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Compute the mean, variance, and both population and sample standard deviation from any list of numbers.
Example
For the data set 2, 4, 4, 4, 5, 5, 7, 9:
Count (n) = 8
Mean = 5
Population SD = 2
Population var = 4
Sample SD = 2.138090
Sample variance = 4.571429
Population SD uses Σ(x−mean)²/n = 32/8 = 4, so σ = 2. Sample SD uses 32/(8−1) = 4.5714, so s ≈ 2.1381.
How it works
Enter your values separated by commas, spaces, or new lines. The calculator parses every finite number and instantly reports the count, mean, variance, and standard deviation for both population and sample.
Good to know
This Standard Deviation Calculator turns a raw list of numbers into a full set of descriptive statistics: count, sum, mean, variance, and both the population and sample standard deviation. You paste or type values separated by commas, spaces, or line breaks, and every result updates as you go without sending anything to a server. It is built for students checking homework, researchers and analysts summarizing measurements, quality-control and finance users gauging spread, and anyone who wants the numbers behind a data set without opening a spreadsheet.
Reach for it when you need to know how tightly or loosely values cluster around their average. A small standard deviation means the numbers sit close to the mean and the data is consistent; a large one means they are widely scattered. Variance is just the standard deviation squared and stays in squared units, so the standard deviation is usually the figure people quote because it shares the same unit as your original data.
The single most important choice is population versus sample. Use the population value when your numbers cover the entire group you care about (for example, the test scores of every student in a specific class). Use the sample value when your numbers are only a subset drawn from a larger group and you want to estimate that group's spread; it divides by n minus 1 and so is always slightly larger than the population figure for the same data.
Practical tips and one caveat:
- Results round to six decimal places, so a displayed value like 2 may be an exact whole number or a tidy rounding; check the variance if you need the full precision.
- Any token that is not a finite number is silently skipped, which keeps messy pasted data usable but also means a mistyped entry can quietly drop out of the calculation, so confirm the count (n) matches how many values you expected.
Frequently asked questions
What is the difference between population and sample standard deviation?
Population SD divides the summed squared deviations by n and is used when your data covers the entire group. Sample SD divides by n minus 1 (Bessel's correction) and is used when your data is only a sample drawn from a larger population, giving an unbiased estimate.
Why does the sample standard deviation show a dash for a single number?
Sample SD divides by n minus 1, which is zero when you have only one value, so it is undefined. You need at least two numbers for a sample standard deviation; the population SD of a single value is always 0.
Is my data uploaded anywhere?
No — this calculator runs entirely in your browser; nothing is uploaded.
Is it free?
Yes, completely free with no sign-up and no limits.
People also ask
How do you calculate standard deviation step by step?
Find the mean of your numbers, subtract the mean from each value and square the result, add those squared differences together, divide that sum by n for the population version or by n minus 1 for the sample version, then take the square root. The result is the standard deviation in the same units as your data.
What is a good standard deviation value?
There is no universal good or bad value because it depends entirely on the scale and context of your data. A smaller standard deviation relative to the mean indicates more consistent, tightly grouped values, while a larger one indicates greater spread; comparing it to the mean (as a coefficient of variation) makes the size meaningful.
What is the difference between standard deviation and variance?
Variance is the average of the squared differences from the mean, while standard deviation is the square root of the variance. Standard deviation is reported in the original units of the data, whereas variance is in squared units, which is why standard deviation is usually easier to interpret.
Should I use population or sample standard deviation?
Use the population formula (divide by n) when your data includes every member of the group you are studying. Use the sample formula (divide by n minus 1) when your data is only a sample meant to represent a larger population, since that adjustment gives an unbiased estimate of the larger group's spread.
What does a standard deviation of zero mean?
A standard deviation of zero means every number in the data set is identical, so there is no spread around the mean. This happens whenever all values are the same, and the population standard deviation of a single value is also zero.
Can standard deviation be negative?
No. Standard deviation is the square root of an average of squared differences, and squaring makes every term non-negative, so the result is always zero or a positive number.
How is standard deviation related to the normal distribution?
In a normal (bell-shaped) distribution, about 68 percent of values fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. This empirical rule only applies to roughly normal data, not to every data set.
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