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- Choose what to calculate: a z-score from a raw value, probabilities for a z-score, or the z-score at a percentile.
- Enter the values. The standard deviation must be greater than 0.
- Read the left-tail, right-tail, between and outside probabilities.
Formel
A z-score counts how many standard deviations a value is from the mean. Probabilities use the standard normal cumulative distribution Φ(z), computed to double precision.
z = (x − μ) / σ
P(Z < z) = Φ(z); P(Z > z) = 1 − Φ(z)
P(−|z| < Z < |z|) = 2Φ(|z|) − 1
Example
- Score 85 on a test with mean 70 and SD 10.
- z = (85 − 70) / 10 = 1.5.
- Φ(1.5) ≈ 0.9332, so the score is at about the 93rd percentile.
Häufige Fragen
What does a negative z-score mean?
The value is below the mean. A z-score of −1 is one standard deviation below average.
Why is z = 1.96 important?
95% of a normal distribution lies between −1.96 and +1.96 standard deviations, which is why 1.96 appears in 95% confidence intervals.
Does my data need to be normally distributed?
A z-score can be computed for any data, but the probabilities and percentiles only apply if the data are approximately normal.