receives equal weight.
Separators
Separate values with commas, spaces, semicolons, tabs, or line breaks. Use a period for decimals; decimal-comma notation is not supported.
Numbers only
Remove headers, labels, currency symbols, percent signs, and notes. Negative and decimal values are accepted; text produces an error rather than being skipped.
Sample minimum
A sample result needs at least two observations for its n - 1 denominator. A one-value population has zero spread.
Finite values
Values must be finite and no larger than 1,000,000,000,000 in absolute value. Use statistical software for very large scientific datasets.
Maximum list size
The browser accepts up to 1,000 values. Use a spreadsheet or statistics package for larger files, missing-data rules, filters, weights, or grouped observations.
Unweighted observations
Every observation has equal weight. Use a weighted method when values represent different group sizes or sampling weights.
Standard deviation calculator FAQ
Formula, data-entry, and interpretation questions.
Use population for a complete group and sample for a subset used to estimate a wider population. The sampling design still matters.
Variance is the mean squared deviation. Standard deviation is its square root and uses the original measurement units.
No. It is zero or positive; zero means every entered value is identical.
Yes. Paste a single row or column. Line breaks, commas, spaces, and semicolons separate values; remove headers and text cells.
No. Standard deviation describes variation among observations. Standard error describes how an estimate varies across repeated samples under a model.
Ready to measure the spread?
Enter your observations, choose the population or sample formula, and review the variance and calculation steps with the result.