Math calculator
Standard Deviation Calculator
Calculate population and sample variance and standard deviation from a numeric dataset.
Sample variance and deviation require at least two values.
Population standard deviation
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What the calculator returns
Type or paste a list of numbers separated by commas, spaces, or new lines. The tool reads every value, finds the mean, and reports five numbers at once: the mean, the population variance, the sample variance, the sample standard deviation, and the count of values you entered. The population standard deviation is shown as the headline result, since that is the more common classroom default.
Sample variance and sample standard deviation need at least two numbers to mean anything — dividing by a sample size of one minus one is dividing by zero. If you enter a single value, those two fields stay blank instead of showing an error, while the population figures still calculate normally because a population of one has a variance of zero.
Every entry has to parse as a plain number; a stray letter or symbol in the list leaves every result blank rather than silently skipping the bad token, so a typo anywhere in a long dataset is easy to spot because nothing calculates at all.
The two formulas
Both formulas start the same way: subtract the mean from every value, square each difference so negatives don't cancel positives, and add the squares up. Where they split is the denominator — population variance divides by the full count N, while sample variance divides by one less, n − 1.
Take five exam scores: 68, 72, 75, 80, 85. The mean is (68 + 72 + 75 + 80 + 85) ÷ 5 = 380 ÷ 5 = 76. The deviations from 76 are −8, −4, −1, 4, and 9, and squaring each gives 64, 16, 1, 16, and 81, which sum to 178. Population variance is 178 ÷ 5 = 35.6, so population standard deviation is √35.6 ≈ 5.9666. Sample variance is 178 ÷ 4 = 44.5, so sample standard deviation is √44.5 ≈ 6.6708. The sample figure is always the larger of the two for the same dataset, because dividing the same sum by a smaller number produces a bigger result.
Picking population or sample — and the outlier trap
Use the population formula only when your numbers are the entire group you care about, such as every quiz score from one specific class this semester. Use the sample formula when your numbers are a subset standing in for a bigger group you can't fully measure, such as 40 surveyed customers representing all customers. Reporting the wrong one is the single most common error with this calculator: a sample treated as a population understates how spread out the true population probably is.
Because every deviation gets squared, one distant value moves the result far more than the same number of ordinary values would. Swap the 85 in the exam-score example for 190 and the mean jumps to (68 + 72 + 75 + 80 + 190) ÷ 5 = 97, while the population standard deviation climbs to about 46.66 — nearly eight times higher than the original 5.9666 — even though only one of the five scores changed.
Where this measure is used
Standard deviation shows up wherever consistency matters more than the raw average:
- Grading on a curve – a class with standard deviation 5.97 has scores clustered tighter than one with standard deviation 15, even with the same average
- Manufacturing tolerances – a machined part's diameters need a low standard deviation so every unit fits the same housing
- Investment risk – two funds can share the same average annual return but very different standard deviations, meaning very different swing sizes
- Weather records – comparing the standard deviation of daily highs shows how variable a city's climate is, not just its average temperature
- Sports consistency – a free-throw shooter's shot-to-shot standard deviation says more about reliability than the season average alone
- Lab measurements – repeated readings of the same sample use standard deviation to report measurement precision
Frequently asked questions
Should I use population or sample standard deviation?
Use population when your list is every member of the group you're studying. Use sample when your list is a subset used to estimate a larger group. The sample formula divides by n − 1 instead of n, which always produces a slightly larger, more cautious estimate.
Why are the sample variance and sample standard deviation blank?
Those two figures need at least two data points, because the sample formula divides by n − 1. Entering exactly one number makes that denominator zero, so the calculator leaves the fields blank instead of dividing by zero.
What's the difference between variance and standard deviation?
Variance is the average of the squared deviations, so it's expressed in squared units — square dollars, square points, and so on, which is hard to interpret. Standard deviation is the square root of variance, which brings the units back to the original scale, like dollars or test points.
Can standard deviation be negative or zero?
It can never be negative, since it's built from a square root of squared numbers. It equals exactly zero only when every value in the list is identical, because then every deviation from the mean is zero.
How much does one outlier change the result?
A lot, because deviations are squared before averaging. In the five-score example, replacing 85 with 190 raises the mean from 76 to 97 and multiplies the population standard deviation almost eightfold, from about 5.97 to roughly 46.66.
Does a bigger dataset always lower the standard deviation?
No. Standard deviation reflects how spread out the values are, not how many there are. Ten scores that all fall between 70 and 80 will have a lower standard deviation than three scores of 0, 50, and 100, regardless of the different counts.