Concept

Empirical rule — where it appears

The 68, 95 and 99.7 per cent of a normal distribution lying within one, two and three standard deviations. All three are integrals of a curve whose centre and width are given, so a band drawn at the same multiples of a sample's own estimates holds less.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

What x-bar plus or minus 2 sample standard deviations holds, at n = 10. The content of the band is a random variable. Across 20,000 normal samples of 10 it averages 91.1%, its fifth percentile is 74.7%, and it falls short of 95% on 59.9% of samples. The band that is drawn to show where 95% of the data lies.

Two standard deviations of what

The 95.45% inside two standard deviations is a fact about a curve whose centre and width are given. Drawn from ten observations, the same band holds 91.1% on average and less than 95% on 59.9% of samples — and the average is the reading that hides it.

estimated · Bands

The normal density at sigma = 1.00. The bands hold 68.27%, 95.45%, 99.73% of the mass. Those figures are integrals of the curve drawn, not the memorised 68-95-99.7.

The shape, and where its mass is

68, 95, 99.7 is recited more often than any other set of numbers in the subject. They are integrals of a specific curve, they are worth computing rather than remembering, and the third one is the one people misuse.

normal · Bands

Named alongside it

The objects these essays reach for when they reach for this one.

Normal distribution Standard deviation Coverage Degrees of freedom Estimated variance Normality Plug in estimate Sample size Tail probability Tolerance interval

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