Normal Distribution Empirical Rule Worksheet- Practice Problems
What Is the Empirical Rule (68-95-99.7 Rule)?
The empirical rule describes how data spreads out in a normal distribution. It tells you what percentage of values fall within 1, 2, and 3 standard deviations from the mean. Here's the breakdown:- 68% of data falls within 1 standard deviation
- 95% of data falls within 2 standard deviations
- 99.7% of data falls within 3 standard deviations
Why You Need Practice Worksheets
Reading about the empirical rule feels easy. Actually applying it feels completely different. Most students freeze when they see a problem asking them to find the percentage of values above a certain point or between two boundaries. Worksheets force you to work through problems step by step. You can't fake understanding when you're solving 20 problems on probability boundaries.Empirical Rule Practice Problems with Solutions
Problem 1: Finding Percentages Within Boundaries
A dataset has a mean of 100 and a standard deviation of 15. What percentage of values fall between 85 and 115?
Solution:
85 is exactly 1 standard deviation below the mean (100 - 15 = 85).
115 is exactly 1 standard deviation above the mean (100 + 15 = 115).
According to the empirical rule, 68% of values fall within this range.
Problem 2: Finding Percentages Outside Boundaries
A dataset has a mean of 250 and a standard deviation of 20. What percentage of values fall above 290?
Solution:
290 is exactly 2 standard deviations above the mean (250 + 2(20) = 290).
95% falls between 210 and 290.
That leaves 5% outside this range. Since the distribution is symmetric, 2.5% falls above 290 and 2.5% falls below 210.
Answer: 2.5%
Problem 3: Finding Values Given Percentages
A dataset has a mean of 80 and a standard deviation of 12. What value separates the top 16% of data?
Solution:
68% falls within 1 standard deviation. That leaves 32% outside.
16% falls above +1 SD and 16% falls below -1 SD.
The top 16% starts at +1 SD from the mean.
80 + 12 = 92
Problem 4: Combined Boundaries
A dataset has a mean of 500 and a standard deviation of 25. What percentage of values fall between 450 and 575?
Solution:
450 is 2 SDs below the mean (500 - 2(25)).
575 is 3 SDs above the mean (500 + 3(25)).
This isn't a clean empirical rule problem. Break it into segments:
- 450 to 500: 2.35% (half of 68% minus half of 95%)
- 500 to 525: 34% (half of 68%)
- 525 to 550: 13.5% (half of 95% minus half of 99.7%)
- 550 to 575: 2.35% (half of 99.7% minus outer 0.3%)
Total: 52.2%
Problem 5: Real-World Application
Test scores are normally distributed with a mean of 72 and standard deviation of 8. If 2,000 students took the test, how many scored below 56 or above 88?
Solution:
56 is 2 SDs below the mean (72 - 16).
88 is 2 SDs above the mean (72 + 16).
95% falls between 56 and 88.
5% falls outside this range. Half of 5% = 2.5% on each tail.
2.5% + 2.5% = 5% of 2,000 = 100 students
Quick Reference Table
| Standard Deviations from Mean | Percentage Within Range | Percentage Outside (Each Tail) |
|---|---|---|
| 1 SD | 68% | 16% each |
| 2 SD | 95% | 2.5% each |
| 3 SD | 99.7% | 0.15% each |
How to Use These Practice Problems Effectively
- Don't look at the solutions first. Attempt each problem with a blank sheet of paper.
- Draw the bell curve. Sketch the mean in the center, mark your standard deviations, and shade the relevant areas.
- Check your answers afterward. If you got it wrong, figure out where your reasoning broke down.
- Repeat until it's automatic. Work through all problems twice before moving on.
Common Mistakes to Avoid
Most errors on empirical rule problems come from a few predictable sources:
- Forgetting to halve percentages. The empirical rule gives you the total outside both tails. You need to halve it for one tail.
- Confusing "between" with "above/below." Read the question carefully. "Between" uses the full percentage. "Above" or "Below" uses half.
- Arithmetic errors. Double-check your mean ± standard deviation calculations before touching the percentages.
- Assuming symmetry breaks down. The empirical rule only applies to normal distributions. If the problem doesn't say "normally distributed," you can't use it.
Where to Find More Practice
Look for worksheets that include:
- Problems with different means and standard deviations
- Questions asking for values, not just percentages
- Combined boundary problems like Problem 4 above
- Real-world context problems involving heights, weights, test scores, or measurements
Work through at least 30-50 problems before your exam. The pattern recognition becomes automatic with enough repetition.
When the Empirical Rule Doesn't Apply
The empirical rule only works for approximately normal distributions. If your data is skewed, has outliers, or doesn't follow a bell curve, the 68-95-99.7 breakdown will be wrong. For non-normal data, you'll need Chebyshev's inequality (which gives weaker bounds but works for any distribution) or actual probability calculations. Your instructor will tell you when to assume normality. When in doubt, ask.