# Sturges' rule

Sturges' rule is a rule for determining how wide to choose bars (i.e. of the bins) when visually representing data by a histogram. It says the data range should be split into k equally spaced classes where

where  is the ceiling operator (meaning take the closest integer above the calculated value).

Though not stated in Sturges (1926), Herbert Sturges considered a histogram of k bins where the number of data values in the ith bin (i = 0,...k-1) is given by

Summing over all bins we get the total number of data values n:

Equation 1

The binomial expansion identity says:

(replacing q with (1-p) we get the binomial equation). Setting p = q = 1 in Equation 1 we get:

Solving for k we get Sturges' formula:

and then we take the nearest integer above this value. Implicit in Sturges' rule is the assumption of a Normally distributed data set that is being well approximated by a Binomial distribution with probability 0.5 (which gives a symmetric distribution). To see that, the expected number of data points falling into the ith class is, from the binomial probability mass function:

Setting  from above, this reduces to  which is Sturges' idealized histogram. Note: Sturges's paper actually gives a class width w as:

where R is the data range and 3.322 is 1/Log10(2), so R/w gives the formula quoted above for n.