Z-Score Calculator

How to use the z-score calculator

1. Choose what you know: a raw score, a z-score, two z-scores or a percentile.
2. Enter the mean (μ) and standard deviation (σ) when converting between raw scores and z-scores.
3. Read the z-score or raw score, the percentile and the tail probabilities. Negative values are allowed throughout.

Z-score formula

A z-score (standard score) says how many standard deviations a value x is above or below the mean μ, where σ is the standard deviation. Positive means above the mean, negative below. Φ is the cumulative distribution function of the standard normal distribution, the bell curve with mean 0 and standard deviation 1; the calculator evaluates it through the incomplete gamma function, accurate to at least 12 significant digits even far into the tails, such as P(Z > 10) ≈ 7.62 × 10⁻²⁴.

Worked example

A student scores 85 on a test where the class mean is 70 and the standard deviation is 10. Then z = (85 − 70) ÷ 10 = 1.5: the score is one and a half standard deviations above average. If the scores are roughly normal, Φ(1.5) = 0.9332, so about 93.32% of the class scored lower and 6.68% scored higher. Going the other way, a z-score of −0.5 corresponds to a raw score of 70 + (−0.5) × 10 = 65.

Standard normal table (z table)

Each cell is Φ(z), the area to the left of z, for z = row + column. For example, z = 1.96 is in row 1.9, column 0.06: 0.9750. For a negative z, use symmetry: Φ(−z) = 1 − Φ(z).

Z-scores for common percentiles

Within ±1: 68.27% · Within ±2: 95.45% · Within ±3: 99.73%. These are the familiar 68–95–99.7 figures. A two-sided 95% range is ±1.9600, while the one-sided 95th percentile is 1.6449.

Tips for using z-scores

• Compare across different scales. A z of 1.5 on one test and 0.8 on another shows the first result is relatively stronger, even if the raw marks are lower.
• Percentiles need normal data. The z-score itself is valid for any data, but turning it into a percentile with Φ assumes a bell-shaped distribution.
• Use the right standard deviation. For a population use σ; when standardizing a sample mean, divide by the standard error σ ÷ √n instead.
• |z| above 3 is rare for normal data (about 0.27% of values), so it is a common outlier flag.

For hypothesis tests, the p-value calculator turns z, t, chi-square and F statistics into p-values, and the standard deviation calculator finds μ and σ from raw data.