Frequency Table

A frequency table is a descriptive statistical summary produced by counting observations by value or class interval. Distinct values can be counted individually for discrete variables. Continuous measurements and numerical variables with many distinct values are usually grouped into intervals to describe concentrations and the spread of a distribution.

Basic quantities

For n observations assigned to k classes, let f_i denote absolute frequency and p_i = f_i/n relative frequency. The ascending cumulative frequency is F_j = Σ(i=1..j) f_i, with cumulative proportion P_j = F_j/n. The number of observations above the same threshold may be written as G_j = n − F_j. For an exhaustive grouping, Σ f_i = n, Σ p_i = 1, and F_k = n. Rounding the displayed proportions does not change these underlying mathematical relationships.

Constructing classes

The number of classes determines the level of detail in a grouped distribution. After finding the minimum, maximum, and range, lower and upper class limits are established. Each midpoint is the arithmetic mean of its two limits. The interval notation should match the measurement resolution; half-open intervals are also useful for continuously valued observations.

Sturges' Rule is one method for estimating the number of classes from sample size. For equal-width bins, histogram heights can be proportional to the class counts. With unequal widths, density histograms use heights f_i/(n × class width), making each bar's area equal to its relative frequency.

Numerical example and implementation

The December 2014 Frequency Table Generation project implements class construction, frequency counting, and relative and cumulative calculations in C#. Its 120-observation dataset is organized into seven classes using the Sturges approach. The class counts are 3, 11, 32, 32, 25, 13, 4, summing to 120. Probability and Statistics develops the theoretical background in distributions and sampling.