Overview
This lecture introduces the empirical rule for normal (bell-shaped) data distributions, explaining how it helps estimate the spread of data using standard deviations and percentages.
The Empirical Rule (68-95-99.7 Rule)
- The empirical rule applies to normal (bell-shaped) distributions of quantitative data.
- About 68% of data falls within one standard deviation (σ) of the mean (μ).
- About 95% falls within two standard deviations.
- About 99.7% falls within three standard deviations.
- The rule is also called the 68-95-99.7 rule because of these percentages.
Calculating Percentages in Quantitative Data
- For categorical data, percentages are calculated using the count with a characteristic out of the total.
- For quantitative data (which is continuous and infinite), percentages are found using probability density curves.
- The area under the curve represents the proportion or percentage of values in a given range.
Probability Density Curves and Area
- The total area under a probability density curve is 1 (representing 100% of the data).
- The area between intervals on the x-axis (the scale of the variable) gives the percent of data in those intervals.
- For normal data, the areas within each standard deviation are consistent.
Example: Women's Wrist Circumference
- Mean (average) wrist circumference (x̄) = 5.067 inches; standard deviation (s) = 0.331 inches.
- One standard deviation above mean: 5.067 + 0.331 = 5.398 inches.
- Two standard deviations above mean: 5.067 + 2×0.331 = 5.729 inches.
- Three standard deviations above mean: 5.067 + 3×0.331 = 6.060 inches.
- Repeat with subtraction for values below the mean.
Standard Scores (Z-scores)
- A z-score represents how many standard deviations a value is from the mean.
- For values at the mean + 1s, the z-score is 1; at mean + 2s, the z-score is 2; at mean - 2s, it is -2.
- The standard normal distribution shows these percentages using z-scores.
Using the Empirical Rule for Questions
- To find the percent greater than a value (e.g., >5.398 inches), add the areas (proportions) in the "right tail" beyond that value.
- Right tail refers to data values greater than the specified cutoff.
Key Terms & Definitions
- Normal Distribution — a symmetrical, bell-shaped curve representing the distribution of a quantitative variable.
- Empirical Rule — states 68%, 95%, and 99.7% of data lie within 1, 2, and 3 standard deviations of the mean, respectively.
- Standard Deviation (σ or s) — a measure of data variability around the mean.
- Probability Density Curve — a curve showing the distribution of data; area under curve equals probability or proportion.
- Z-score — a standardized value indicating the number of standard deviations a data point is from the mean.
- Standard Normal Distribution — a normal distribution with a mean of 0 and a standard deviation of 1.
Action Items / Next Steps
- Practice applying the empirical rule to sample problems.
- Review how to calculate z-scores.
- Prepare for a future session on using software to calculate precise proportions for normal data.