Z-Score Calculator with Percentile

Score & Distribution

Fill the top three to standardise a value; use the bottom field to reverse the process.

Raw score (x)

Population mean (μ)

Standard deviation (σ)Must be greater than zero

Reverse: z-score to convert backGives the raw score at this z

Z-SCORE

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Percentile rank—

P(X < x) — area to the left—

P(X > x) — area to the right—

Two-tailed p-value—

Distance from mean—

Interpretation—

Reverse lookup

Raw score at that z—

Percentile at that z—

Middle interval at ±z—

Area outside ±z—

Probabilities assume the values follow a normal distribution. On strongly skewed data a z-score is still a valid distance measure, but the percentile attached to it will be wrong.

What the number actually means

z = (x − μ) / σ. Subtracting the mean re-centres the distribution on zero; dividing by the standard deviation rescales it so one unit equals one standard deviation. An IQ of 128 against a mean of 100 and an SD of 15 gives z = 1.87 — nearly two standard deviations up, around the 97th percentile.

Because the units cancel, z-scores make unlike things comparable. A z of 1.5 on a maths test and a z of 1.5 on a swimming time describe the same relative standing even though one is marks and the other is seconds.

The empirical rule, and where it stops working

For a normal distribution, about 68% of values fall within ±1 SD, 95% within ±1.96, and 99.7% within ±3. This is why 1.96 turns up everywhere in statistics — it is the z that leaves exactly 2.5% in each tail.

The rule depends entirely on normality. Income, wait times and insurance claims are right-skewed, so the real percentage below +1 SD is nothing like 84%. The z-score is still computable and still tells you the distance; it is the percentile conversion that quietly becomes fiction.

One tail or two

Use a one-tailed probability when only one direction would interest you — testing whether a new process is faster. Use two-tailed when a difference in either direction matters, which is most of the time. The two-tailed p is simply double the smaller tail, which is why the same z looks half as significant under it.

Frequently Asked Questions

What is a good or bad z-score?

There is no universally good value — it depends on what you are measuring. As a rough guide, |z| under 1 is unremarkable, between 1 and 2 is notable, above 2 is unusual (about 5% of values), and above 3 is rare (about 0.3%).

Can a z-score be negative?

Yes, and it simply means the value is below the mean. The magnitude is what indicates how unusual it is; a z of −2.5 is exactly as extreme as +2.5, just on the other side.

How do I convert a z-score back into a raw score?

Multiply the z by the standard deviation and add the mean: x = μ + zσ. The reverse field above does this, which is how you find, for example, the exam mark that sits at the 90th percentile.

Should I use the sample or population standard deviation here?

A true z-score uses the population standard deviation. If you only have a sample estimate and n is small, the correct statistic is a t-score rather than a z-score, and the tail probabilities come from the t-distribution instead.

Where these numbers come from

Sources

Official publications only. Links open the original document in a new tab.

National Institute of Standards and Technology Special Publication 811 — Guide for the use of the SI Significant figures and rounding conventions
National Institute of Standards and Technology CODATA fundamental physical constants Values of the fundamental physical constants

Z-Score & Percentile Calculator

Score &amp; Distribution