Standard Deviation and Variance: What Spread Really Tells You
Published · 3 min read · By The Samsung Calculator editorial team
What it measures
Standard deviation is the typical distance of a value from the mean of its data set. Two classes can both average 70% on a test: one where everyone scored between 68 and 72, and one where half scored 40 and half scored 100. The means are identical and the standard deviations — about 1.5 and 30 — tell you they are nothing alike.
Why square, then square-root
Deviations from the mean sum to zero by definition, so averaging them directly gives nothing. Squaring makes every deviation positive and, usefully, penalises large deviations more than small ones. The average of the squared deviations is the variance.
Variance is in squared units — squared marks, squared dollars — which is not interpretable. Taking the square root returns the figure to the original units, and that is the standard deviation. Absolute deviation would also work and is easier to explain; squaring won because it has better mathematical properties, not because it is more intuitive.
Sample or population, and why n−1
Divide by n when your data is the entire population. Divide by n−1 when it is a sample and you are estimating the population's spread.
The reason is that a sample's own mean sits closer to its own data than the true population mean does, so squared deviations from the sample mean systematically understate the real spread. Dividing by n−1 rather than n corrects the bias exactly. This is Bessel's correction, and it matters most for small samples — at n = 5 it changes the answer by over 10%; at n = 500 it is invisible.
For a roughly normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. That is the empirical rule, and it is the fastest way to sanity-check whether an observation is unusual.
Where it misleads
Standard deviation assumes the spread is meaningfully summarised by a single number, which is only true for roughly symmetric distributions. For skewed data — incomes, house prices, waiting times — the mean and standard deviation both mislead, and the median with an interquartile range describes reality far better.
It is also sensitive to outliers by construction, because it squares them. One mistyped value can double a standard deviation. Always plot the data before summarising it.
Compute it on your own data
The standard deviation calculator shows the mean, variance, sum of squares and the five-number summary together, so you can see whether the standard deviation is even the right summary for your data.
Questions this answers
- how to calculate analysis of variance
- how to calculate analysis of variance in excel
Run your own numbers
Compute standard deviation, variance, mean, sum of squares and the full five-number summary of any data set.
About this article
Written and reviewed by The Samsung Calculator editorial team. Every calculator is written against a published formula, reviewed against at least one independent reference implementation, and dated when it changes. Last updated July 13, 2026. Spotted an error? Tell us.