The z-score calculator converts between raw scores, z-scores (standard deviations from the mean), and percentile ranks. Enter a raw value with mean and SD to find the z-score and percentile, or enter a z-score to find the percentile.
Z-Score Calculator
How to Use the Z-Score Calculator
The z-score (standard score) tells you how many standard deviations a value is from the mean, allowing comparison across different distributions.
Formula and Interpretation
z = (x − μ) / σ. If exam scores have mean=75, SD=10: a score of 90 has z = (90−75)/10 = 1.5. This means the score is 1.5 standard deviations above average, corresponding to the 93.3rd percentile.
Converting Z-Score to Percentile
The percentile is P(Z ≤ z) = Φ(z), from the standard normal cumulative distribution. z=0: 50th percentile. z=1: 84th. z=−1: 16th. z=1.96: 97.5th percentile (used in 95% confidence intervals). z=2.576: 99.5th percentile (99% CI).
Reverse Lookup
To find what raw score corresponds to the 90th percentile: z = 1.282, x = μ + z×σ. With mean=75, SD=10: x = 75 + 1.282×10 = 87.82. Use the Z-score → Raw mode for this direction.
FAQ
What is a z-score?
A z-score measures how many standard deviations a value is from the mean. z = (x − μ) / σ. A z-score of +2 means the value is 2 SDs above the mean. A z-score of −1.5 means 1.5 SDs below the mean. Z-scores allow comparison across different distributions.
How do you interpret a z-score?
In a normal distribution: z=0 is exactly average (50th percentile). z=1 → 84th percentile. z=2 → 97.7th percentile. z=3 → 99.9th percentile. z=−1 → 16th percentile. If your exam score has z=1.5, you scored better than about 93.3% of test-takers.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All calculations run locally. Nothing is transmitted.
What is the percentile for a given z-score?
The percentile is P(Z < z) × 100 from the standard normal table. z=0: 50%. z=1.65: 95%. z=1.96: 97.5% (used for 95% confidence intervals). z=2.58: 99.5% (used for 99% confidence). The calculator uses the error function approximation for precise results.