How this calculator works
The mean uses all values; the median uses the ordered centre; modes identify repeated values at the highest frequency. Spread summaries describe the extremes and the middle half. Sample variance divides squared deviations by n - 1, while population variance divides by n. The displayed standard error SD/sqrt(n) assumes independent observations; when the data describe an entire population it is only that algebraic quantity, not evidence of sampling uncertainty.
Formula
mean = Σx/n; variance = Σ(x - mean)²/(n - 1) for a sample, or /n for a population; SD = sqrt(variance); IQR = Q3 - Q1.
Worked example
For 1, 2, 3, 4, 5, the count is 5, total 15, mean and median 3, and sample variance 10/4 = 2.5. Every value appears once, so this tool reports no mode.
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Frequently asked questions
Should I choose sample or population?
Choose population when the list is the entire group you intend to describe. Choose sample when estimating spread in a broader population.
Why is a singleton rejected in sample mode?
The n - 1 divisor would be zero. Population mode can describe one observation with variance zero.
Do these numbers replace a graph?
No. Similar summaries can hide different shapes, clusters and outliers. Inspect the observations as well.