What Is the Optimal Bin Size of a Histogram: An Informal Description

A natural way to estimate the probability density function of an unknown distribution from the sample of data points is to use histograms. The accuracy of the estimate depends on the size of the histogram’s bins. There exist heuristic rules for selecting the bin size. In this paper, we show that these rules indeed provide the optimal value of the bin size. 1 Formulation of the Problem Need to estimate pdfs. One of the most frequent ways to describe a probability distribution is by specifying its probability density function (pdf) ρ(x) def = dp dx = lim h→0 Prob(X ∈ [x, x+ h]) h . In many practical situations, all we know about a probability distribution is a sample of data points corresponding to this distribution. How can we estimate the pdf based on this sample? Enter histograms. A natural way to estimate the limit when h tends to 0 is to consider the value of the ratio corresponding to some small h: ρ(x) ≈ Prob(X ∈ [x, x+ h]) h . To use this expression, we need to approximate the corresponding probabilities Prob(X ∈ [x, x + h]). By definition, the probability of an event is the limit of this event’s frequency when the number of data points increases. In particular, Prob(X ∈ [x, x+ h]) = lim n→∞ n([x, x+ h]) n ,