Statistics Calculator
Mean, median, mode, range, variance and standard deviation for a data set.
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Your details
Non-numeric entries are ignored.
Result
- Mean
- 14.3750
- Median
- 15.0000
- Mode
- 15.0000
- Standard deviation (sample)
- 3.6228
- Variance (sample)
- 13.1250
- Count
- 8
- Sum
- 115.0000
- Range
- 9.0000 to 20.0000
- Quartiles
- Q1 11.750 · Q3 15.750
Uses the n−1 denominator
Span of 11.0000
IQR 4.000
Mean, median and which to report
The mean is the arithmetic average; the median is the middle value once sorted. They agree on symmetric data and diverge sharply on skewed data — which is why median income is reported rather than mean income, since a handful of very high earners drag the mean upward.
When the mean and median differ substantially, the distribution is skewed and the median is usually the more honest summary.
Sample versus population variance
This calculator uses the sample formula, dividing by n−1 rather than n. That correction (Bessel's correction) exists because a sample's spread systematically underestimates the population's, and dividing by the smaller number compensates.
Use the sample formula when your numbers are a subset drawn from a larger group — which is almost always. The population formula applies only when you have measured every member.
Standard deviation in practice
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This is what makes standard deviation useful: it converts a spread into a statement about how typical any given value is.
A value more than two standard deviations from the mean is unusual but not remarkable in a large sample. Beyond three, it is worth checking whether it is a genuine observation or a data-entry error.
Outliers and the interquartile range
The mean and standard deviation are both sensitive to extreme values; a single mistyped figure can shift them substantially. The median and interquartile range are resistant, which is why they are preferred for skewed or messy data.
A common rule flags values more than 1.5 times the interquartile range beyond the quartiles as outliers. Both quartiles and the IQR are shown above so you can apply it directly.
Formula & methodology
What this calculator assumes
- Standard deviation and variance use the n − 1 (sample) denominator, which is the unbiased estimator for a population from which the data is a sample. Divide by n instead only when the data is the entire population.
- The mean is sensitive to outliers and the median is not; for skewed data the median is usually the more honest summary.
- Several quartile conventions exist and give slightly different answers on small samples. This uses linear interpolation, matching Excel’s PERCENTILE and NumPy’s default.
Sources
Frequently asked questions
Why does my calculator give a different standard deviation?
It is probably using the population formula (dividing by n). Most scientific calculators offer both, labelled σ for population and s for sample.
When should I use median over mean?
Whenever the data is skewed or contains outliers. Income, house prices and response times are all routinely reported as medians for this reason.
Can a data set have more than one mode?
Yes. If several values tie for the highest frequency the set is multimodal, and all of them are shown. If no value repeats, there is no mode at all.