Definition: The normal distribution is a symmetric, bell-shaped model described by its mean and standard deviation, where about 68, 95, and 99.7 percent of values fall within 1, 2, and 3 standard deviations.
What is a normal distribution?
It is a smooth bell curve centered at the mean, symmetric on both sides, with spread set by the standard deviation. Many natural measurements, such as heights and measurement errors, follow it closely.
What is the empirical rule?
- About 68% of values lie within 1 standard deviation of the mean.
- About 95% lie within 2 standard deviations.
- About 99.7% lie within 3 standard deviations.
Worked example: adult heights have mean 64 inches and standard deviation 2.5 inches. About 68% fall between 61.5 and 66.5 inches. For a height above 69 inches, z = (69 – 64)/2.5 = 2. About 95% lie within two standard deviations, so the remaining 5% splits evenly and roughly 2.5% are taller than 69 inches.
What mistake occurs?
Using the rule on skewed data. Mathbot suggests drawing a histogram first. Also, the area under the whole curve is 1, so probabilities are areas, not heights.