How this calculator works
Each class is represented by its midpoint, repeated conceptually according to its frequency. The weighted mean and squared deviations then follow the ordinary variance calculation. This avoids expanding a frequency table into a long list. Results are estimates when the original observations vary within each interval, but exact when the supplied midpoints are the actual distinct values. Frequencies are whole counts, including zero.
Formula
N = Σf; mean ≈ Σ(fm)/N; variance ≈ Σ(f(m - mean)²)/(N - 1) for a sample, or /N for a population; SD = sqrt(variance).
Worked example
Midpoints 1 and 3 with frequencies 1 and 3 represent 1, 3, 3, 3. The mean is 2.5 and the squared deviations total 3, giving sample SD = sqrt(3/3) = 1.
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Frequently asked questions
Can I use an open-ended class?
Not without a defensible representative value. An interval such as 65 and over has no finite midpoint supplied by its boundaries.
Do class widths need to match?
No. Use each class midpoint and its count. Do not substitute frequency density for the count.
Why can grouped SD differ from raw-data SD?
Values inside a class are replaced by one midpoint, losing within-class detail. Wider classes generally lose more information.