Free statistics calculator — compute mean, median, mode, standard deviation, variance, and range from any dataset. Essential for students, researchers, and analysts who need quick descriptive statistics without software like Excel or SPSS.
Mean, median, mode, standard deviation & more
Descriptive statistics summarize and describe the key features of a dataset. The mean (average), median (middle value), and mode (most frequent value) tell you where data is centered. Standard deviation and variance measure how spread out values are around the mean. Range shows the distance from the minimum to the maximum value. Together, these measures paint a complete picture of any dataset — whether you're analyzing test scores for 30 students, comparing monthly sales figures for a small business, or processing survey results from a research study. This statistics calculator handles datasets of any size: just paste your numbers (comma or space separated) and click Calculate.
The difference between sample standard deviation (s) and population standard deviation (σ) matters for research validity. When your data represents a sample from a larger population — which is almost always the case in practice — use sample standard deviation (s), which divides by n−1 (Bessel's correction). This calculator computes both. The US Census Bureau, for instance, reports income statistics using sample data from hundreds of thousands of households — the 2024 median US household income was approximately $80,610, a figure where median is far more useful than mean because a handful of billionaires would skew the average income dramatically upward. The mode is especially useful in retail and manufacturing quality control, where you want to know the most frequently occurring measurement or defect type.
The arithmetic mean sums all values and divides by count. It's sensitive to outliers — add one extreme value and the mean shifts dramatically. For the dataset {1, 2, 3, 100}, the mean is 26.5 but the median is 2.5, which better represents the "typical" value in a skewed distribution.
The middle value when sorted in order. More robust than mean when outliers exist. The US Census and Federal Reserve both report median household income and median home prices specifically because medians resist distortion from extreme values — the same reason realtors cite median sale price instead of average price.
Measures dispersion — how spread out values are from the mean. A standard deviation of 0 means all values are identical. In a normal distribution, 68% of data falls within 1 standard deviation of the mean, and 95% within 2 standard deviations. Low std dev = consistent data. High std dev = highly variable data.
Standard deviation squared (s² or σ²). While standard deviation is expressed in the same units as your data (making it intuitive), variance is used mathematically in ANOVA, regression, portfolio risk analysis, and statistical testing. In finance, portfolio variance drives modern portfolio theory and the Sharpe ratio calculation.
Descriptive statistics summarise a dataset with a measure of centre and a measure of spread, and the choice between them matters. The mean uses every value and is therefore sensitive to outliers; the median is the middle value and is robust to them. When mean and median diverge sharply the distribution is skewed, which is why median household income is always quoted rather than mean. For spread, the critical technical detail is Bessel's correction: dividing by n−1 rather than n when the data are a sample rather than the whole population, because a sample's spread systematically understates the population's.
Mean x̄ = Σx ÷ nSample variance s² = Σ(x − x̄)² ÷ (n − 1)Population variance σ² = Σ(x − x̄)² ÷ nStandard deviation = √variancewhere:
Assumptions: Use the sample formula unless you genuinely have every member of the population. The two diverge most at small n — with 7 observations the difference is around 8%, and it becomes negligible above about 100.
Compute mean, median and both standard deviations for a small sample.
ResultMean 20.0, median 18, sample SD 6.976
The mean exceeds the median by 2, indicating a mild right skew — the 31 pulls the average up while leaving the middle untouched. The two standard deviations differ by 8%, which is exactly the small-sample effect Bessel's correction exists to address.