Calculate z-scores, convert them back to raw values, and use the empirical rule and standard normal percentiles to find probabilities. 15 problems, answer key at the bottom.
Z-Scores and the Normal Distribution
Z-Score Formula
z = (x − μ) ÷ σ
How many standard deviations a value x sits from the mean μ.
Empirical Rule (68-95-99.7)
±1σ: 68% · ±2σ: 95% · ±3σ: 99.7%
The percent of data within 1, 2, and 3 standard deviations of the mean in a normal distribution.
Percentiles (area to the left)
z=1: 84.13% · z=1.5: 93.32% · z=2: 97.72% · z=2.5: 99.38% · z=3: 99.87%
By symmetry, the area to the left of a negative z equals 100% minus the area to the left of the positive z.
Name: _______________________
Date: _______________________
AP Statistics — Z-Score Practice
15 Problems
1. A student scores 85 on a test with mean 75 and standard deviation 5. Calculate the z-score.
2. A data value has a z-score of −1.5 in a distribution with mean 50 and standard deviation 8. Find the actual value.
3. Test scores are normal with mean 70 and sd 10. Using the empirical rule, what percent of scores fall between 60 and 80?
4. Same distribution (mean 70, sd 10). What percent of scores fall between 50 and 90?
5. Same distribution (mean 70, sd 10). What percent of scores fall above 100?
6. Using P(Z < 1) = 0.8413, what percent of data falls above z = 1?
7. Adult men's heights are normal with mean 70 in and sd 3 in. Find the z-score for a man who is 76 inches tall.
8. Using P(Z < 2) = 0.9772, what percentage of men are shorter than 76 inches (from the previous problem)?
9. A z-score of 0 always corresponds to what value in a distribution?
10. Student A scores 82 on an exam with mean 75, sd 6. Student B scores 88 on an exam with mean 80, sd 5. Who performed better relative to their peers?
11. A distribution has mean 500 and sd 100. Find the z-score for a score of 650.
12. Using P(Z < 1.5) = 0.9332, what percentile is a score of 650 (from the previous problem)?
13. In a standard normal distribution, what is P(−1 < Z < 1)?
14. A negative z-score indicates what about a data value relative to the mean?
15. If P(Z < z) = 0.5, what must z equal?
The exam almost never just asks you to compute a z-score — it asks you to compare two values from different distributions, which only a z-score makes possible. A raw score of 88 sounds better than 82, but z-scores can reverse that ranking entirely once you account for each distribution’s own mean and spread.
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