Descriptive Statistics Calculator
Compute key summary statistics — mean, median, variance, and standard deviation — from any list of numbers. Ideal for quickly summarizing datasets in research, class assignments, or data analysis.
About this calculator
Descriptive statistics condense a dataset into a few interpretable numbers. The mean is the arithmetic average: x̄ = (Σxᵢ) / n. The median is the middle value when data are sorted; for an even count it is the average of the two central values. Variance measures spread: for a population, σ² = Σ(xᵢ − x̄)² / n; for a sample, s² = Σ(xᵢ − x̄)² / (n − 1), where dividing by n − 1 (Bessel's correction) produces an unbiased estimate. Standard deviation is simply the square root of variance (σ or s). Choosing between population and sample formulas matters: use the population formula only when you have data for every member of the group; otherwise use the sample formula. These statistics together describe center, spread, and shape of a distribution.
How to use
Enter the dataset: 4, 8, 6, 5, 3. Step 1 — Mean: (4+8+6+5+3)/5 = 26/5 = 5.2. Step 2 — Sorted data: 3, 4, 5, 6, 8; Median = middle value = 5. Step 3 — Sample variance: deviations² = (4−5.2)²+(8−5.2)²+(6−5.2)²+(5−5.2)²+(3−5.2)² = 1.44+7.84+0.64+0.04+4.84 = 14.8; s² = 14.8/4 = 3.7. Step 4 — Sample standard deviation: s = √3.7 ≈ 1.92. Select your desired statistic from the dropdown to see each result instantly.
Frequently asked questions
What is the difference between population variance and sample variance?
Population variance divides the sum of squared deviations by n (the total count), and is correct only when your data include every member of the group being studied. Sample variance divides by n − 1, applying Bessel's correction to compensate for the fact that a sample's spread tends to underestimate the true population spread. Using the wrong formula can produce a biased estimate. In practice, most real-world datasets are samples, so n − 1 is the appropriate denominator for inferential purposes.
When should I use the median instead of the mean to describe a dataset?
Use the median when your data are skewed or contain outliers, because the mean is sensitive to extreme values while the median is not. For example, household income data are right-skewed due to very high earners; the median income better represents a typical household than the mean. The mean is preferable for symmetric, roughly normal distributions where no extreme outliers are present. When in doubt, report both: a large gap between mean and median signals skewness worth investigating.
How do I interpret standard deviation in plain language?
Standard deviation tells you, on average, how far individual data points stray from the mean. A small standard deviation means the values cluster tightly around the average; a large one means they are widely spread. For data that follow a roughly normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, and about 95% within two. This makes standard deviation a practical yardstick for judging whether a particular value is typical or unusually extreme for a given dataset.