When is the data-set mode useful?

When you have raw scores and want each row's z directly — for example, normalising a feature before feeding it to a model or comparing students against the class mean.

Does the calculator use the sample or population SD?

In data-set mode it uses the sample SD (divisor n − 1). In single-value mode you supply σ directly, so the calculator uses whatever value you provide.

How are the tail probabilities computed?

From Φ(z), the standard normal CDF, using the 7-decimal Abramowitz & Stegun 7.1.26 approximation for erf. No network calls.

What does a two-tailed p-value of 0.1336 mean?

If the null hypothesis is true (the value comes from the assumed distribution), there is about a 13.36% chance of seeing a value at least this far from the mean in either direction.

What if every value in the data set is identical?

The sample SD is 0, so z-scores are undefined. The tool surfaces an explicit error rather than dividing by zero.

Z-Score Calculator

Calculate z = (x − μ) / σ and the left, right, and two-tail probabilities from the standard normal table.

Written by Golam Rabbani, Founder & Lead Engineer

This interactive tool requires JavaScript. Read the formula and worked example below; you can compute the result by hand.

How to use this z-score calculator

Pick the mode: a single value (need x, μ, σ) or a data set (paste numbers).
Fill in the fields. For a data set, the calculator uses the sample SD (divisor n − 1).
Press Calculate to get the z-score plus left-tail, right-tail, and two-tail probabilities.
Copy the result or Reset to clear inputs.

About this z-score calculator

A z-score (also called a standard score) tells you how many standard deviations a value is from its mean: z = (x − μ) / σ. A z of 0 sits exactly on the mean, +1 is one SD above, −2 is two SDs below, and so on. The calculator also reports tail probabilities from the standard normal distribution using a 7-decimal Abramowitz & Stegun approximation, so you can read off a p-value or percentile directly.

Worked example. A test has μ = 70, σ = 10. You score x = 85. Then z = (85 − 70) / 10 = 1.5. From the standard normal CDF, P(Z ≤ 1.5) ≈ 0.9332, so you outperformed about 93.32% of the distribution; P(Z ≥ 1.5) ≈ 0.0668 is the right-tail probability; and the two-tail probability 2 · min(P(Z ≤ z), P(Z ≥ z)) ≈ 0.1336 is the relevant p-value for a two-sided test of "is x unusual?" If you switch to data-set mode and paste the actual exam scores, the calculator produces a z-score for every row using the sample SD.

FAQ

When is the data-set mode useful?: When you have raw scores and want each row's z directly — for example, normalising a feature before feeding it to a model or comparing students against the class mean.
Does the calculator use the sample or population SD?: In data-set mode it uses the sample SD (divisor n − 1). In single-value mode you supply σ directly, so the calculator uses whatever value you provide.
How are the tail probabilities computed?: From Φ(z), the standard normal CDF, using the 7-decimal Abramowitz & Stegun 7.1.26 approximation for erf. No network calls.
What does a two-tailed p-value of 0.1336 mean?: If the null hypothesis is true (the value comes from the assumed distribution), there is about a 13.36% chance of seeing a value at least this far from the mean in either direction.
What if every value in the data set is identical?: The sample SD is 0, so z-scores are undefined. The tool surfaces an explicit error rather than dividing by zero.