Standard Deviation Calculator

How to use this calculator

1. Paste data in any formatCommas, spaces, tabs or new lines all work, so you can paste a column straight out of a spreadsheet without reformatting it.
2. Choose sample unless your data is everythingIf you measured 30 people to say something about a population, that is a sample. Only if you measured every member of the group is it a population.
3. Read both figures anywayThe breakdown shows both, along with the percentage difference between them. On small datasets that gap is substantial.
4. Check the range and median tooA standard deviation on its own can hide skew and outliers. Comparing the mean against the median is a quick check for both.

Why divide by n − 1

This is the part of introductory statistics most people memorise and few can explain, and the explanation is genuinely intuitive once stated.

When you calculate the spread of a sample, you measure deviations from the sample mean — because you do not know the true population mean. But the sample mean is, by construction, the point that minimises the sum of squared deviations for that particular sample. It sits right in the middle of your data.

The true population mean almost certainly sits somewhere slightly else. Deviations measured from it would be, on average, slightly larger.

So using the sample mean systematically underestimates the spread. Not by chance — every time, in the same direction. That is a bias rather than noise.

Dividing by n−1 rather than n inflates the result by exactly enough to correct that bias on average. This is Bessel's correction, and n−1 is called the degrees of freedom: once the mean is fixed, only n−1 of the values are free to vary, because the last one is determined by the requirement that they average to the mean.

The size of the correction depends on the sample. With 6 values, dividing by 5 instead of 6 inflates the variance by 20% and the standard deviation by about 9.5%. With 100 values the difference is 0.5%. With 1,000 it is negligible.

So the choice matters most exactly where data is scarcest — which is also where people are most likely to be working by hand and most likely to get it wrong.

Worked example: 4, 8, 15, 16, 23, 42

Six values. Sum: 4 + 8 + 15 + 16 + 23 + 42 = 108. Mean: 108 ÷ 6 = 18.

Now the deviations from the mean: −14, −10, −3, −2, 5, 24.

Notice they sum to zero. They always do, which is why you cannot simply average the deviations — the positives and negatives cancel exactly. Squaring is what solves that.

Squared deviations: 196, 100, 9, 4, 25, 576. Sum: 910.

Sample variance: 910 ÷ 5 = 182. Sample standard deviation: √182 = 13.49.

Population variance: 910 ÷ 6 = 151.67. Population standard deviation: √151.67 = 12.32.

The two differ by 1.18, which is 9.5% — a substantial gap, and entirely a consequence of six being a small n.

The square root at the end matters for interpretation. Variance is in squared units, so if these were heights in centimetres the variance would be in square centimetres, which means nothing physically. Taking the root returns the figure to centimetres, which is why standard deviation is the number people actually quote.

One more thing this dataset shows. The mean is 18 but the median is 15.5, and the largest value, 42, is nearly twice the next largest. That single value contributes 576 of the 910 total — 63% of the entire sum of squares from one observation out of six. Squaring makes outliers dominate, and a standard deviation reported without checking for them can be badly misleading.

What standard deviation tells you

The 68-95-99.7 rule only holds for roughly normal data. Chebyshev's inequality gives weaker bounds that hold for any distribution whatsoever, which is what to fall back on when normality is doubtful. Skewed data, bimodal data and data with heavy tails all break the first three rows while still obeying the last two.

Where standard deviation misleads

Standard deviation summarises spread in one number, and any single number will hide things.

It assumes the mean is a sensible centre. For skewed data — income, house prices, waiting times — the mean is pulled toward the tail and the standard deviation describes spread around a point that few observations are near. The median and interquartile range describe such data far better.

It is dominated by outliers, because deviations are squared before averaging. One extreme value can double a standard deviation, as the worked example above shows. Always look at the data, or at least at the range and median, before reporting a standard deviation.

It cannot distinguish distributions with the same spread but different shapes. A bimodal dataset with clusters at 10 and 30 and a uniform spread from 10 to 30 can produce identical standard deviations while describing completely different situations. A histogram takes seconds and shows what the summary cannot.

Comparing standard deviations between datasets with different means is misleading. A standard deviation of 5 means something very different around a mean of 10 than around a mean of 1,000. The coefficient of variation — standard deviation divided by the mean, expressed as a percentage — makes such comparisons meaningful, and it is in the breakdown for that reason.

Standard deviation is also not standard error. Standard deviation describes the spread of the data; standard error describes the precision of the sample mean as an estimate, and it shrinks as the sample grows. Error bars on a chart labelled only "± error" are ambiguous between the two, and the difference is large.

Finally, a standard deviation calculated from data that is not a random sample estimates nothing in particular. The arithmetic will run happily on any numbers; whether the result means anything depends on how they were collected.

Assumptions and limitations

• Requires at least two values — a single observation has no spread to measure
• Assumes the mean is a meaningful centre, which fails for strongly skewed data
• Heavily influenced by outliers, because deviations are squared
• Cannot distinguish distributions with the same spread but different shapes
• The 68-95-99.7 rule applies only to approximately normal data
• Does not detect whether your data is a random sample, which determines whether the estimate means anything
• Standard deviation is not standard error — the breakdown gives both

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