Statistics Calculator

Sample (divides by n−1) when your numbers are a subset standing in for something bigger — survey responses, measurements, most homework. Population (divides by n) only when the data is literally everything — all 30 students in the one class you're describing. When a problem just says 'find the standard deviation' of sampled data, it means sample.

By linear interpolation on the sorted data (the same method as Excel's QUARTILE.INC and most stats software). Textbooks sometimes use the median-of-halves method, which can differ slightly on small datasets — if your answer key disagrees in the decimals, that's the reason, not an error.

Shape — the thing summary numbers hide. Symmetric and single-peaked suggests normal-ish data; a long tail means skew (and explains why mean and median disagree); two peaks suggest two mixed groups; a lone bar far from the rest is an outlier worth investigating before trusting the mean.

When every value appears exactly once, no value is 'most frequent' — the calculator reports none rather than pretending. Ties produce multiple modes (bimodal data), all listed.