Z-Score Calculator
Calculate the z-score from a raw score, mean, and standard deviation. Instantly find the percentile, probability above/below, and confidence intervals.
Z-Score
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Percentile —
Probability Below —
Probability Above —
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Z-Score
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Percentile —
P(X < x) —
P(X > x) —
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Z-Score & Probability
Z-Score —
Percentile —
P(X < x) — one-tailed left —
P(X > x) — one-tailed right —
p-value (two-tailed) —
Critical Values
Critical z at α=0.10 (two-tailed) —
Critical z at α=0.05 (two-tailed) —
Critical z at α=0.01 (two-tailed) —
Hypothesis Testing
Standard Error of Mean —
Z-Test Statistic (x̄ vs μ) —
Cohen's d (Effect Size) —
How to Use This Calculator
Enter a raw score, the mean, and the standard deviation. The calculator returns the z-score, percentile, and probability of a value falling below or above that score in a normal distribution.
Formula
z = (x − μ) / σ • Percentile = Φ(z) × 100 • P(X < x) = Φ(z)
Example
x=75, μ=70, σ=10 → z = 0.5 → percentile ≈ 69.15% → P(below) ≈ 0.6915
Frequently Asked Questions
- A z-score measures how many standard deviations a data point is from the mean: z = (x − μ) / σ. A z-score of 0 means the value equals the mean.
- z = 1 means the value is 1 standard deviation above the mean (≈84th percentile). z = −1 is 1 SD below (≈16th percentile). z = ±2 covers ≈95% of a normal distribution.
- Use the standard normal cumulative distribution function Φ(z). For example, z = 1 → Φ(1) ≈ 0.8413, so the 84th percentile.
- A one-tailed p-value is the probability in one direction (above or below z). A two-tailed p-value is twice the smaller tail, used when testing for any deviation.
- A confidence interval gives a range expected to contain the true population mean. At 95% confidence with n=30 and σ=10, the interval is x̄ ± 1.96×(σ/√n).