Your statistics worksheet is due tomorrow and every question looks the same: a mean, a standard deviation, a number, and "find the z-score" or "find the probability". The formula itself is short: z = (x − μ) / σ. The hard part is knowing which version to use and what to do with the answer. Below are six z-score example problems with full solutions. They move from the basic calculation to percentiles, comparisons, probabilities between two values, working backwards, and sample means. Each one shows every step, so you can match it to your own question.
How z-scores work
A z-score measures distance from the mean in units of standard deviation. A z-score of 0 is exactly average. A z-score of +2 is two standard deviations above average, and −1 is one below.
There are two formulas you'll meet:
- Single value: z = (x − μ) / σ
- Sample mean: z = (x̄ − μ) / (σ / √n), where σ / √n is the standard error
Once you have z, a standard normal table (or the normal CDF on your calculator) gives the area to the left, which is the proportion of values below it. As OpenStax Introductory Statistics explains, in a normal distribution about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. That "empirical rule" is a quick way to sanity-check your answers.
All table values below are rounded to four decimal places.
Example 1: Basic z-score and percentile
Problem. A class's exam scores have a mean of 70 and a standard deviation of 8. Maya scored 82. What is her z-score, and roughly what percentile is she in?
Solution.
- Write down the values: x = 82, μ = 70, σ = 8.
- Subtract the mean: 82 − 70 = 12.
- Divide by the standard deviation: 12 / 8 = 1.5.
- Look up z = 1.50 in the table: 0.9332.
Answer. z = 1.5. About 93.32% of scores are below Maya's, so she sits around the 93rd percentile.
Check. A z of 1.5 is between 1 and 2 standard deviations above the mean. By the empirical rule, that should land between the 84th and 97.5th percentiles, and it does.
Example 2: A negative z-score
Problem. Daily commute times in a city have a mean of 30 minutes and a standard deviation of 6 minutes. Today's commute took 22 minutes. Find the z-score and the proportion of commutes shorter than this.
Solution.
- x = 22, μ = 30, σ = 6.
- 22 − 30 = −8.
- −8 / 6 = −1.33 (rounded to two decimals for the table).
- Table value for z = −1.33: 0.0918.
Answer. z ≈ −1.33. Only about 9.18% of commutes are shorter.
Common slip. Students often drop the minus sign. The negative tells you the value is below the mean, and it changes which side of the table you read.
Example 3: Comparing scores from different tests
Problem. Sam scored 78 on a maths test (mean 65, SD 10) and 85 on an English test (mean 80, SD 4). On which test did Sam do better relative to the class?
Solution.
| Test | x | μ | σ | z = (x − μ) / σ |
|---|---|---|---|---|
| Maths | 78 | 65 | 10 | (78 − 65) / 10 = 1.30 |
| English | 85 | 80 | 4 | (85 − 80) / 4 = 1.25 |
Answer. The maths score has the higher z-score (1.30 vs 1.25), so Sam did slightly better in maths relative to classmates, even though the raw English score is higher.
Why this matters. Raw scores on different scales can't be compared directly. Z-scores put them on the same scale, which is the main reason the idea exists.
Example 4: Probability between two values
Problem. A brand of battery lasts a mean of 500 hours with a standard deviation of 40 hours, and lifetimes are approximately normal. What proportion of batteries last between 450 and 550 hours?
Solution.
- Lower z: (450 − 500) / 40 = −50 / 40 = −1.25.
- Upper z: (550 − 500) / 40 = 50 / 40 = 1.25.
- Table: P(Z ≤ 1.25) = 0.8944 and P(Z ≤ −1.25) = 0.1056.
- Subtract: 0.8944 − 0.1056 = 0.7888.
Answer. About 78.9% of batteries last between 450 and 550 hours. (Software that skips rounding the table gives 0.7887. Either is accepted in most courses.)
Shortcut. Because the interval is symmetric around the mean, you can also use 1 − 2 × 0.1056 = 0.7888.
Example 5: Working backwards from a percentile
Problem. Using the exam from Example 1 (mean 70, SD 8), what score do you need to be in the top 10%?
Solution.
- The top 10% means 90% of scores are below the cut-off, so you need the z with an area of 0.90 to its left.
- Search the body of the table for 0.9000. The closest value is 0.8997 at z = 1.28. (A calculator's inverse normal gives 1.2816.)
- Rearrange the formula: x = μ + zσ.
