A z-score measures how many standard deviations a value is from the mean. It is the foundation of statistical inference, allowing you to convert any normal distribution into a standardized scale where you can find probabilities using a universal normal table or calculator.
The Formula
z = (x - μ) / σ
Where:
- x = the value you're evaluating
- μ (mu) = population mean
- σ (sigma) = population standard deviation
A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative z-scores are below. The magnitude tells you the distance in standard deviations.
Worked Example
A college entrance exam has mean 500 and standard deviation 100. You score 650. What is your z-score?
z = (650 - 500) / 100 = 150 / 100 = 1.5
Your score is 1.5 standard deviations above the mean. Using the standard normal table, P(z ≤ 1.5) ≈ 0.9332, meaning about 93.32% of test-takers scored below you.
Using Z-Score Tables
After calculating z, you look up its probability in a standard normal table, which gives cumulative probabilities P(Z ≤ z). Tables show:
- One-tailed probabilities: P(Z ≤ z) or P(Z ≥ z)
- Two-tailed probabilities: useful for confidence intervals and hypothesis tests
For example, z = 1.96 corresponds to P(Z ≤ 1.96) ≈ 0.975. The area in both tails beyond z = ±1.96 is 0.05, which is why 1.96 is the critical value for 95% confidence intervals.
Common Z-Score Cutoffs
| Z-Score | Cumulative Probability | Percentile |
|---|---|---|
| -3 | 0.0013 | 0.13th |
| -2 | 0.0228 | 2.28th |
| -1 | 0.1587 | 15.87th |
| 0 | 0.5000 | 50th |
| 1 | 0.8413 | 84.13th |
| 2 | 0.9772 | 97.72nd |
| 3 | 0.9987 | 99.87th |
When to Use
Z-scores are essential for:
- Comparing values from different distributions
- Finding probabilities using the normal distribution
- Identifying outliers (usually |z| > 3)
- Hypothesis testing and confidence intervals
- Standardizing test scores
Tips
Z-scores only work for normally distributed data. If your distribution is severely skewed or has heavy tails, z-scores will be misleading. Also, remember the difference between z (population parameter) and t (sample statistic) — use z when σ is known, t when you estimate it from the sample.
Use our Z-Score Calculator to convert scores to z-scores and find probabilities instantly.