Statistics · Normal Distribution
Z-Score
Calculator
Convert raw scores to z-scores, find all 5 probability types from a z-score, calculate probabilities between two z-scores, and look up percentiles — with normal curve visualization.
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Z-Score
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P(X < z)
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Percentile
Z-Score Calculator
x
μ
σ
Must be > 0
z
Probability Type to Highlight
Z₁
Z₂
P
P = cumulative probability P(X < z)
Z-Score
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standard deviations from mean
P(X < z)
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Percentile
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P(X < z)
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P(X > z)
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P(-|z|<X<|z|)
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P(0 to |z|)
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📊 All Probability Results
📈 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)
Understanding Z-Scores
A z-score (standard score) measures how many standard deviations a data point is from the mean: z = (x − μ) / σ. Z-scores standardize different scales so you can compare across datasets — like comparing a math test score to a reading test score when they use different scales.
Key Z-Score Formulas
Z-score: z = (x − μ) / σ
Raw score: x = μ + z × σ
Probability: P(X < z) = Φ(z) = CDF of standard normal
Left tail: P(X < z) = Φ(z)
Right tail: P(X > z) = 1 − Φ(z)
Symmetric: P(-|z| < X < |z|) = 2Φ(|z|) − 1
Between Z1,Z2: P(Z1 < X < Z2) = Φ(Z2) − Φ(Z1)
What does a z-score of 1.96 mean?
A z-score of 1.96 means the value is 1.96 standard deviations above the mean. P(X < 1.96) = 0.975, meaning 97.5% of values in a normal distribution fall below this point. The range −1.96 to +1.96 contains exactly 95% of all values — which is why 1.96 is the critical value for the 95% confidence interval in statistics. In two-tailed tests, α = 0.05 corresponds to z = ±1.96.
How do you convert a z-score to a percentile?
The percentile equals P(X < z) × 100. For z = 1.5: P(X < 1.5) ≈ 0.9332, so you're at the 93rd percentile — scoring higher than 93.32% of the population. For z = 0: you're at the 50th percentile (exactly at the mean). For z = −1: you're at about the 16th percentile (15.87%).
What is the normal distribution CDF?
The Cumulative Distribution Function (CDF) Φ(z) gives P(X < z) for the standard normal distribution. It doesn't have a closed-form expression — it's computed numerically using the error function: Φ(z) = (1 + erf(z/√2)) / 2. The z-table is a lookup table for Φ(z). Modern calculators (including this one) compute it using the rational approximation of erf for high accuracy.
When is a z-score considered an outlier?
A common rule of thumb: |z| > 2 is a mild outlier (about 5% of data), |z| > 3 is a strong outlier (about 0.3% of data), and |z| > 4 is extremely rare (0.006% of data). In practice, the threshold depends on the field. Finance often uses |z| > 2, while experimental physics may require |z| > 5 (a "5-sigma" result, as used for the Higgs boson discovery).