Enter a data set to calculate its central tendency, spread, and sorted order in one pass, including both population and sample measurements.
How to interpret the results
Start with the sorted data and count so you can spot missing entries or a value entered twice. Then choose a measure of center and a measure of spread that fit the data:
- Mean uses every value, so a very large or small observation can pull it toward that extreme.
- Median is the middle of the sorted data and is less affected by outliers.
- Mode is the most frequent value. A data set can have more than one mode or no repeated value.
- Range is quick to understand but depends only on the minimum and maximum.
- Standard deviation uses every observation and describes typical distance from the mean in the original unit.
NIST describes the range, variance, and standard deviation as common measures of scale. A small standard deviation means values cluster more closely around the mean; a large one means they are more dispersed. It does not, by itself, say whether the values are good, bad, or normally distributed.
Sample or population?
Use the population result when the list contains every member you intend to
describe—for example, the final scores of all 12 people in one class. Use the
sample result when those observations stand in for a larger group. Dividing by
N − 1 makes the sample variance larger than the population variance for the
same list.
If you only need one statistic, the Average Calculator, Standard Deviation Calculator, and Mean, Median, Mode & Range Calculator show more focused working steps.
Preparing data for calculation
Paste numbers separated by commas, spaces, semicolons, or line breaks. Remove labels, currency symbols, and units first, and keep every value in the same unit. Decimals and negative numbers are accepted. At least two finite values are required because sample variance cannot be calculated from a single observation.
The geometric mean is meaningful for positive values and is often useful for multiplicative change, such as growth factors. If the list contains zero or a negative value, this calculator reports zero for geometric mean to match the reference behavior; do not interpret that display as a general mathematical definition for signed data.
Examples
Eight-value data set
For 10, 2, 38, 23, 38, 23, 21, 23, the count is 8, sum is 178,
mean is 22.25, median and mode are both 23, and the range is 36.
The population standard deviation is 11.508149286484; the sample standard
deviation is 12.302729081677.
Population versus sample
If those eight values are every member of the group, use population variance 132.4375. If they are a sample, use sample variance 151.35714285714.
Formula
For N values, the arithmetic mean is Σx / N. The median is the middle sorted
value (or the average of the two middle values), and the range is
maximum − minimum.
Population variance divides squared deviations by N, while sample variance uses N − 1:
σ² = Σ(x − μ)² / N
s² = Σ(x − x̄)² / (N − 1)
Standard deviation is the square root of variance. For positive values, the
geometric mean is (x1 × x2 × … × xN)^(1/N).
Sources
Statistical interpretation
These descriptive statistics summarize only the values entered. Sample formulas use N−1 where shown, geometric mean requires appropriate positive values, and mode can be nonunique. None of the outputs establishes causation, representativeness, normality, or statistical significance; inspect the sampling process and distribution as well as the summary.