Key takeaway: The empirical rule says that in a bell-shaped distribution about 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean.
Grade 11 – Algebra 2S-ID.A.4Statistics and Data
Definition of Empirical Rule
The empirical rule is a quick way to estimate proportions for data that follow a bell curve. Instead of consulting a table, remember three landmarks measured outward from the mean in steps of one standard deviation.
Within one step on either side you find roughly two thirds of the data. Within two steps you find nearly everything, and within three steps you find almost literally all of it. Values beyond three steps are so rare that they are often treated as outliers. The rule is also called the 68-95-99.7 rule, after its three percentages.
Formula and notation
For a normal distribution with mean \(\mu\) and standard deviation \(\sigma\):
\[P(\mu-\sigma\le X\le\mu+\sigma)\approx 0.68,\quad P(\mu-2\sigma\le X\le\mu+2\sigma)\approx 0.95,\quad P(\mu-3\sigma\le X\le\mu+3\sigma)\approx 0.997.\]
Key facts
- Only about 0.3% of values fall beyond three standard deviations, split evenly between both tails.
- The empirical rule describes approximately bell-shaped data; it is unreliable for skewed data.
- Because of symmetry, half of each percentage lies below the mean and half above.
- The exact figures are 68.27%, 95.45% and 99.73%.
Empirical Rule example
Loaves from a bakery weigh 500 grams on average, with a standard deviation of 12 grams, and the weights are roughly bell-shaped. Between what weights do 95% of loaves fall, and what share of loaves weigh more than 536 grams?
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For 95%, go two deviations out: \(500\pm 2(12)=500\pm 24\), so 476 to 524 grams.
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Find where 536 sits: \(536-500=36=3\times 12\), exactly three deviations above the mean.
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About 99.7% lie within three deviations, so 0.3% lie outside, and half of that is above: \(0.3\%: 2=0.15\%\).
Ninety-five percent weigh 476 to 524 grams, and only about 0.15% exceed 536 grams.
Common mistake
Watch out
Gabe says that 95% of the loaves lie within one standard deviation, 476 to 524, because the interval is 24 grams away.
Why it is wrong
One standard deviation here is 12 grams, so 488 to 512 holds about 68%. The 24-gram distance is two standard deviations. Before using the empirical rule, divide each distance by the standard deviation to count steps.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Statistics and Data