What are the sources for these z-score calculations?

The z formula is OpenStax Introductory Statistics 2e, section 6.1: https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution The probability uses erf from NIST DLMF 7.2.1 (https://dlmf.nist.gov/7.2.E1) and the power series DLMF 7.6.1 (https://dlmf.nist.gov/7.6.E1), with Φ(z) = 1/2 + 1/2 erf(z/√2). NIST notes that the normal integral has no simple closed form: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm

Z-Score Calculator

Enter a value, a mean, and a standard deviation for the z-score and a normal-curve probability.

Smart Tips

• z is positive when the value is above the mean and negative when it is below.

• The probability is for a standard normal curve, mean 0 and standard deviation 1.

• P(Z ≤ 1.96) is about 0.9750. That is the familiar two-sided 95 percent point.

• This page does not run a t-test and it does not use a t table.

The Formula

z = (x − μ) / σ Φ(z) = 1/2 + 1/2 erf(z / √2)

x Observed value
The number you entered

μ Mean
Center of the distribution

σ Standard deviation
Must be greater than 0

Φ(z) Left-tail probability
Share of the curve at or below z

The normal integral has no elementary closed form. erf is the NIST DLMF error function, summed with the power series in DLMF 7.6.1. For a continuous curve, P(Z ≤ z) and P(Z < z) are the same number.

Key Insights

A labeled curve

The result names the percent of the standard normal curve at or below your z-score, and whether the value sits above or below the mean.

Both tails

The left figure is P(Z ≤ z). The right figure is the remaining probability, P(Z ≥ z).

Not a t-test

A t-test needs a sample size and a different reference distribution. This page stops at z and the normal curve.

Known checkpoint

At z = 1.96 the series matches the table value 0.9750 to the digits that table shows.

Frequently Asked Questions

z = (value − mean) / standard deviation. It counts how many standard deviations the value sits from the mean. Positive is above the mean. Negative is below it. Zero means the value is the mean.

P(Z ≤ z) for a standard normal curve, and the right-tail probability P(Z ≥ z). The sentence under the z-score states the percent of the curve at or below that z. This is not a t-test.

The standard normal cumulative distribution is an integral. NIST notes that the integral has no simple closed form, so this page computes it from the error function.

The series returns a value within a very small tolerance of 0.9750021, which rounds to the familiar table value 0.9750.

The z-score would divide by zero. The page asks for a standard deviation greater than zero.

The z formula is OpenStax Introductory Statistics 2e, section 6.1, The Standard Normal Distribution. The probability uses erf from NIST DLMF 7.2.1 and the power series in DLMF 7.6.1, with Φ(z) = 1/2 + 1/2 erf(z/√2). NIST notes that the normal integral has no simple closed form: Normal Distribution.

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Z-score

Enter values to calculate

P(Z ≤ z)

P(Z ≥ z)

Note:The probability assumes a normal curve with mean 0 and standard deviation 1 after the z conversion. It is a numerical series for the error function, not a t-test and not a clinical cutoff.