Sample mean (x̄)

How to read it

OpenStax’s Introductory Statistics describes the symbol as “an x with a bar over it (pronounced “x bar”)”, and gives the sample mean as its example of a statistic, a number calculated from a sample, which estimates the population mean μ. The letter under the bar is whichever letter names the data. Penn State’s STAT 500 course notes write x̄ for measurements called x and ȳ for measurements called y, and NIST’s Engineering Statistics Handbook writes the mean of its data as Ȳ. Unicode has no single character for x̄: it is the letter x followed by the combining macron, U+0304.

Also searched as: average symbol, mean symbol, sample mean symbol, x̄, x bar, x bar meaning, x bar symbol.

x̄ and μ

x̄ belongs to the sample and μ to the population. OpenStax names one requirement for x̄ to be a good estimate of μ: the sample must be truly random. The law of large numbers, in OpenStax’s words, says that as samples get larger the sample mean x̄ is very likely to get closer and closer to μ. Fisher himself wrote m for the mean of the normal distribution from 1912 and only switched to μ in 1936, Miller records, but for the sample mean he kept x̄ throughout.

Mean or average?

They are the same thing in everyday use. OpenStax says the words “mean” and “average” are often used interchangeably: the technical term is “arithmetic mean”, and among non-statisticians “average” is commonly accepted for it. NIST’s handbook uses average as a synonym for the mean.

Look-alikes

NIST writes the median of the same data with a tilde, Ỹ, where the mean has a bar, Ȳ. Unicode has two combining marks that draw a line over a letter: the combining macron (U+0304), and the combining overline (U+0305), which Unicode also calls the vinculum and which connects on the left and right, so that several in a row make one unbroken line. The bar is not the hat of p̂ or ŷ.