Z-Score Calculator
Convert a raw value to a z-score, read off its percentile and probability, calculate the area between two z-scores, or work backward from a percentile — all with a live bell-curve chart and clickable z-table.
z = (x−μ)/σ
The Formula
68-95-99.7
Empirical Rule
0-100th
Percentile Scale
▤ Z-Score Calculator
Must be > 0
Z-Score
0.50
0.50 SD above mean
Above average — 69.1462% below this point
Typical range
P(X < Z)
69.1462%
Percentile
69.15th
P(−|Z| < X < |Z|)
38.2925%
Combined Tails
61.7075%
All Probability Results
P(X < Z)
69.1462%
Left tail = percentile
P(X > Z)
30.8538%
Right tail
P(0 < X < |Z|)
19.1462%
From center to |z|
P(−|Z| < X < |Z|)
38.2925%
Symmetric around mean
P(X < −|Z| or X > |Z|)
61.7075%
Both tails combined
Percentile
69.14625th
z = 0.5000
Normal Distribution Curve
68–95–99.7 Empirical Rule
95% Exact
z = ±1.960
99% Exact
z = ±2.576
Top 5%
z ≥ 1.645
Top 1%
z ≥ 2.326
Z-Table — P(X < z) Cumulative Probabilities (Click any cell to load)
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Understanding Z-Scores
A z-score (also called a standard score) measures how many standard deviations a single data point sits from the mean of its distribution. Instead of reporting a raw number in whatever units it started in — a test score, a height, a temperature — a z-score puts that value on a universal, unit-free scale: 0 means right at the mean, positive means above it, negative means below it, and the size of the number tells you how extreme that position is. Once two values are standardized this way, you can compare them directly even if they came from completely different datasets.
🧮 The Key Formulas
Z-score: z = (x − μ) / σ | Raw score: x = μ + z × σProbability: P(X < z) = Φ(z) = CDF of the standard normal distributionLeft tail: P(X < z) = Φ(z) | Right tail: P(X > z) = 1 − Φ(z)Symmetric: P(−|z| < X < |z|) = 2Φ(|z|) − 1📊 Interpreting the Sign and Magnitude
The sign of a z-score tells you direction: negative means the value falls below the mean, positive means it falls above. The magnitude tells you how extreme it is — a z-score near 0 is typical, while a z-score with a large absolute value (commonly beyond ±2 or ±3) is unusual relative to the rest of the distribution. Under the empirical rule, about 68% of values sit within ±1 standard deviation of the mean, about 95% within ±2, and about 99.7% within ±3, which is why those thresholds are the standard checkpoints for flagging unusual or outlier values.
🎓
Exam Scores
Standardize a test score against its class mean and SD to see relative performance, independent of how hard the test was.
🏭
Quality Control
Flag manufactured parts whose measurements sit too many standard deviations from spec — a common Six Sigma technique.
📈
Finance
Compare an asset's return against its historical mean and volatility to judge whether a move is ordinary or extreme.
🧪
Research
Standardize variables measured in different units before combining or comparing them in a statistical model.
— FAQ