Z-Score Calculator

Standard deviations from the mean, and the percentile that follows.

Written and maintained by Mohit PatelLast checked August 4, 2026How we build these

Your value

Population mean (μ)

Standard deviation (σ)

Converting a z-score to a percentile assumes a normal distribution. Income, house prices and response times are all strongly skewed, and for those the z-score is still correct while the percentile it implies can be badly wrong.

z-score

+2.000

2.00 standard deviations above the mean

Value130.00

Distance from the mean+30.00

z-score+2.000

Percentile97.72%

Proportion below97.72%

Proportion above2.28%

Two-tailed p-value0.0455

A z-score of +2.000 says the value sits 2.00 standard deviations above the mean. The percentile that follows — 97.7% — is the part that carries an assumption: it is only meaningful if the data is roughly normally distributed. The z-score itself is always valid; the percentile is not.

How the Z-Score Calculator works

A z-score says how far a value sits from the mean, measured in standard deviations: z = (x − μ) / σ. An IQ of 130 against a mean of 100 and a standard deviation of 15 is exactly two, which is why 130 is the number people quote. The percentile that follows is the part worth being careful about, because it assumes the data is normally distributed, and plenty of real data is not.

Also known as: standard score calculator · z score to percentile · how to calculate z score · z score formula

The calculation itself

z = (x − μ) / σ. Subtract the mean, divide by the standard deviation, and the result is a distance measured in standard deviations rather than in the original units.

That rescaling is the whole point. It makes a test score and a blood pressure reading comparable, because both become 'how far from typical, in units of typical spread'.

In practice

An IQ of 130 against a mean of 100 and a standard deviation of 15 gives (130 − 100) ÷ 15 = exactly 2.00. That is why 130 is the number people quote — it is the round z-score, not a round IQ.

The percentile follows from the normal distribution: a z of 2 sits at the 97.72nd percentile, so 2.28% of the population scores higher. The two-tailed p-value is 0.0455, which is the figure a significance test would report.

A value below the mean gives a negative z, and the sign is doing real work: −1.5 and +1.5 are equally unusual, in opposite directions.

Where this goes wrong

The z-score is a rescaling and is always arithmetically valid. The percentile is not. It assumes the data follows a normal distribution, and a great deal of real data does not.

Income is the standard example. It is strongly right-skewed, so an income two standard deviations above the mean is nowhere near the 97.7th percentile; the long tail of very high earners inflates the standard deviation and the normal curve gets the shape wrong. House prices and response times behave the same way.

The rule of thumb: trust the z-score, question the percentile, and ask what the distribution actually looks like before quoting one.

Where to go next

The Z-Score question rarely arrives on its own. These are the ones that usually come with it:

Confidence Interval Calculator — The range around an estimate, for a mean or a proportion.
Average Calculator — Mean, median, mode, range, and standard deviation.
Percentage Calculator — Every common percentage question in one place.
Age Calculator — Exact age in years, months, days, and total units.

Sources

NIST/SEMATECH e-Handbook of Statistical Methods

Frequently asked questions

How do I calculate a z-score?

Subtract the mean from your value, then divide by the standard deviation. An IQ of 130 with a mean of 100 and a standard deviation of 15 gives (130 − 100) ÷ 15 = 2.00.

What is a good z-score?

There is no good or bad in the abstract — it depends entirely on what is being measured. A z-score of +2 is high, reached by about 2.3% of a normal population. A z-score of −2 is equally unusual in the other direction.

How do I convert a z-score to a percentile?

Through the standard normal cumulative distribution, which this page computes. A z of 2 corresponds to the 97.7th percentile, meaning 97.7% of the distribution falls below that value. This step, not the z-score itself, is where the normality assumption enters.

Can a z-score be negative?

Yes, and it simply means the value is below the mean. The sign carries the direction and the magnitude carries the distance, so −1.5 and +1.5 are equally far from typical.

What does a z-score of 0 mean?

The value is exactly the mean, which puts it at the 50th percentile. Half the distribution falls below it.

Is a z-score valid for skewed data?

The z-score itself is, since it is only a rescaling. The percentile is not: for income, house prices or response times — all strongly skewed — the implied percentile can be badly wrong even though the z-score is arithmetically correct.

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