Z-Score Calculator
Calculate the z-score from a value, or find the value from a z-score. Includes percentile and probability. See also Standard Deviation Calculator and Percentile Calculator.
How to Calculate Z-Score
A z-score (standard score) tells you how many standard deviations a value is from the mean. A positive z-score means the value is above the mean; a negative z-score means it is below. Z-scores are used to compare values from different distributions and to find percentiles in a normal distribution.
Z-Score Formulas
Find Z-Score: z = (x − μ) / σ
Find Value: x = μ + z × σ
Where: x = value, μ = mean, σ = standard deviation
Example Calculation
A student scores 85 on a test with mean 75 and std dev 10.
z = (85 − 75) / 10 = 10 / 10 = 1.0
Percentile ≈ 84.13% (the student scored better than ~84% of students)
Z-Score Reference Table
| Z-Score | Area (Left) | Percentile |
|---|---|---|
| -3.0 | 0.0013 | 0.13% |
| -2.5 | 0.0062 | 0.62% |
| -2.0 | 0.0228 | 2.28% |
| -1.5 | 0.0668 | 6.68% |
| -1.0 | 0.1587 | 15.87% |
| -0.5 | 0.3085 | 30.85% |
| 0.0 | 0.5000 | 50.00% |
| 0.5 | 0.6915 | 69.15% |
| 1.0 | 0.8413 | 84.13% |
| 1.5 | 0.9332 | 93.32% |
| 2.0 | 0.9772 | 97.72% |
| 2.5 | 0.9938 | 99.38% |
| 3.0 | 0.9987 | 99.87% |
Solved Examples
Example 1: Comparing Test Scores Across Exams
A student scores 78 on a biology exam (class mean = 72, SD = 8) and 85 on a chemistry exam (class mean = 80, SD = 10). On which exam did the student perform relatively better?
Biology Z = (78 - 72) / 8 = 6/8 = 0.75
Chemistry Z = (85 - 80) / 10 = 5/10 = 0.50
Answer: The biology z-score (0.75) is higher than the chemistry z-score (0.50), so the student performed relatively better on biology despite the lower raw score. Z-scores allow comparison across different scales.
Example 2: Identifying Outliers
A factory produces widgets with mean weight 500g and standard deviation 12g. A widget weighs 540g. Is it an outlier?
Z = (540 - 500) / 12 = 40/12 = 3.33
|Z| = 3.33 > 3 (common outlier threshold)
Answer: With a z-score of 3.33, this widget is more than 3 standard deviations from the mean. Under the empirical rule, only 0.27% of data falls beyond 3 SD. This widget should be flagged for quality investigation.
Example 3: Finding Raw Score from Z-Score
SAT scores have mean 1060 and standard deviation 195. What score corresponds to the 90th percentile (z = 1.282)?
X = mean + z x SD
X = 1060 + 1.282 x 195
X = 1060 + 250 = 1310
Answer: A score of 1310 corresponds to the 90th percentile. This means 90% of test-takers score below 1310.
Practice Questions
Question 1
Adult male heights are normally distributed with mean 175 cm and SD 7 cm. What is the z-score for a man who is 168 cm tall?
Answer: Z = (168 - 175)/7 = -7/7 = -1.0. He is exactly one standard deviation below the mean, placing him at approximately the 16th percentile.
Question 2
A dataset has mean 50 and SD 5. What percentage of values fall between z = -1.5 and z = 2.0 (assume normal distribution)?
Answer: P(Z < 2.0) = 0.9772. P(Z < -1.5) = 0.0668. P(-1.5 < Z < 2.0) = 0.9772 - 0.0668 = 0.9104 or 91.04%.
Question 3
Two athletes: Runner A finishes a marathon in 3:15 (mean 3:30, SD 0:12). Swimmer B swims 1500m in 18:00 (mean 19:30, SD 1:00). Who performed better relative to their sport?
Answer: Runner A: Z = (195 - 210)/12 = -1.25. Swimmer B: Z = (18 - 19.5)/1 = -1.5. Lower times are better, so more negative z-scores indicate better performance. Swimmer B (z = -1.5) performed relatively better than Runner A (z = -1.25).
Common Mistakes
Confusing z-score with probability
A z-score of 1.96 does not mean 1.96% probability. You must look up the z-score in a standard normal table or use a calculator. Z = 1.96 corresponds to the 97.5th percentile (P = 0.975).
Using z-scores with non-normal data
Z-scores can be calculated for any data, but converting z-scores to probabilities assumes normality. For heavily skewed data, a z-score of 2 may not correspond to the 97.7th percentile as it would for normal data.
Forgetting the sign indicates direction
A negative z-score means below the mean, and positive means above. When comparing performance where lower is better (race times, error rates), a more negative z-score is actually better performance.
Key Takeaways
- Z-score = (X - mean) / SD tells you how many standard deviations a value is from the mean.
- Z-scores standardize data to a common scale, enabling comparison across different units and distributions.
- For normal distributions: about 68% of z-scores fall between -1 and 1, 95% between -2 and 2, and 99.7% between -3 and 3.
- Z-scores beyond +/-3 are considered extreme outliers in most applications.
- To convert back to raw score: X = mean + (Z x SD).
Frequently Asked Questions
What does a z-score of 0 mean?
A z-score of 0 means the value is exactly equal to the mean. It is at the 50th percentile.
What is a "good" z-score?
It depends on context. In testing, a z-score of 1.0 or higher means above average. In quality control, values within ±2 standard deviations (z between −2 and 2) are typically considered normal.
Can z-scores be negative?
Yes. A negative z-score means the value is below the mean. For example, z = −1.5 means the value is 1.5 standard deviations below the mean.
What is the empirical rule (68-95-99.7)?
In a normal distribution, about 68% of values fall within ±1 standard deviation, 95% within ±2, and 99.7% within ±3 standard deviations of the mean.