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Free Online Z-Score Calculator With the Percentile

Turn a value into a z-score and into the percentile that goes with it, or work backwards from a z-score to the value. You can type the mean and standard deviation, or paste a list of numbers and have them worked out, with the sample and population choice made explicit.

Free Forever Nothing Uploaded Percentile Included Both Directions

Free Online Z-Score Calculator With the Percentile

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🎚️ Z-Score Calculator Z-score and percentile together · or straight from a list

The sum runs in your browser. Nothing you type is sent anywhere.

Where the figures come from

Which way

🎚️ The figures

A z-score says how many standard deviations a value sits from the mean.

The value, x

The mean

The standard deviation Has to be above zero. A standard deviation of nothing means every value is the same.

Try one of these

Z-score

—

In words—

Percentile—

Below this value—

Above this value—

Further out than this, either side—

Mean and standard deviation used—

Step	What happens

Where your z-score sits on the usual scale, and what share of normal data falls inside each band.

Z-score	The value there	Percentile	Inside this many deviations

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How to Work Out a Z-Score

Three steps, and the third one is the answer people actually want.

Give it the figures, or the numbers themselves Either type the mean and standard deviation, or paste a list and have both worked out. From a list you also choose whether the numbers are a sample or the whole population, which changes the answer.

Type the value, or the z-score if you are going backwards Both directions are on the same page, because working out which mark corresponds to the top five per cent is the same relationship read the other way.

Read the percentile, and the warning beside it The percentile is what most people came for, and it relies on the data being normally distributed. The z-score itself does not. Where the data is clearly not normal, the page says so.

What to Know About Z-Scores

What the percentile assumes, and when it is not safe.

The z-score needs no assumptions. The percentile does. Dividing the distance from the mean by the standard deviation is just a change of scale and is always valid. Turning that into a percentile requires the data to follow a normal distribution, because the percentile is read off the normal curve. Income, waiting times, prices and counts are not normal, so for those the z-score is fine and the percentile is wrong. Almost every calculator gives the percentile without mentioning this.

Sample or population changes the answer, and the difference is biggest on small lists. A sample standard deviation divides by one less than the count, which makes it larger, which makes every z-score smaller. On ten numbers the two differ by about five per cent; on five numbers by about twelve. Sample is the right default, because you almost always have a sample, and both are shown so the choice is visible.

A negative z-score is not a bad one, it just means below the mean. A z of minus one is one standard deviation below average, which is the 15.9th percentile. There is nothing wrong with it arithmetically, and on something like a golf score or a cholesterol reading it is the good direction. The sign tells you which side of the mean you are on and nothing more.

The familiar percentages are not round numbers. About 68.27 per cent of normal data falls within one standard deviation, 95.45 within two and 99.73 within three. The figure people remember as 95 per cent is actually 1.96 deviations, not 2, and that difference is why confidence intervals use 1.96. Both are in the scale table.

A standard deviation of zero has no z-score at all. If every value is identical there is no spread to measure against, and the formula divides by zero. Some calculators return zero or infinity here; neither is meaningful. The honest answer is that the question does not apply, which is what you get.

The tail probabilities are accurate a long way out. At a z of 5 the chance of being further out on one side is about 2.87 in ten million, and this page gets that right. The quick approximations that circulate for the normal curve are accurate to about seven decimal places, which is enough near the middle and useless in the tail, where the answer itself is smaller than their error.

Key Features & Capabilities

What it does, and what it deliberately does not.

Both directions Value to z-score, or z-score back to the value, which is what you need for a cut-off at the top five per cent.

Straight from a list Paste the numbers and the mean and standard deviation are worked out, with the sample or population choice explicit.

Percentile and both tails Below, above, and further out on either side, since which one you want depends on the question.

Says when the percentile is unsafe The percentile assumes normal data. From a list, the skew is measured and you are told when it looks wrong.

The scale beside your answer Minus three to plus three with the value at each point, so your z-score has something to sit against.

Nothing is uploaded It runs in this tab and keeps working offline.

About Z-Scores

A z-score answers a question that raw numbers cannot: is 85 good? On its own there is no way to tell. If the mean is 70 and the standard deviation 10 then 85 is one and a half deviations above average, and now it can be compared with a score of 112 from a different test with a different scale. That is the whole purpose, and it is why z-scores are the common currency of statistics.

The formula is one line, so a page that only computes it would not be worth building. What people actually want is the next step: the percentile. One and a half deviations above the mean puts you at about the 93rd percentile, meaning roughly 93 per cent of the group is below you, and that is the sentence someone came here to be able to say.

The step from z-score to percentile is where the honesty matters, because it quietly assumes the data follows a normal distribution. The z-score needs no such assumption; it is a change of units and nothing more. The percentile is read off the normal curve, and if the data is not shaped like that curve the percentile is simply wrong while still looking perfectly reasonable. Incomes, waiting times, house prices and counts of things are all distinctly not normal. So the assumption is written next to the percentile here, and when you paste a list the skew is measured and you are told if it looks like a problem.

The other choice worth making consciously is sample against population. Working out the standard deviation from a list divides by either the count or one less than it, and the second gives a larger figure, which makes every z-score smaller. On a list of ten that is about five per cent; on a list of five, twelve. Sample is the sensible default because a list of numbers is nearly always a sample of something larger, but both are shown so the decision is yours and not hidden.

Frequently Asked Questions

Percentiles, negative scores, sample against population.

The value sits one and a half standard deviations above the mean. If the data is normally distributed that places it at about the 93rd percentile, so roughly 93 per cent of the group is below it.

It just means below the mean. Whether that is bad depends entirely on what is being measured; for a golf score or a blood pressure reading, below average is the good direction. The sign carries no judgement.

Because the percentile is read off the normal curve. The z-score itself is only a change of scale and needs no assumption, but converting it to a percentage of the group below depends on the shape of the distribution. For skewed data such as income the percentile will be wrong.

Sample if your numbers represent a larger group, which is almost always. Population only if you genuinely have every member. Sample divides by one less than the count, giving a larger standard deviation and therefore smaller z-scores, and the difference matters most on small lists.

About 1.645 for the top five per cent on one side, and 1.96 if you mean the outer five per cent split between both tails. Those two get confused constantly, which is why both the one tail and two tail figures are shown.

There is no z-score. Every value being identical means there is no spread to measure against, and the formula would divide by zero. Returning zero or infinity, as some calculators do, would be inventing an answer.

No. Everything is worked out in your browser, so a pasted list of real figures stays in the tab, and the page works offline once loaded.