Paste in a list of numbers to get the count, mean, median, minimum, maximum and range, plus population and sample standard deviation and variance — the full spread picture in one go.
Standard Deviation Calculator
How the Standard Deviation Calculator Works
Averages alone can mislead. Two classes can both average 80 marks and be completely different: one bunched tightly around 80, the other spread from 40 to 100. The standard deviation measures that spread — how far, on average, the numbers sit from their own mean. This calculator takes any list of numbers you type in and reports the mean, the median, the range and the standard deviation, for both the population and the sample versions, along with the variance behind each one.
The calculation has four steps. First the mean: add all the numbers and divide by how many there are. Second, take each number’s distance from the mean and square it, so that below-average and above-average distances count equally and big gaps count more than small ones. Third, average those squared distances — that average is the variance. Fourth, take the square root, which converts the answer back into the original units: that square root is the standard deviation. The single decision is what to divide by in step three. The population version divides by the count itself (N); the sample version divides by one less (N − 1), because a sample underestimates the spread of the full population it was drawn from, and the correction compensates.
The same input list also gives the companion figures. The median is the middle value when the list is sorted, or the average of the two middle values when the count is even. The range is the largest minus the smallest. Both are reported alongside, because the median sometimes describes a skewed list better than the mean, and the range gives a quick sense of the extremes even before you look at the standard deviation.
How to Use This Calculator
- Enter your numbers in the box, separated by commas, spaces or new lines.
- Choose population or sample (or read both — this calculator shows both).
- Press Calculate.
- Read the mean, median and range first for the overall picture.
- Read the standard deviation and variance: the standard deviation is in your original units and is the headline spread figure; the variance is its square, useful mainly for further statistics.
Example
Example 1. The numbers 2, 4, 4, 4, 5, 5, 7, 9. The count is 8 and the total is 40, so the mean is 5. The squared distances from 5 are 9, 1, 1, 1, 0, 0, 4 and 16, which sum to 32. The population variance is 32 ÷ 8 = 4, so the population standard deviation is 2.0. The sample variance is 32 ÷ 7 ≈ 4.57, so the sample standard deviation is 2.138. The median is 4.5 (the average of the two middle values, 4 and 5), and the range is 9 − 2 = 7.
Example 2. The numbers 70, 80, 90. The mean is 80 and the median is also 80. The squared distances from 80 are 100, 0 and 100, which sum to 200. The population variance is 200 ÷ 3 ≈ 66.67, so the population standard deviation is 8.16. The sample variance is 200 ÷ 2 = 100, so the sample standard deviation is exactly 10.
Common Mistakes to Avoid
The first mistake is choosing the wrong version. If your numbers are the whole group you care about — every exam mark in the class, every month’s sales for the year — use the population figure. If they are a sample drawn from something bigger, use the sample figure; the difference is small for large lists and large for tiny ones, as the examples show. Second, do not read the standard deviation as the average distance from the mean: because distances are squared before averaging, a single far-off value pulls the standard deviation up more than its raw distance would suggest. Third, remember that one outlier dominates: add a single 900 to a list of numbers around 90 and the standard deviation explodes, while the median barely moves — which is why this calculator reports both. Fourth, keep the units straight: the standard deviation is in your original units, but the variance is in those units squared, so a variance of 100 for heights in centimetres means a standard deviation of 10 cm, not 100 cm.
Standard Deviation Calculator FAQs
Which should I use, population or sample?
Population when your list is everything you care about; sample when it is a subset of a larger group. Most classroom and research data are samples, so the sample figure is the usual default — but this calculator shows both so you can compare.
Can the standard deviation be zero or negative?
Zero, yes — when every number is identical, there is no spread at all. Negative, never: it is a square root, and square roots of the (non-negative) variance are never negative.
How do I enter the numbers?
Type them separated by commas, spaces or new lines — any mix works. The calculator ignores extra spaces, so “1, 2, 3” and “1 2 3” give the same result.
Why does the order of my numbers not matter?
Because the calculation only uses the sum, the count and the squared distances — all of which are the same regardless of order. Sorting the list changes nothing except which value the median picks out, and the median is order-independent too once the list is sorted.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean; standard deviation is its square root. Variance is the working step, standard deviation is the answer in real units. For most purposes, the standard deviation is the number to quote.
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