Overview
This lecture explains percentiles in the context of a normal distribution, illustrating how to find both the percentile for a given value and the value for a given percentile using the empirical rule.
Understanding Percentiles
- Percentile represents the percentage of data that falls below a specific value in the dataset.
- The 25th percentile means 25% of data points are below that value.
- Percentiles are often interpreted with context, such as test scores or weights.
Using the Empirical Rule with Percentiles
- The empirical rule describes data distribution in a normal curve: approximately 68% within one standard deviation, 95% within two, 99.7% within three.
- For the weights example, key values on the curve are labeled: 56, 82, 108, 134, 160, 186, and 212 pounds.
Finding the Percentile for a Given Value
- To find the percentile for 160 pounds, consider the area under the curve to the left of 160.
- The mean is the midpoint, so 50% of data is below the mean.
- From the mean to one standard deviation above is 34%; add to 50% for a total of 84% below 160 pounds.
- Thus, 160 pounds is at the 84th percentile.
Finding the Value for a Given Percentile
- To find what weight represents the 16th percentile, shade from the left up to where 16% of the area is covered.
- The cumulative area up to one standard deviation below the mean (108 pounds) equals 16% of the data.
- Therefore, the 16th percentile corresponds to 108 pounds.
Key Terms & Definitions
- Percentile — The percentage of data values that fall below a specific value.
- Empirical Rule — States that for a normal distribution: 68% is within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean.
- Standard Deviation (SD) — A measure of the spread or dispersion of a set of data points.
Action Items / Next Steps
- Practice finding percentiles for specific values using the empirical rule and normal distribution.
- Review how to find values corresponding to specific percentiles on a normal curve.