Z-Score Calculator

How to use the z-score calculator

Pick what you want to find from the first menu:

• Z-score of a value: enter the value (x), the mean (μ) and the standard deviation (σ). You get the z-score, the percentile and the probability of falling below or above x.
• Probability for a z-score: enter z and choose the area you need: the left tail P(Z < z), the right tail P(Z > z), the middle P(−z < Z < z) or both tails.
• Z-score for a percentile: enter a percentile such as 95 to get z = 1.6449. Keep a mean and standard deviation filled in to convert it to your own scale.
• Value from a z-score: turn a z-score back into a raw value with x = μ + zσ.

Z-score formula

z = (x − μ) ÷ σ

• x = the value you are standardizing
• μ = the mean of the distribution (use x̄ for a sample)
• σ = the standard deviation (use s for a sample)

Rearranged, x = μ + z × σ turns a z-score back into a value. Standardizing does not change the shape of the data. It only shifts the mean to 0 and rescales the spread to 1, which is what lets one table of probabilities serve every normal distribution. If you still need the mean and standard deviation of your data, the standard deviation calculator gives both.

Worked example: how good is a score of 85?

A class test has a mean of 70 and a standard deviation of 10. You scored 85.

1. z = (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5. Your score is 1.5 standard deviations above the mean.
2. The area to the left of z = 1.5 under the standard normal curve is Φ(1.5) = 0.933193.
3. If scores are roughly normal, about 93.32% of the class scored below you and 6.68% scored above.

The same z-score means the same thing on any scale. A score of 1.5 standard deviations above average on an IQ test (mean 100, SD 15) is 122.5, at the same 93rd percentile.

Z-table: common z-scores and probabilities

Printed z-tables usually give four decimal places for z in steps of 0.01. This calculator works out the exact area for any z, including values far out in the tails, such as P(Z > 6) = 9.866 × 10⁻¹⁰.

Critical values for confidence intervals and tests

A two-sided interval that holds the middle C% of a normal distribution runs from −z to +z, where z = Φ⁻¹(0.5 + C/200):

For a hypothesis test, the two-tailed p-value for a z statistic is the “both tails” area. A test statistic of z = 1.96 gives p = 0.049996, just under the usual 0.05 threshold.

How to interpret a z-score

• |z| < 1: within one standard deviation, where about 68.27% of values lie. Very typical.
• 1 ≤ |z| < 2: noticeably above or below average, but common. About 95.45% of values lie within ±2.
• 2 ≤ |z| < 3: unusual. Only 4.55% of values fall beyond ±2.
• |z| ≥ 3: rare. Only 0.27% fall beyond ±3, so check for data errors or a genuine outlier.

These percentages assume the data are roughly bell-shaped. For skewed data, such as incomes or waiting times, a z-score still measures distance from the mean, but the percentiles from the normal curve can be badly off. The probability calculator covers counts of successes, where the binomial distribution applies instead.

How the probabilities are calculated

The standard normal cumulative distribution Φ(z) has no simple closed form, so it is computed numerically. This calculator uses George Marsaglia’s series for |z| < 3 and Laplace’s continued fraction beyond that, which agree with high-precision reference values to about 15 significant digits. Percentiles are turned into z-scores with Peter Acklam’s rational approximation, refined with a Halley step to the same precision. Results match Excel’s NORM.S.DIST and NORM.S.INV to at least six decimal places.