Statistical Measures Calculator
Enter a list of numbers to calculate mean, median, mode, standard deviation, and variance.
Formulas
Mean: x̄ = (Σxᵢ) / N
Median: Middle value of sorted data. If N is even, average of the two middle values.
Mode: Value(s) that appear most frequently in the dataset.
Population Variance: σ² = Σ(xᵢ − μ)² / N
Sample Variance: s² = Σ(xᵢ − x̄)² / (N − 1) (Bessel's correction)
Standard Deviation: σ = √σ² or s = √s²
Range: max(x) − min(x)
Assumptions & References
- Input values must be real numbers (integers or decimals).
- At least 2 data points are required for sample variance/standard deviation.
- Sample standard deviation uses Bessel's correction (N−1) to provide an unbiased estimate of the population variance from a sample.
- Population formulas (divide by N) are used when the entire population is known.
- If all values are unique, there is no mode; if multiple values share the highest frequency, all are reported.
- Reference: Freedman, Pisani & Purves, Statistics (4th ed.); NIST/SEMATECH e-Handbook of Statistical Methods.