Z-Score Calculator
Calculate z-scores for the standard normal distribution.
Calculate Z-Score
Z to Probability
Quick Examples
Understanding Z-Scores
A z-score measures how many standard deviations a data point is from the mean. It standardizes scores for comparison across different distributions.
Interpreting Z-Scores
- Z = 0: Score equals the mean
- Z > 0: Score is above the mean
- Z < 0: Score is below the mean
- |Z| > 2: Unusual score (beyond 95% of data)
- |Z| > 3: Very unusual (outlier candidate)
Z-Score Properties
| Range | % of Data |
|---|---|
| ±1σ (Z between -1 and 1) | 68.27% |
| ±2σ (Z between -2 and 2) | 95.45% |
| ±3σ (Z between -3 and 3) | 99.73% |
Z-Score Formulas
For a single value from a population with mean μ and standard deviation σ:
For a sample mean x̄ from a sample of size n, the denominator becomes the standard error (σ/√n):
Reversed, the formula recovers the original value from its z-score:
Worked Examples
Example 1: A test score
You score 85 on a test where the class mean is 70 and the standard deviation is 10. z = (85 − 70)/10 = 1.5 — your score sits 1.5 standard deviations above the mean.
Example 2: Reverse — from z-score back to a value
A z-score of 2 on an IQ-style scale (μ = 100, σ = 15) corresponds to x = 100 + 2×15 = 130.
Example 3: A sample mean
A sample of 25 widgets averages 52 g when the population mean is 50 g with σ = 10 g. The standard error is 10/√25 = 2, so z = (52 − 50)/2 = 1.0. The sample mean is one standard error above the population mean.
Z-Scores and Percentiles
Because z-scores place every normal distribution on the same scale, they convert directly to percentiles via the standard normal table. A z of 1.5 corresponds to roughly the 93rd percentile — about 93.3% of values fall below it. A z of 1 is about the 84th percentile; a z of −1, about the 16th.
Common Mistakes
- Using the sample SD where σ belongs: the basic z formula assumes the population σ is known; with an estimated s from a small sample, the value behaves more like a t-statistic.
- Dividing by variance instead of standard deviation: the denominator is σ, not σ².
- Assuming normality: z-score percentiles and the 68–95–99.7 rule only hold for roughly bell-shaped data.
- Reading |z| = 2 as "exactly 95%": about 95.45% of data falls within ±2; the common "95%" shortcut actually corresponds to ±1.96.
Quick Answers
Can z-scores be compared across different tests? Yes — that is their main purpose. Raw scores live on different scales; z-scores do not.
What is the difference between a z-score and a percentile? A z-score measures distance from the mean in standard-deviation units; a percentile measures the share of values below it. For normal data, one converts to the other.
Why does the sample-mean formula divide by σ/√n instead of σ? Sample means cluster more tightly than individual values — the average of 25 values varies 5 times less — so the same raw gap from the mean is more surprising for a sample mean, and the smaller denominator accounts for that.
Reading a Z-Table
A standard normal table turns z-scores into probabilities. Find your z along the margins and read the body entry: for z = 1.5 the table gives 0.9332, meaning 93.32% of values fall below it. To find the share between two z-scores, subtract: between z = −1 and z = 1 lies 0.8413 − 0.1587 = 0.6826, or about 68.3% — the familiar "68%" of the empirical rule, derived rather than memorized.
Z-Scores in Real Life
- Standardized testing: exams are scaled so scores from different years share a mean and standard deviation — the z-score is what makes a 2024 score comparable to a 2025 one.
- Quality control: control charts flag any process reading beyond ±3σ — about 3 in 1,000 by chance — as a signal to investigate.
- Finance: expressing a return in standard-deviation units separates a genuinely exceptional month from ordinary volatility.
- Medicine: growth charts report a child's height as a z-score against age norms, so a pediatrician sees "tall for her age" as a number.
Z-Score vs T-Score
When the population σ is unknown and estimated from a small sample, the same ratio follows a t-distribution with heavier tails instead of the normal curve — the t value behaves like a z-score but with extra caution baked in. As the sample grows, the t-distribution converges to the normal, and the distinction fades.
More Quick Answers
What z-score counts as unusual? Beyond ±2 is unusual (about 1 in 20 by chance); beyond ±3 is very unusual (about 1 in 370).
Can a z-score be larger than 3? Yes — there is no upper bound. A z of 5 simply means an extraordinarily far-out value.
Do z-scores have units? No. Dividing by the standard deviation cancels the units, which is exactly why scores from different scales become comparable.