Normal distribution calculator
The z-score of a value and the share of a normal distribution below and above it.
Enter your values and press Calculate. Your inputs stay in the address, so you can switch language or share the link.
Formula
z = (x − μ) ÷ σ P(X ≤ x) = Φ(z)
How it works
The z-score measures how many standard deviations a value is from the mean. Converting it to a probability uses the normal distribution's cumulative curve (Φ), approximated here to about 7 decimal places.
Worked example
The same calculation with the default values:
- Mean (μ) 100
- Standard deviation (σ) 15
- Value (x) 115
- z = (115 − 100) ÷ 15 = 1
- P(X ≤ 115) = Φ(1) = 84.134474%
Result: z-score: 1 · P(X ≤ x): 84.13% · P(X > x): 15.87%
Φ is computed with a standard numerical approximation, accurate to about 7 decimal places — more than enough for coursework.
Common mistakes
- Applying this calculator to data that isn't actually normally distributed - the z-score and Φ(z) lookup only give meaningful probabilities when the underlying data genuinely follows (or closely approximates) a normal, bell-shaped distribution.
- Confusing the z-score itself with the probability - a z-score is how many standard deviations a value is from the mean; the probability (from Φ(z)) is a separate, derived number between 0 and 1.
- Using the sample standard deviation when the population standard deviation is required by the specific problem, or vice versa - these can differ, especially for small samples, and the formula's z-score is sensitive to which one is used.
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