Z-Score Calculator
A z-score measures how many standard deviations a value is from the mean: z = (x − μ) / σ. A z-score of 1.5 means the value sits 1.5 standard deviations above average, roughly the 93rd percentile.
Enter a value, the mean, and the standard deviation, get the z-score, with steps and plain-English meaning.
How to calculate a z-score
Enter the value (x), the distribution’s mean (μ), and its standard deviation (σ), then press Calculate. The calculator applies z = (x − μ) ÷ σ, shows the substitution and arithmetic, and translates the result into plain English, how unusual the value is.
The standard deviation must be positive. A z-score of 0 means “exactly average”; positive means above the mean, negative below.
Beyond the number itself, the calculator explains what the score means, typical, unusual, or outlier, using the empirical rule.
Z-score formula and interpretation
A z-score answers: “how many standard deviations is this value from the mean?” Subtract the mean to get the raw distance, then divide by the standard deviation to express that distance in “typical spread” units. This standardization is what makes values from different distributions comparable, a z of 1.5 means the same relative standing whether the data is test scores or heights.
For roughly normal data, the empirical rule applies: ~68% of values fall within z = ±1, ~95% within ±2, and ~99.7% within ±3. Beyond ±2 is “unusual”; beyond ±3 is outlier territory.
Pro tips
Need the SD? compute it first with the data set, never guess. Comparing across tests: convert every score to z before ranking; raw points lie when tests differ in difficulty. z vs t: with small samples (n < 30) and unknown population SD, statisticians use t-scores instead, same idea, heavier tails.
Worked example
Example: test score 85, mean 70, SD 10
Step 1, Formula: z = (x − μ) ÷ σ.
Step 2, Substitute: z = (85 − 70) ÷ 10.
Step 3, Calculate: z = 15 ÷ 10 = 1.5.
Interpretation: the score is 1.5 standard deviations above the mean, better than roughly 93% of a normal distribution. Solid, but not freakish.
More examples: x=60, μ=70, σ=10 → z=−1.0; x=100, μ=70, σ=15 → z=2.0 (unusual).
Common mistakes
- Using variance instead of SD: the formula needs σ (standard deviation), not σ² (variance). If you have variance 100, σ = 10.
- Sign confusion: z = −1.2 means 1.2 SDs below the mean, the sign is information, not an error.
- Assuming normality: the z-score itself is always valid arithmetic, but the “68-95-99.7” interpretation assumes roughly normal data.
- SD of zero: with no spread, every value equals the mean and z-scores are undefined, the calculator rejects σ = 0.
When to use it
Comparing scores across different tests or scales, identifying outliers, quality control (is this measurement within tolerance?), and as the gateway to normal-distribution probabilities. Need the SD first? The standard deviation calculator computes it with steps; for centers, see mean, median & mode.
Full walkthrough: comparing two scores
Score A: 88 with mean 75, SD 8 gives z = (88-75)/8 = 1.625. Score B: 92 with mean 80, SD 5 gives z = (92-80)/5 = 2.4. Score B is stronger relative to its group – exactly what z-scores are for: fair comparison across scales.
Key takeaway: z = (x – mean)/SD counts standard deviations from average. Beyond 2 is unusual; beyond 3, an outlier.
Frequently asked questions
The number of standard deviations a value lies from the mean: z = (x − μ) ÷ σ. A z-score of 1.5 means the value sits 1.5 standard deviations above average.
“Good” depends on context. In a normal distribution, |z| < 2 covers about 95% of values (typical); |z| > 2 is unusual; |z| > 3 suggests an outlier.
Yes, negative just means below the mean. A z of −1.2 is 1.2 standard deviations below average, about the 12th percentile of a normal distribution.
Then every value equals the mean, there is no spread, and z-scores are undefined (division by zero). The calculator asks for a positive SD.
About 1.645 for a normal distribution (1.96 for the two-sided 95% interval). A z of 1.645 means the value beats roughly 95% of the distribution.
Standardized testing (comparing scores across years), manufacturing quality control, medical lab ranges, and finance (how extreme is this return?). Anywhere “how unusual is this?” needs a number.
A z-score counts standard deviations from the mean; a percentile states what share of values fall below. They convert via the normal curve: z = 1.0 ≈ 84th percentile, z = 2.0 ≈ 97.7th percentile.