- x = 70 + 1.2816 × 8 = 70 + 10.25 = 80.25.
Answer. You need about 80.3 or higher to be in the top 10%. With z = 1.28 from the table, you get 80.24, which rounds to the same answer.
Common slip. Using 0.10 instead of 0.90. If you get a negative z for a "top" question, you've read the wrong tail.
Example 6: Z-score for a sample mean
Problem. A machine fills bags of rice with a mean of 500 g and a standard deviation of 12 g. A quality check weighs a random sample of 36 bags and finds an average of 503 g. How likely is a sample mean of 503 g or more if the machine is working normally?
Solution.
- This is about a sample mean, so use the standard error: σ / √n = 12 / √36 = 12 / 6 = 2.
- z = (503 − 500) / 2 = 1.5.
- Table: P(Z ≤ 1.5) = 0.9332, so P(Z ≥ 1.5) = 1 − 0.9332 = 0.0668.
Answer. There's about a 6.7% chance of a sample mean this high or higher by chance alone.
Why the standard error? Averages vary less than single values. The central limit theorem says the spread of sample means is σ / √n, and with n = 36 the sampling distribution is close to normal. Dividing by 12 instead of 2 would give z = 0.25, which is a very different (and wrong) answer.
How to do your own
Every z-score problem follows the same five steps. Run through them on your own question.
1. Name the parts
Write x (or x̄), μ, σ and, if it's a sample, n. Underline what the question asks: a z-score, a probability, a percentile or a raw value.
2. Pick the formula
Single value: divide by σ. Sample mean: divide by σ / √n. Finding x from a percentile: use x = μ + zσ.
3. Calculate z to two decimals
Most printed tables go to two decimal places.
4. Read the right area
Tables give the area to the left. For "greater than", subtract from 1. For "between", subtract the two left areas.
5. Sense-check
Is the sign right? Does the answer fit the 68–95–99.7 rule? A probability can never be above 1 or below 0.
When you're stuck between two methods, sketch a bell curve and shade the area the question asks about. Most mistakes become obvious once the picture is in front of you.
To memorise the formulas before a test, it helps to turn each one into a question-and-answer card and test yourself over several days rather than re-reading the night before. There are more practice guides like this one on the Maths Problem blog.
How Maths Problem can help
When you've worked a problem and want to check it, Maths Problem lets you photograph the question, whether handwritten, printed or from a textbook, and see a step-by-step solution with each formula explained in plain English. It covers descriptive statistics, normal distributions, probability and hypothesis testing, so the same app works when your course moves on from z-scores to t-tests and confidence intervals. It can also turn a topic into flashcards for revising formulas. Use it to check your method and learn from the steps, not to replace your own working, and follow your school's rules on using AI for coursework. It's available on iPhone.
Frequently asked questions
What is the formula for a z-score?
For a single value, z = (x − μ) / σ, where x is the value, μ is the mean and σ is the standard deviation. For a sample mean, divide by the standard error instead: z = (x̄ − μ) / (σ / √n). The result is how many standard deviations the value sits above or below the mean.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. For example, z = −1.33 means the value is 1.33 standard deviations below average. The sign only shows direction; the size of the number shows how unusual the value is.
What is a good z-score?
It depends on the question. For a test score, a positive z-score means above average, and z = 2 is higher than about 97.7% of a normal distribution. For spotting outliers, many courses treat values beyond about ±2 or ±3 as unusual.
How do I turn a z-score into a percentile?
Look up the z-score in a standard normal table, or use a calculator's normal CDF function. The value you get is the proportion of the distribution below that z-score. Multiply by 100 for the percentile. For z = 1.5, the table gives 0.9332, about the 93rd percentile.
When should I not use a z-table?
The z-table assumes the data, or the sampling distribution of the mean, is approximately normal. With heavily skewed data and a small sample, or when the population standard deviation is unknown and the sample is small, your course will usually ask you to use a t-distribution instead.
Conclusion
Z-score problems look different on the page, but they all come back to one idea: how far a value is from the mean, measured in standard deviations. Find z with (x − μ) / σ, or with the standard error for a sample mean, then use the table to turn it into a probability or percentile. Work backwards with x = μ + zσ when you're given a percentile. Write out each step, sketch the curve, and check your answer against the 68–95–99.7 rule. With practice, these questions become some of the quickest marks on a statistics exam.