Last reviewed 22 Sept 2026 · 6 min read
Frequency distributions
Raw data is grouped into classes with frequencies. For a class such as 20–30: the class mark (midpoint) is 25, the class size is 10, and the cumulative frequency is the running total of the frequencies up to that class. In an exclusive distribution (10–20, 20–30) the upper limit belongs to the next class; in an inclusive one (10–19, 20–29) it does not, and the limits must be corrected by half the gap before any formula is used.
The three averages
- Ungrouped:
- Grouped: with the class mark
- Assumed mean: , where
- Step deviation: , where
- Combined mean of two groups:
Ungrouped (data arranged in order):
- odd → the th value; even → the mean of the two middle values.
- Mode is the most frequent value; a set may have none or several.
Grouped:
where is the lower limit of the median or modal class, , the cumulative frequency before it, its frequency, the class size, and the frequencies of the classes before, at and after the modal class.
Empirical relation: .
Which one to use: the mean uses every value but is dragged by extremes; the median ignores extremes and suits skewed data such as incomes; the mode is the only average for a non-numeric category.