Z-Score Calculator

The z-score formula is z = (x − μ) / σ, where x is your raw value, μ is the mean, and σ is the standard deviation.

Enter these three values into the calculator to get the z-score instantly, along with the corresponding percentile and probability.

Convert any z-score to a percentile rank using the cumulative standard normal distribution.

• A z-score of 0 = 50th percentile (the mean).
• A z-score of 1.0 = 84th percentile.
• A z-score of 1.96 = 97.5th percentile (used in 95% confidence intervals).

Z-scores follow the standard normal distribution (mean=0, SD=1).

The 68-95-99.7 rule states that:

• 68% of data falls within z=±1
• 95% within z=±2
• 99.7% within z=±3

Our calculator visualizes your z-score on the normal distribution curve.

A z-score, also called a standard score, is the number of standard deviations a data point lies from the mean of its distribution.

A z-score of 0 sits exactly at the mean, +1.5 lies one and a half standard deviations above it, and −2 lies two standard deviations below.

Because it rescales any variable to a common unit, the z-score lets you compare values measured on different scales — a test score against a height, for example.

According to Wolfram MathWorld, the standard score standardizes a normal variable to a distribution with mean 0 and standard deviation 1, the foundation of nearly every parametric statistical test.

The z-score is computed as z = (x − μ) / σ, where x is the raw observation, μ is the population mean, and σ is the population standard deviation.

You subtract the mean to center the data on zero, then divide by the standard deviation to express the distance in standardized units. When you only have a sample, substitute the sample mean x̄ and sample standard deviation s.

For a score of 85 with mean 70 and SD 10, z = (85 − 70) / 10 = 1.5.

Khan Academy notes this transformation preserves the shape of the distribution while standardizing its center and spread.

To turn a z-score into a probability, evaluate the cumulative distribution function (CDF) of the standard normal distribution, Φ(z), which gives P(Z < z).

For z = 1.5, Φ(1.5) ≈ 0.9332, so about 93.32% of values fall below it and 6.68% above.

The CDF is defined through the error function, erf, as documented in the NIST Digital Library of Mathematical Functions (DLMF §7). Because the normal curve is symmetric, Φ(−z) = 1 − Φ(z).

The calculator handles this integration automatically, so you never need to read a printed z-table.

Several z-scores recur constantly in statistics.

• A z of 1.0 marks the 84th percentile and −1.0 the 16th.
• The critical value 1.645 corresponds to the 95th percentile (one-tailed), while 1.96 gives the 97.5th percentile and defines the familiar 95% two-sided confidence interval.
• A z of 2.576 marks the 99.5th percentile, used for 99% confidence.

Under the 68-95-99.7 rule described by Encyclopaedia Britannica, roughly 68% of data lies within ±1, 95% within ±2, and 99.7% within ±3 standard deviations of the mean. Memorizing these anchors speeds up interpretation.

Z-scores appear far beyond the classroom.

• Standardized exams report them so scores from different test forms can be compared fairly.
• In manufacturing quality control, engineers flag any measurement beyond ±3 standard deviations as an outlier, since fewer than 0.3% of a normal process should fall there.
• Finance uses z-scores in models such as the Altman Z-score for bankruptcy risk, and healthcare relies on them for growth charts and bone-density scores.

Any time you must compare observations drawn from different scales or datasets, standardizing them into z-scores, as Khan Academy illustrates, makes the comparison meaningful.

To reverse the process and find a z-score from a percentile, apply the inverse normal (probit) function, Φ⁻¹(p).

Enter the percentile as a proportion between 0 and 1:

• the 90th percentile returns a z of about 1.282
• the 95th percentile returns 1.645
• the 99th percentile returns roughly 2.326

Values below the 50th percentile yield negative z-scores, reflecting positions below the mean.

The NIST/SEMATECH e-Handbook of Statistical Methods tabulates these quantiles of the standard normal, and this calculator's reverse mode computes them directly from any percentile you enter.

In a z-test, the test statistic is a z-score measuring how far a sample mean sits from a hypothesized value under the null hypothesis.

You compare it against critical values:

• for a two-tailed test at the 5% level, reject the null if |z| exceeds 1.96
• at the 1% level if |z| exceeds 2.576

Equivalently, convert the z-score to a p-value using the standard normal CDF and reject when p is below your significance threshold α.

Britannica notes the z-test assumes a known population standard deviation and an approximately normal sampling distribution, conditions often met by the Central Limit Theorem for large samples.

• The most frequent error is dividing by the variance instead of the standard deviation — remember σ is the square root of the variance, so always divide by σ.
• Another mistake is mixing up population and sample statistics: use μ and σ for a full population but x̄ and s for a sample.
• People also confuse a z-score with a percentile; a z of 1.0 is not the 100th percentile but the 84th.
• Finally, z-scores only carry their usual probability meaning when the data is approximately normal.

As Wolfram MathWorld cautions, applying normal-curve probabilities to heavily skewed data produces misleading percentiles.