The Empirical Rule (68-95-99.7)
Master the empirical rule (68-95-99.7) for NSW Year 12 Mathematics Standard 2. For normally distributed data, about 68% of the values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three - the fast way to read percentages off a bell curve.
You will learn to find the interval that holds a given percentage, work out the percentage in a one-sided or across-the-mean region using the 34%, 13.5% and 2.35% segments, and estimate how many of a batch fall in a range with expected frequency - core Standard 2 skills for heights, masses, test scores and machine-fill volumes.
Theory
The empirical rule (68-95-99.7) tells you how normally distributed data spreads around the mean. This Year 12 Standard 2 (NSW) guide shows how to find the percentage of data within 1, 2 or 3 standard deviations, handle more-than and less-than regions, and estimate how many of a batch fall in a range.
The empirical rule (also called the 68-95-99.7 rule) describes how normally distributed data spreads around the mean. For a bell-shaped data set, a fixed percentage of the values falls within each whole number of standard deviations (SD) of the mean.
About 68% of the data lies within 1 SD of the mean, about 95% within 2 SD, and about 99.7% within 3 SD. Because the curve is symmetric, each half breaks into segments of 34% (mean to 1 SD), 13.5% (1 to 2 SD) and 2.35% (2 to 3 SD).
This Year 12 Standard 2 (NSW) tool lets you read off the percentage of data in a region, work out a more-than or less-than percentage, or estimate how many of a batch fall in a range \((\text{percentage}\times n)\) - all without a \(z\)-table, as long as the boundaries sit at whole standard deviations.
Take the mean \(\mu\) and standard deviation \(\sigma\). The middle band that reaches \(k\) standard deviations each side of the mean is:
The percentage in each segment (one side of the mean) is fixed:
| Segment | mean to 1 SD | 1 to 2 SD | 2 to 3 SD |
|---|---|---|---|
| Percentage | 34% | 13.5% | 2.35% |
Each half of the curve holds \(50\%\), so for a one-sided region start from the mean and add or subtract segments:
Using the empirical rule
- Write down the mean \(\mu\) and standard deviation \(\sigma\).
- Count the SDs. For each boundary work out \(k = \dfrac{\text{value}-\mu}{\sigma}\); it should be a whole number \(1\), \(2\) or \(3\).
- Sketch a bell curve and mark the region you want.
- Add the segments. Use \(34\%\), \(13.5\%\), \(2.35\%\) for each band, and \(50\%\) for a whole half; for a one-sided region start at the mean.
- For a count, multiply the percentage by the number of items \(n\).
The middle \(95\%\) reaches two SD each side of the mean.
| \(2\,\text{SD}\) | \(=\) | \(2 \times 0.4 = 0.8\) |
| \(\text{lower}\) | \(=\) | \(3.4 - 0.8 = 2.6\) |
| \(\text{upper}\) | \(=\) | \(3.4 + 0.8 = 4.2\) |
About \(95\%\) have a mass between 2.6 kg and 4.2 kg.
\(35 = 30 + 1\times 5\), so \(35\) min is one SD above the mean.
| \(\text{within }1\,\text{SD}\) | \(=\) | \(68\%\) |
| \(\text{both tails}\) | \(=\) | \(100\% - 68\% = 32\%\) |
| \(\text{one tail}\) | \(=\) | \(\dfrac{32\%}{2} = 16\%\) |
About 16% of the commutes take more than \(35\) minutes.
Find how many SDs each boundary is from the mean, then add the segments.
| \(190\) | \(=\) | \(200 - 1\times 10\ \ (1\,\text{SD below})\) |
| \(220\) | \(=\) | \(200 + 2\times 10\ \ (2\,\text{SD above})\) |
| \(\text{total}\) | \(=\) | \(34\% + 34\% + 13.5\%\) |
| \(\) | \(=\) | \(81.5\%\) |
About 81.5% of the cups hold between \(190\) and \(220\) mL.
\(68\) and \(92\) are two SD each side of the mean, so use \(95\%\).
| \(68\) | \(=\) | \(80 - 2\times 6,\ \ 92 = 80 + 2\times 6\) |
| \(\text{within }2\,\text{SD}\) | \(=\) | \(95\%\) |
| \(\text{count}\) | \(=\) | \(0.95 \times 600 = 570\) |
About 570 of the \(600\) plants are between \(68\) and \(92\) cm tall.
Common pitfalls
Frequently asked questions
What is the 68-95-99.7 rule?
It is the empirical rule for normally distributed data: about 68% of the values lie within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations.
How do I find the range that holds 95% of the data?
Go two standard deviations each side of the mean. The 95% range is from mean minus 2 times SD up to mean plus 2 times SD. For 68% use one SD each side, and for 99.7% use three SD each side.
What percentage is between the mean and one standard deviation above it?
About 34%. The 68% within one SD is split evenly by the symmetric curve, so each side holds 34%. The next band (1 to 2 SD) holds 13.5% and the outer band (2 to 3 SD) holds 2.35%.
How do I work out a 'more than' or 'less than' percentage?
A one-sided region splits at the mean, where 50% lies on each side. Start from 50% and add or subtract the 34%, 13.5% and 2.35% segments up to your boundary. For example, more than one SD above the mean is (100% minus 68%) divided by 2, which is 16%.
When can I not use the empirical rule?
Only when the boundary is a whole number of standard deviations from the mean (1, 2 or 3 SD). If the value falls between whole SDs you must calculate a z-score and use a z-table or calculator instead.
How do I find how many items are expected in a range?
Work out the percentage of data in the range using the empirical rule, then multiply by the total number of items n. For example, 95% of 600 items is 0.95 times 600, which is 570 items.