Z-Score Practice Problems: 10 Examples With Full Solutions
Reading about z-scores is easy. Doing them under exam pressure is where marks leak. Below are ten problems in the order teachers actually ask them — plain forward calculations first, then reverse problems, probabilities, and one comparison question that catches everyone.
Try each before reading the solution. Every solution shows all the arithmetic, because "and then you get z = 1.5" is where students get abandoned. Check any answer with the z score calculator or the z score table.
Problems 1–3: the forward calculation
Problem 1. A quiz has mean 70 and standard deviation 10. You score 85. Find your z-score and percentile. Solution: z = (85 − 70) / 10 = 15 / 10 = 1.50. Table value for 1.50 is 0.9332 → 93.32nd percentile. You beat about 93% of the class.
Problem 2. Same quiz (mean 70, SD 10). Another student scores 62. Solution: z = (62 − 70) / 10 = −8 / 10 = −0.80. Look up the positive twin: 0.80 → 0.7881, so the mirror gives 1 − 0.7881 = 0.2119 → 21.19th percentile. Below average, but well within the normal range — nothing alarming.
Problem 3. Adult heights in a group have mean 68 inches and SD 3 inches. How unusual is someone 72 inches tall? Solution: z = (72 − 68) / 3 = 4 / 3 = 1.33. Table: 1.33 → 0.9082 → 90.82nd percentile. Taller than about 91% of the group — noticeably tall, not remarkably so.
Problems 4–5: working backwards
Problem 4. On a test with mean 500 and SD 100, a student has z = 2.25. What was the raw score? Solution: Reverse the formula: x = μ + zσ = 500 + (2.25 × 100) = 500 + 225 = 725. (That's the 98.78th percentile, for the curious — z of 2.25 leaves only about 1.22% above.)
Problem 5. Find the z-score that marks the 97.5th percentile. Solution: Area = 0.9750. Search the table body for 0.9750 → row 1.9, column 0.06 → z = 1.96. This is the famous one: the two-tailed 95% cutoff, because 2.5% sits in the tail above it and 2.5% below −1.96.
Problems 6–7: probabilities
Problem 6. For a standard normal variable, what is the probability of a value above z = 1.5? Solution: The table gives the left side: 0.9332. Above means the complement: 1 − 0.9332 = 0.0668, about 6.68%. The most-missed step in all of z-score work is forgetting this subtraction — see the other traps in common z-score mistakes.
Problem 7. What proportion of values fall between z = −1 and z = +1? Solution: P(Z ≤ 1) − P(Z ≤ −1) = 0.8413 − 0.1587 = 0.6827, or 68.27%. Recognize it? That's the middle band of the empirical rule, derived from the table instead of memorized.
Problems 8–9: real scales
Problem 8. An IQ score of 115 (scale mean 100, SD 15). Find z and the percentile. Solution: z = (115 − 100) / 15 = 15 / 15 = 1.00 → 84.13th percentile. One clean standard deviation above average.
Problem 9. A score of 78 on a hard exam with mean 84 and SD 6. Solution: z = (78 − 84) / 6 = −6 / 6 = −1.00 → 15.87th percentile. Notice the raw score (78) looks fine while the standing is low — the exam was easy for everyone else. Raw numbers lie; z-scores contextualize.
Problem 10: the comparison (the one that catches everyone)
Problem. Ana scored 72 in math, where the class mean was 65 and SD was 7. Ben scored 82 in English, where the mean was 70 and SD was 8. Who performed better relative to their class?
Solution: Raw scores can't answer this — different tests, different scales. - Ana: z = (72 − 65) / 7 = 7 / 7 = 1.00 (84.13th percentile) - Ben: z = (82 − 70) / 8 = 12 / 8 = 1.50 (93.32nd percentile)
Ben's relative performance was stronger, even though the raw gap between their scores (10 points) tells you nothing on its own. Comparing across groups is the entire reason the z score formula exists.
Finished all ten? Then you can handle the standard exam repertoire: forward, backward, tails, bands, and comparisons. If one type kept tripping you, redo just that pair tomorrow — spacing the retry is what makes it stick.
Frequently asked questions
How can I check my z-score answers without a solutions page?
Use the sign-and-size sanity check first: above the mean must be positive, and answers beyond ±3 should make you suspicious. Then verify the percentile in a z-table or calculator. If the z is right but the percentile looks wrong, the error is almost always in the table lookup, not the formula.
Do practice problems use population or sample standard deviation?
Most textbook problems hand you the standard deviation directly, so it's simply σ in the formula — no decision needed. When a problem gives raw data and says "sample," divide by n − 1 when computing the standard deviation. The z-score formula itself doesn't change either way.
Why do some problems ask for probability instead of percentile?
They're the same quantity at different scales. A percentile of 93.32 is a left-tail probability of 0.9332. Problems phrase it as probability when they want a decimal (and often a tail statement like "the probability of scoring above"), and as percentile when they want a rank. Convert by multiplying or dividing by 100.
What if a problem gives the percentile and asks for the score?
Work backwards in two moves: convert the percentile to a z-score using the inverse table lookup, then reverse the formula with x = μ + zσ. Problem 5 above shows the first move; a cutoff example — 90th percentile, mean 70, SD 8 — gives z = 1.28 and x ≈ 80.2.