Standard Deviation Calculator
Paste or type your data as a list of numbers to find the sample and population standard deviation, along with the mean and variance.
Standard deviation measures how spread out a set of numbers is around their mean. Find the mean, sum the squared differences from it, divide by n for population or nโ1 for a sample, then take the square root. For the data 2, 4, 4, 4, 5, 5, 7, 9 (mean 5), the population standard deviation is 2 and the sample standard deviation is about 2.14.
How to Use This Calculator
Type or paste your numbers into the box, separated by commas, spaces or line breaks โ the calculator ignores the formatting and reads the values. It returns two figures: the sample standard deviation and the population standard deviation, plus the mean and variance in the working.
Choose sample or population based on your data. Use the population standard deviation when your list contains every member of the group you care about (for example, the test scores of all students in one class you are describing). Use the sample standard deviation โ the one dividing by nโ1 โ when your numbers are a sample drawn from a larger population and you want to estimate that population's spread. In practice, sample standard deviation is the more common choice.
How the Calculation Works
Standard deviation is the square root of the variance, and variance is the average of the squared distances from the mean. First compute the mean. Then, for each value, subtract the mean and square the result โ squaring removes the sign so deviations above and below the mean both count. Sum those squared deviations. Divide by N for the population variance, or by Nโ1 for the sample variance (the nโ1, called Bessel's correction, compensates for the fact that a sample tends to underestimate the true spread). Finally, take the square root to return to the original units.
Worked Example
Take the classic data set 2, 4, 4, 4, 5, 5, 7, 9. The mean is (2+4+4+4+5+5+7+9) รท 8 = 40 รท 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Population variance = 32 รท 8 = 4, so population SD = โ4 = 2. Sample variance = 32 รท 7 = 4.571, so sample SD = โ4.571 โ 2.14. The sample value is slightly larger, as it always is, because of the nโ1 divisor.
What the Result Means
A small standard deviation means the values cluster tightly around the mean; a large one means they are widely scattered. Because it is in the same units as your data, standard deviation is directly interpretable โ a standard deviation of 2 on test scores means typical scores fall about 2 points from the average. For roughly normal data, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three (the empirical rule). Standard deviation also feeds into z-scores, confidence intervals and many other statistics.
Assumptions and Limitations
Standard deviation summarizes spread but is sensitive to outliers, because squaring gives extreme values a large influence โ a single unusual number can inflate it substantially. It is most meaningful for roughly symmetric, unimodal data; for very skewed data the interquartile range may describe spread better. Make sure you pick the correct version: reporting a population SD when you meant a sample (or vice versa) is a common mistake. The calculation itself is exact for the numbers you enter.
Who Uses a Standard Deviation Calculator
Students, analysts, and quality teams use this to measure how spread out a data set is, with both sample and population results. Run it to summarize variability, complete coursework, or judge consistency in a process. Paste a list of numbers to get the standard deviation, variance, and mean.
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Frequently Asked Questions
What is the difference between sample and population standard deviation?
Population standard deviation divides by N and is used when you have every value in the group. Sample standard deviation divides by Nโ1 and is used when your data is a sample estimating a larger population. The sample value is always slightly larger.
Why does sample standard deviation divide by nโ1?
Because a sample tends to underestimate the true population spread. Dividing by nโ1 (Bessel's correction) makes the sample variance an unbiased estimator of the population variance.
What does a high standard deviation mean?
It means the data values are widely spread out from the mean. A low standard deviation means they are clustered closely around the mean. It is measured in the same units as the data.
How is standard deviation related to variance?
Standard deviation is the square root of the variance. Variance is in squared units, while standard deviation is in the original units, which makes it easier to interpret.
What is the empirical rule?
For roughly normal data, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It gives a quick sense of how unusual a value is.
Do outliers affect standard deviation?
Yes, strongly. Because deviations are squared, a single extreme value can substantially increase the standard deviation. For heavily skewed data with outliers, the interquartile range may be a better measure of spread.
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