Z-score Calculator
Turn a raw score into a z-score — how many standard deviations it sits above or below the mean — then read off the percentile and the probabilities P(X < x) and P(X > x) from the standard normal distribution.
TheCalculators — Z-score Calculator
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Z-score
1
x = 5 is 1 std dev above the mean · percentile 84.13%
- Raw score (x)
- 5
- Mean (μ)
- 3
- Std dev (σ)
- 2
- Z-score = (x − μ)/σ
- 1
- P(X < x) = Φ(z)
- 0.841345
- P(X > x)
- 0.158655
- Percentile
- 84.134%
- Distance from mean
- 1 σ above
Where your z-score sits (standard normal)
1.0σ
Average (−1σ to 1σ)
Step-by-step solution
- 1Subtract the meanx − μ = 5 − 3 = 2
- 2Divide by the standard deviationz = 2 / 2 = 1
- 3Look up the probabilityP(X < x) = Φ(1) = 0.841345 → percentile 84.13%
- 4ComplementP(X > x) = 1 − 0.841345 = 0.158655
Z-scores let you compare values from different distributions: a z of +2 means "two standard deviations above average" no matter the original units.
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Formula
z = (x − μ) / σ · P(X < x) = Φ(z) · P(X > x) = 1 − Φ(z)
Pro tips
- • z = 0 means exactly average; ±1 covers the middle ~68% of a normal distribution.
- • z beyond ±2 is unusual (~5% of data); beyond ±3 is rare (~0.3%).
- • Percentile = Φ(z) × 100 — how much of the distribution lies below your score.
FAQ
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