Frequency Tables

A precise guide to frequency, relative and cumulative frequency tables, class intervals, and validation.

Touching bins show frequencies across adjacent numeric intervals. A histogram uses adjacent equal-width bins on value-interval and frequency axes. One highlighted bin makes its interval and height easier to compare with neighbouring intervals.

Precise definition

A frequency table lists values or categories with their counts. Relative frequency is $f_i/n$ and percentage is $100f_i/n$. For ordered categories or numerical classes, cumulative frequency adds counts up to and including a boundary.

Notation and mathematical language

For grouped quantitative data, class intervals must be mutually exclusive and collectively cover the intended range. Notation such as $[10,20)$ includes 10 but excludes 20. Class width, boundaries, and treatment of endpoints must be explicit.

Conceptual picture

A table is a lossless summary for categorical counts but grouping numerical data sacrifices exact values within intervals. Cumulative columns reveal thresholds and approximate percentiles, while relative columns allow fair comparison across different sample sizes.

Fully worked example

Interpretation and application

Frequency tables support survey summaries, quality control, and distribution plots. Group choices can conceal clusters, gaps, or extreme observations, so retain raw data and document bins.