Z Score Calculator

CalculateZScore turns any raw score into a z-score, percentile, and probability — instantly, with every step shown. Free, no signup. This z score calculator also works as a standard score calculator, z value calculator, z statistic calculator, normal score calculator, and standardized score calculator: three ways to enter, one clear answer.

Calculate your z-score

From raw score From z-score Percentile → z

Raw score (x)

Mean (μ)

Standard deviation (σ)

Z-score

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Percentile

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P(X ≤ x) left-tail

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P(X > x) right-tail

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Enter your three values above and the results appear here instantly.

The shaded region is the left-tail area — the percentile of your score.

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Results update as you type. Your numbers never leave your browser — no signup, no tracking, no data sent anywhere.

Z-score

Area type (which tail to shade) Left tail Right tail Two tails Middle area

Percentile

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P(Z ≤ z) — left-tail probability

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P(Z > z) — right-tail probability

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Enter a z-score and pick an area type to see the probabilities.

Your step-by-step working will appear here.

This mode is a z score to percentile calculator and z score probability calculator in one — the area mode picker chooses which tail probability the curve shades.

Percentile (0–100)

Enter the percentile rank, e.g. 95 for the 95th percentile.

Z-score

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Left-tail probability

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Right-tail probability

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Enter a percentile to find its z-score.

Your step-by-step working will appear here.

This inverse z score calculator runs the normal distribution backwards: from an area under the curve back to the z-score that produces it.

What is a z-score? A plain-English explanation

A z-score is simply a way of describing where one number stands compared to a whole group of numbers. Test scores are the classic example: scoring 90 on an exam means little on its own, but if the class average (the mean) was 80 with a standard deviation of 5, your score is clearly above the pack — and the z-score turns that "clearly above" into an exact number.

Statisticians call the z-score a standard score because it standardizes any value onto one universal scale: the standard normal distribution, the familiar bell curve. On this scale, 0 is always the mean, +1 and −1 mark one standard deviation above and below it, and so on. Once a value is converted, you can read off its percentile (what share of values fall below it) and its probability directly. The z score meaning never changes: distance from average, measured in standard-deviation units.

That is why the z-score shows up everywhere: teachers grading on a curve, researchers running hypothesis testing and significance tests, analysts building confidence intervals, and quality-control teams flagging unusual measurements. Any time you need to compare numbers that come from different scales — say, a math score out of 100 and a verbal score out of 800 — converting both to z-scores puts them on common ground. In short, if you have ever asked what is a z score in plain language, that is the answer: a universal translator that turns any number into a position on the bell curve. And because every probability on this page comes straight from the normal curve, the site doubles as a normal distribution calculator for the standard normal case.

The z-score formula, piece by piece

• x — your raw score (the value). This is the number you want to evaluate: a test score, a measurement, a data point. In our worked example below, x = 90.
• μ — the mean (mu, the average). Add up all the values in the group and divide by how many there are. The mean is the center of the bell curve. Here, μ = 80.
• σ — the standard deviation (sigma). This measures spread: how far values typically deviate from the mean. A small σ means values cluster tightly; a large σ means they scatter widely. Here, σ = 5. The standard deviation formula itself averages the squared deviations from the mean and takes the square root — but for z-scores you only need its result.
• The subtraction (x − μ) tells you the raw distance from average: 90 − 80 = 10 points above the mean.
• The division by σ converts that distance into standard-deviation units: 10 / 5 = 2.00. The score is two standard deviations above average — that is the z-score.

How to calculate a z-score — step by step

Worked example: you scored 90 on a test. The class mean was 80 and the standard deviation was 5. What is the z-score, and what percentile is it?

1. Step 1 — Write down your three values.
   x = 90 (your raw score), μ = 80 (the mean), σ = 5 (the standard deviation).
2. Step 2 — Subtract the mean from your score.
   90 − 80 = 10. You scored 10 points above the average.
3. Step 3 — Divide by the standard deviation.
   10 / 5 = 2.00. Your z-score is 2.00 — two standard deviations above the mean.
4. Step 4 — Convert the z-score to a percentile.
   Look up z = 2.00 in a z-table (or use the calculator above): the left-tail probability is 0.9772, so the score sits at the 97.72nd percentile. Only about 2.28% of the class scored higher.
5. Step 5 — Sanity-check the sign and size.
   Positive z means above average — correct, 90 > 80. A z-score of 2.00 is high but not extreme (extreme starts around ±3). Everything checks out.

Which area do you need? Left tail, right tail, two tails, middle area

A z-score alone does not give you a probability — you also have to choose which area under the curve you want. The calculator's area type picker offers four area modes, and picking the wrong one is the most common source of wrong answers in z score examples:

• Left tail — the area to the left of your z-score. This is the left-tail probability, and × 100 it is the percentile. Use it when the question asks "what percent scored at or below this?"
• Right tail — the area to the right. Use it for "what is the chance of scoring above this value?" For z = 2.00 the right-tail probability is 0.0228, i.e. a 2.28% chance.
• Two tails — both extreme ends combined. This is the standard choice in two-sided significance tests: the tail probability of a result at least this unusual in either direction.
• Middle area (the middle-area mode) — everything between −z and +z. Use it for "what percent of values fall within this range?" — the 95% in the middle of a 95% confidence interval is a middle area.

