Overview
This lecture explains how to use the mean and standard deviation with the empirical rule to identify unusual data in unimodal, symmetric (normal) distributions.
Normal Distributions and Key Concepts
- A unimodal and symmetric graph is called a normal distribution.
- The mean is the center or typical value in a normal distribution.
- The standard deviation measures how data spreads out from the mean.
The Empirical Rule
- The empirical rule only applies to unimodal, symmetric (normal) distributions.
- About 68% of data falls within one standard deviation of the mean.
- About 95% of data falls within two standard deviations of the mean.
- About 99.7% of data falls within three standard deviations of the mean.
- These percentages are used to determine if a data point is unusual.
Application Example
- Given mean = 134 pounds, standard deviation = 26 pounds (weights of women aged 18–25).
- The mean (134) is placed at the center of the normal curve.
- Tick marks at each standard deviation: 134 ± 26, 160, 186, 212 to the right; 108, 82, 56 to the left.
- 68% of the data is between 108 and 160 pounds (one standard deviation from the mean).
- 95% of the data is between 82 and 186 pounds (two standard deviations from the mean).
Key Terms & Definitions
- Mean — The average value of a data set; the center of a normal distribution.
- Standard Deviation — A measure of the spread of data values around the mean.
- Normal Distribution — A unimodal, symmetric bell-shaped distribution.
- Empirical Rule — States that 68%, 95%, and 99.7% of data lie within 1, 2, and 3 standard deviations of the mean, respectively.
Action Items / Next Steps
- Review the empirical rule and practice finding percentage ranges using mean and standard deviation.
- Complete any assigned homework or example problems related to applying the empirical rule to data sets.