Apply the 68–95–99.7% rule

Ten observations are 8, 8, 9, 9, 10, 10, 11, 11, 12 and 12. Six lie between 8.51 and 11.49. What percentage is that?

These ten observations have mean 10 and sample standard deviation about 1.49. Predict the percentage within one standard deviation. Try k = 1, 2 and 3, counting the dots between the boundaries before reading the totals. Displayed boundaries are rounded to two decimal places

What does the observed data show?

Observed percentages and model estimates

For a supplied data set, calculate the endpoints and count observations inside them. If only an approximately symmetric, bell-shaped distribution is described, the empirical rule estimates the share inside each central interval

1. 1Observed percentage Divide the number inside by the total number of observations, then multiply by 100.
2. 2A central interval The endpoints are equally far below and above the mean: mean minus k standard deviations and mean plus k standard deviations
3. 3Check the shape Symmetry alone is not enough. Do not apply the rule to a strongly skewed, flat or two-peaked distribution

A distribution is symmetric but has two separated peaks. Can symmetry alone justify using the 68–95–99.7% rule?

This is a bell-shaped model, not the ten-point data set. Keep k = 1 and change s from 2 to 10. Then try k = 2 and 3. Compare the shaded area and endpoints. Percentages describe area, not the height of the curve

What changes the estimated percentage in the central shaded interval?

The 68–95–99.7% rule

Use these approximate central percentages only for an approximately symmetric, bell-shaped distribution. The endpoints are mean ± k standard deviations

1. 1One standard deviation About 68% lies between mean − s and mean + s.
2. 2Two standard deviations About 95% lies between mean − 2s and mean + 2s
3. 3Three standard deviations About 99.7% lies between mean − 3s and mean + 3s

Mean

\(\displaystyle 50\)

Standard deviation

\(\displaystyle 5\)

An approximately bell-shaped distribution has mean 50 and standard deviation 5. Use the empirical rule: what percentage lies between 45 and 55?

Identify the interval or side being asked about before calculating

Mean

\(\displaystyle 50\)

Standard deviation

\(\displaystyle 5\)

For the same bell-shaped distribution, what percentage lies between 40 and 60?

Identify the interval or side being asked about before calculating

Mean

\(\displaystyle 50\)

Standard deviation

\(\displaystyle 5\)

For the same bell-shaped distribution, what percentage lies between 35 and 65?

Identify the interval or side being asked about before calculating

Mean (minutes)

\(\displaystyle 40\)

Standard deviation (minutes)

\(\displaystyle 6\)

Central percentage

\(\displaystyle 95\%\)

Bell-shaped delivery times have mean 40 minutes and standard deviation 6 minutes. Find the lower endpoint of the central 95% interval

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Mean (minutes)

\(\displaystyle 40\)

Standard deviation (minutes)

\(\displaystyle 6\)

Central percentage

\(\displaystyle 95\%\)

Find the upper endpoint of that central 95% interval, with mean 40 minutes and standard deviation 6 minutes

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Central percentage

\(\displaystyle 95\%\)

Approximately 95% lies in a central interval. What percentage lies outside it, counting both tails?

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Both tails combined

\(\displaystyle 5\%\)

A symmetric bell-shaped distribution has 5% outside its central 95% interval. What percentage is in just one tail?

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Runner times are approximately bell-shaped with mean 28 minutes and standard deviation 4 minutes. Use the rule to estimate the percentage taking more than 36 minutes

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For approximately bell-shaped runner times with mean 28 minutes and standard deviation 4 minutes, estimate the percentage taking less than 36 minutes

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A student answers 2.5% for the percentage below mean + 2s. What went wrong?

Approximately bell-shaped runner times have mean 28 minutes and standard deviation 4 minutes. Estimate the percentage between 24 and 28 minutes

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For approximately bell-shaped runner times with mean 28 minutes and standard deviation 4 minutes, estimate the percentage between 28 and 36 minutes

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For approximately bell-shaped runner times with mean 28 minutes and standard deviation 4 minutes, estimate the percentage between 24 and 36 minutes

Identify the interval or side being asked about before calculating

Total runners

\(\displaystyle 2000\)

Estimated percentage

\(\displaystyle 81.5\%\)

There are 2000 runners. If about 81.5% finish between 24 and 36 minutes, estimate the number of runners in that interval

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Mean (seconds)

\(\displaystyle 60\)

Standard deviation (seconds)

\(\displaystyle 8\)

