Statistics Calculator
Comprehensive descriptive statistics and summary analysis.
Centre, spread, and shape
Describing a dataset takes three things, and reporting only the first is how most statistics mislead. Centre is the mean, median or mode. Spread is the range, interquartile range, variance or standard deviation. Shape is the skew and the presence of outliers. Two datasets can share an identical mean and behave nothing alike.
Which measures to use depends on the shape. For roughly symmetric data the mean and standard deviation are efficient and informative. For skewed data or data with outliers, the median and interquartile range are far more robust — a single extreme value can move a mean arbitrarily far while barely touching a median. Income data is the standard example and is essentially always reported as a median for exactly this reason.
Two distinctions worth keeping. A population is everything you care about; asample is a subset used to estimate it, and sample variance divides byn − 1 to correct for the resulting bias. And descriptive statistics summarise the data in front of you, while inferential statistics generalise beyond it — the second requires assumptions about how the sample was drawn that the first does not.
Frequently Asked Questions
What statistical summary metrics are computed?
Mean, Median, Mode, Range, Minimum, Maximum, Sum, Sample Variance, and Standard Deviation.
What is the difference between variance and standard deviation?
Variance is the mean squared deviation from the mean; standard deviation is its square root. Standard deviation is usually reported because it is in the same units as the data - variance of a height dataset is in square centimetres, which means nothing intuitively.
What does skewness tell me?
Which direction a distribution leans. Positive skew has a long right tail and pulls the mean above the median - income distributions are the classic case. Negative skew does the opposite. Near-zero skew suggests symmetry, in which case mean and median roughly coincide.
What is the empirical rule?
In an approximately normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. It is the quickest sanity check on whether an observation is unusual, though it only holds for roughly bell-shaped data.