Standard Deviation Calculator
Find the mean, variance and both population and sample standard deviation of any list of numbers.
Divides by n − 1. Use when your data is a sample of a larger group.
8
18
24
27.4286
4.899
5.2372
Formula
Examples
| Input | Result |
|---|---|
| Population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9 | Mean 5, variance 4, standard deviation 2 |
| Sample standard deviation of 2, 4, 4, 4, 5, 5, 7, 9 | Mean 5, variance 4.5714, standard deviation 2.1381 |
| Population standard deviation of 4, 8, 15, 16, 23, 42 | Mean 18, standard deviation 12.3153 |
| Sample standard deviation of 4, 8, 15, 16, 23, 42 | Mean 18, standard deviation 13.4907 |
About this calculator
A standard deviation calculator measures how spread out a set of numbers is around their average. A small standard deviation means the values cluster tightly near the mean, while a large one means they are scattered widely. It is one of the most important descriptive statistics, summarising the typical distance of a data point from the centre of the data.
The calculation has a clear sequence. First it finds the mean (the sum of all values divided by how many there are). Then for each value it takes the difference from the mean and squares it, so positive and negative deviations both count as positive. Those squared differences are added up and divided to get the variance, and the standard deviation is simply the square root of the variance, which returns the result to the original units. The crucial choice is the divisor: dividing by N gives the population standard deviation, while dividing by N minus 1 gives the sample standard deviation. The N minus 1 version, known as Bessel's correction, compensates for the fact that a sample tends to underestimate the spread of the wider population it was drawn from.
To use the calculator, enter your numbers separated by commas or spaces, then choose whether your data represents an entire population or just a sample. The tool returns the mean, the variance, and the standard deviation, so you can see every stage of the working rather than just the final figure.
Interpreting the result depends on context. In an approximately normal distribution, about 68 percent of values fall within one standard deviation of the mean and about 95 percent within two. So a standard deviation that is small relative to the mean signals consistency, while a large one signals volatility or inconsistency in the data.
The most common mistake is picking the wrong divisor: use population (N) only when your data covers every member of the group you care about, and sample (N minus 1) when you are generalising from a subset. Other pitfalls include forgetting to square the deviations, leaving out a data point, and being misled by outliers, which inflate the standard deviation because their deviations are squared.
Frequently asked questions
Population standard deviation divides the sum of squared deviations by N, the full count, and is used when your data covers every member of the group. Sample standard deviation divides by N minus 1, which gives a slightly larger value and is used when generalising from a subset to a wider population.
This is Bessel's correction. A sample usually has slightly less spread than the full population it came from, so dividing by N would underestimate the true variability. Using N minus 1 corrects for this bias and produces a more accurate estimate.
Variance is the average of the squared deviations from the mean, so its units are squared. Standard deviation is the square root of the variance, which returns the figure to the same units as the original data and makes it far easier to interpret.
It describes the typical distance of values from the mean. In a roughly normal distribution, about 68 percent of values lie within one standard deviation of the mean and about 95 percent within two, so it quantifies how consistent or scattered your data is.
Yes, strongly. Because each deviation is squared, a single extreme value contributes a disproportionately large amount to the total. A large standard deviation relative to the mean often signals the presence of outliers or genuine high variability.
It can be zero, which happens only when every value in the set is identical and there is no spread at all. It can never be negative, because it is the square root of a sum of squared (non-negative) quantities.
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