One-tail percentage

\(\displaystyle 2.5\%\)

Bell-shaped task times have mean 60 seconds and standard deviation 8 seconds. About 2.5% exceed a threshold. Use the rule to estimate that threshold

Identify the interval or side being asked about before calculating

Mean (seconds)

\(\displaystyle 60\)

Standard deviation (seconds)

\(\displaystyle 8\)

One-tail percentage

\(\displaystyle 2.5\%\)

For approximately bell-shaped task times with mean 60 seconds and standard deviation 8 seconds, about 2.5% fall below a threshold. Estimate that threshold

Identify the interval or side being asked about before calculating

Does the empirical rule guarantee exactly 95 of every 100 observations are within two standard deviations?

Distinguish an observed result from a model estimate

Six recorded durations are 4, 4, 5, 5, 6, 6 seconds. Calculate their mean and sample standard deviation, then count the observations within one and two sample standard deviations. Use inclusive endpoints and unrounded calculations. Report counts and exact fractions before percentages, and draw and inspect a dot plot before considering an empirical-rule estimate

1. 1A separate model Separately assume a bell-shaped normal model for 400 future packing durations, with mean 50 seconds and standard deviation 5 seconds. Using the 68–95–99.7% rule, estimate percentages and counts from 40 to 60 seconds, above 60 seconds and from 45 to 60 seconds. Find the threshold with about 2.5% above it
2. 2Choose and verify technology Choose a spreadsheet or statistical tool and explain the choice. Record inputs, sample-standard-deviation settings, interval inequalities and formulas. Verify raw-data counts manually and model percentages by combining central bands and tails. A cumulative normal function uses a different approximation from the stated empirical rule; identify which calculation the task requests
3. 3Interpret the result Explain why the six observed percentages need not equal 68% and 95%, and why predicted counts are not guarantees. For a separate model group of 50, report the unrounded expected count in the 45–60-second interval, then a nearest-whole-number estimate

Model response: count first, estimate under an assumption

The recorded durations are 4, 4, 5, 5, 6, 6 seconds. A separate bell-shaped model has mean 50 seconds and standard deviation 5 seconds, with groups of 400 and 50. The six records total 30 seconds, giving mean 5 seconds. Deviations −1,−1,0,0,1,1 have squared sum 4, so sample variance is 4/5 second² and sample standard deviation is √(4/5) seconds, approximately 0.8944 seconds

1. 1Observed intervals The one-standard-deviation interval is approximately [4.1056,5.8944] seconds; only the two 5s lie inside. The exact observed share is 2/6=1/3, approximately 33.3%. The two-standard-deviation interval is approximately [3.2111,6.7889]; all six lie inside, giving 100%. Compare unrounded endpoints when counting. A dot plot retains two records at each of 4,5,6; symmetry alone does not establish a bell-shaped model
2. 2Model intervals and counts For the separate model, 40 and 60 are two standard deviations from 50. The central estimate is 95%, giving 380 of 400. Above 60 is one tail: (100−95)/2=2.5%, giving 10. From 45 to 60 combines 34% below the mean and 47.5% above it: 81.5%, giving 326. The upper 2.5% threshold is 50+2×5=60 seconds
3. 3A reproducible raw-data workflow Enter the six observations in A2:A7. Calculate COUNT, AVERAGE and STDEV.S of that range, and store the mean and sample deviation in labelled cells. Calculate mean−k×s and mean+k×s for k=1,2. Use COUNTIFS with >=lower and <=upper criteria, dividing by COUNT for each observed proportion. Check the result by listing the included records and independently verifying the squared-deviation sum 4 and divisor 5
4. 4A reproducible model workflow Store the stated central percentages 68,95,99.7 with k=1,2,3. Calculate endpoints as mean±k×standard deviation, tails as (100−central)/2, and expected counts as percentage×group size/100. Record the model assumption and a labelled sketch with mean 50 and boundaries 40,45,60 seconds. These percentages concern areas, not the curve height. A numerical normal CDF would give slightly different values and does not replace the requested empirical-rule arithmetic
5. 5Limits and rounding For a model group of 50, 81.5% gives expected count 40.75, or about 41 to the nearest whole observation. A fractional expectation is legitimate; an observed count is an integer. Neither 41 nor any of the 400-group estimates is guaranteed. The small raw sample and the assumed packing model are separate examples, so do not substitute the model percentages for the observed counts

The six recorded durations have 100% within two sample standard deviations. What follows?