The Normal Distribution & the Empirical Rule
The normal distribution is the symmetric bell curve; the empirical rule places about \(68\%\), \(95\%\) and \(99.7\%\) of data within \(1\), \(2\) and \(3\) standard deviations of the mean.
Part of the NSW Year 12 Mathematics Advanced course, in the Statistical analysis area of study (Random variables focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
The normal distribution is the ideal bell curve, and the empirical rule gives 68-95-99.7 percent within one, two and three standard deviations. This Year 12 Mathematics Advanced topic (MAV-12-07).
For \(X\sim N(\mu,\sigma^{2})\): about \(68\%\) of values lie within \(1\sigma\), \(95\%\) within \(2\sigma\), and \(99.7\%\) within \(3\sigma\). The curve is symmetric with mean \(=\) median \(=\) mode.
Method
- Find \(\sigma\) (take the square root of the variance).
- Match the values to \(\mu\pm1\sigma,\pm2\sigma,\pm3\sigma\).
- Use symmetry for one-sided proportions.
| \(40,60\) | \(=\) | \(\mu\pm\sigma\Rightarrow 68\%\) |
| \(30,70\) | \(=\) | \(\mu\pm2\sigma\Rightarrow 95\%\) |
| \(\dfrac{100-68}{2}\) | \(=\) | \(16\%\) |
| \(0.16\times1000\) | \(=\) | \(160\) |
Common pitfalls
Frequently asked questions
What is the empirical rule?
For a normal distribution about 68 percent of values lie within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three.
What does N(mu, sigma squared) mean?
A normal distribution with mean mu and variance sigma squared, so the standard deviation is the square root of the second number.
How do you find the percentage above one standard deviation?
By symmetry it is (100 minus 68) over 2, which is 16 percent.
Is the normal curve symmetric?
Yes, it is symmetric about the mean, which equals the median and the mode.