Relative Frequency

Relative frequency is the proportion of observations in a dataset that fall into a given category or class — the frequency of that category divided by the total number of observations, usually written as a fraction, decimal, or percentage. If 14 of 40 surveyed customers choose blue, the relative frequency of blue is 14 divided by 40, which is 0.35 or 35 percent. Where the raw frequency says how many, the relative frequency says what share, and that change of question is what makes counts from datasets of different sizes directly comparable.

Relative frequency removes the dependence on sample size that makes raw counts awkward to compare: 35 blue answers out of 100 and 350 out of 1,000 differ tenfold in frequency but are identical in relative frequency. The calculation has two guaranteed properties that serve as checks. Every relative frequency lies between 0 and 1 (or 0 and 100 percent), and the relative frequencies of all the categories sum to exactly 1 (or 100 percent), apart from small rounding discrepancies when percentages are rounded for display. A column of relative frequencies that sums to 103 percent signals a counting or transcription error, or categories that overlap because respondents could choose more than one answer, which changes what the percentages mean.

The deepest role of relative frequency is as the bridge between observed data and probability, and this is the frequentist interpretation of probability: the probability of an event is the value its relative frequency settles toward as the number of repeated trials grows without limit. Jacob Bernoulli's law of large numbers, published posthumously in 1713, is the first rigorous statement that relative frequency converges on the underlying probability. The practical consequence is that a relative frequency from a finite dataset is an estimate, often called an empirical or experimental probability, carrying sampling error that shrinks roughly with the square root of the number of observations. A coin that lands heads in 6 of 10 tosses has a relative frequency of 0.6, but that says little about the coin; the same ratio over 10,000 tosses would say a great deal.

Relative frequencies appear throughout applied work, often under other names. Polling results, market share, conversion rate, defect rate, and error rate are all relative frequencies of some event among some population. In machine learning, class priors are relative frequencies of each label in the training set, a naive Bayes classifier estimates its probabilities from them, and the predicted class probabilities of a well-calibrated model should match observed relative frequencies among cases given the same prediction. In epidemiology, prevalence is the relative frequency of a condition in a population, and in genetics, allele frequency, the share of a gene's copies that are a particular variant, is the central quantity of population genetics. Relative frequency histograms, with bar heights rescaled to proportions, let datasets of different sizes be drawn on a common axis.

The classic pitfall is ignoring the denominator. A relative frequency is only as meaningful as the base it is taken over, and the same count yields very different percentages depending on whether the base is all respondents, only those who answered, or only those who were eligible. A headline such as "a 50 percent increase" can hide a change from two cases to three. A related error is treating small-sample percentages as precise: 3 of 8 is 37.5 percent, but a figure to one decimal place conveys false confidence for so few observations. Finally, percentages of percentages must not be averaged without weighting by group size.

Relative frequency is distinct from the cumulative frequency, which adds counts progressively, and from the frequency distribution, which is the full listing of every category with its count or share. It is one specific rescaling, count over total, and it is the form in which a frequency distribution most naturally approximates a probability distribution, which is why the term probability mass function describes the theoretical version of exactly the table of relative frequencies that data produce.

Read relative frequency as a count with its sample size divided out: a fraction of the whole that makes datasets comparable, always sums to one, and serves as the empirical stand-in for probability — one that sharpens only as the number of observations behind it grows.