Empirical Rule (68-95-99.7)
In short: The empirical rule is a memory trick for bell-shaped data: roughly 68%, 95% and 99.7% of the values land inside one, two and three steps of spread around the average.
Algebra 2StatisticsS-ID.A.4
What is Empirical Rule?
The empirical rule is a pencil-and-paper shortcut for bell-shaped data. No table and no calculator are needed to say how much of a data set hugs the middle; you only count how many spreads away from the average you travel.
With center \(\mu\) and spread \(\sigma\), the interval \([\mu – \sigma, \mu + \sigma]\) holds roughly 68% of the data, \([\mu – 2\sigma, \mu + 2\sigma]\) holds roughly 95%, and \([\mu – 3\sigma, \mu + 3\sigma]\) holds roughly 99.7%. Treat these as estimates that work best for symmetric, single-peaked data.
How to spot it
- A problem gives a center and a spread for bell-shaped data and asks what percent of values lie between two marks.
- The marks in the question are whole-number multiples of the spread away from the center, so the empirical rule fits.
- Words such as roughly or about appear, and no table or technology is mentioned.
- A count of individuals, such as how many of 1,200 students, is requested from a percentage.
The method
- Write down \(\mu\) and \(\sigma\), and compute the interval ends \(\mu \pm \sigma\), \(\mu \pm 2\sigma\), \(\mu \pm 3\sigma\).
- Locate the boundaries of the question on those marks.
- Add the matching slices: 34% per side for the first band, 13.5% for the second, 2.35% for the third, 0.15% for the tails.
- Multiply a percentage by the group size when a count is needed.
Empirical Rule: solved example
Problem
Across a district of 1,200 students, heights form a bell centered at 66 inches with a spread of 2.5 inches. Roughly how many students stand between 61 and 71 inches?
Solution
Measure the marks from the center: \(61 = 66 – 2(2.5)\) and \(71 = 66 + 2(2.5)\). Both sit two spreads away, and that window captures about 95% of the students.
Then \(0.95 \times 1200 = 1140\). About 1,140 students fall in that range.
Try it
Question 1
Bell-shaped scores are centered at 100 with spread 15. What percent land above 130?
Show the answer
The mark 130 is two spreads up. The middle 95% leaves 5% for the two tails combined, so each tail holds 2.5%.
Question 2
A battery model lasts 400 hours on average, with spread 20 hours and a bell-shaped pattern. What percent run between 380 and 440 hours?
Show the answer
The mark 380 is one spread below and 440 is two above. Add slices: \(34\% + 34\% + 13.5\% = 81.5\%\).
Empirical Rule vs. Normal Distribution
The curve is the model; the empirical rule is a memory aid about it. The rule tells you three percentages at three distances, and nothing else. For a boundary at 1.4 standard deviations, say, the shortcut does not apply, and you need a z-score and exact areas instead.
Keep going
Lessons
Worksheets and quizzes
- The Normal Distribution – Worksheet
- The Normal Distribution – Quiz
- z-Scores and the Normal Distribution – Worksheet
- z-Scores and the Normal Distribution – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Statistics