Every shaded region on the calculator's curve matches the area type you picked, so you always see exactly which areas your number describes.

Reading your result: what the number means

This quick z score interpretation guide covers the values you will meet most often:

Z-score interpretation bands

Z-score	Meaning
z = 0	The value equals the mean exactly — perfectly average.
z = +1.00	One standard deviation above average — about the 84.13th percentile.
z = −1.00	One standard deviation below average — about the 15.87th percentile.
z = +2.00	Two standard deviations above average — about the 97.72nd percentile. Unusual.
z = −2.00	Two standard deviations below average — about the 2.28th percentile. Unusual.
|z| > 3	Very unusual — fewer than 0.3% of values fall this far out.

Positive z-scores mean above average; negative z-scores mean below average — a negative z-score is not an error, it is simply a below-mean value. The further from zero in either direction, the more unusual the value. Percentiles make this concrete: a z-score of 1.00 (84.13th percentile) beats roughly 84 out of 100 values, while −1.00 (15.87th percentile) beats only about 16.

Quick reference: common z-scores and their probabilities

Memorize these and half of all z-score questions answer themselves. Every number below is a left-tail probability unless noted:

Common z-score values

Z-score	Left-tail probability	Percentile	Why it matters
1.28	0.8997	89.97th	One-tailed 10% cutoff
1.645	0.9500	95.00th	One-tailed 5% cutoff
1.96	0.9750	97.50th	Two-tailed 5% cutoff — 95% confidence
2.00	0.9772	97.72nd	Right-tail probability 0.0228 (2.28%)
2.576	0.9950	99.50th	Two-tailed 1% cutoff — 99% confidence
−1.96	0.0250	2.50th	Lower 95% confidence boundary

The z critical values 1.645, 1.96, and 2.576 are the backbone of confidence intervals — and the 1.96 z score is the most-used number in all of statistics: the significance level 0.0500 pairs with ±1.96 (leaving a tail probability of 0.0250 on each side), which is exactly why 95% of the normal distribution sits inside those confidence interval boundaries.

The 68-95-99.7 rule (the empirical rule)

For data that follows a normal distribution — the classic bell curve — the spread follows a beautifully simple pattern called the empirical rule, or the 68-95-99.7 rule (also written as the 68 95 99.7 rule):

• About 68% of values fall within one standard deviation of the mean (z between −1 and +1) — precisely 68.27%.
• About 95% fall within two standard deviations (z between −2 and +2).
• About 99.7% fall within three standard deviations (z between −3 and +3).

This is why z-scores of ±1.96 matter so much in statistics: 1.96 is the point where exactly 2.5% of the distribution lies beyond it in each tail (a tail probability of 0.0250), leaving 95% in the middle — the foundation of the 95% confidence interval. The calculator above shades these regions on the curve so you can see the rule in action.

Common mistakes to avoid

• Mixing up the tails. A left-tail probability P(X ≤ x) and a right-tail probability P(X > x) answer different questions. Our 97.72nd-percentile example means 97.72% score at or below — the "percent above" is only 2.28%. Always check which tail you need before reporting a number.
• Thinking a negative z-score is a mistake. It is not. A z-score of −1.50 simply means the value sits 1.5 standard deviations below the mean — about the 6.68th percentile. Negatives are normal and expected.
• Confusing percentile with percent. The 97.72nd percentile does not mean "97.72% correct" — it means the score beats 97.72% of the group. Percentile is a rank, not a grade.
• Assuming every dataset is normal. Z-scores are most meaningful when data roughly follows a normal distribution. Heavily skewed data (like incomes) can make z-scores misleading — check the shape first.
• Using the wrong standard deviation. The formula needs the population standard deviation (σ). If you only have a small sample, use the sample standard deviation (s) as an estimate — and note the distinction in your working.

Where z-scores are used

• Grading and test scores. Teachers convert raw exam scores to z-scores to grade on a curve fairly, even when different test versions have different difficulties. Standardized tests report results this way so a score means the same thing every year.
• Hypothesis testing. Researchers compare a test statistic against critical values like ±1.96 (the 5% significance level). If the z-score falls beyond the critical value, the result is statistically significant — unlikely to have happened by chance.
• Confidence intervals. A 95% confidence interval for a mean is built as the sample mean ± 1.96 standard errors — the same 1.96 that appears throughout z-score work, because 95% of the normal distribution sits within ±1.96 standard deviations.
• Quality control. Manufacturers track measurements (parts dimensions, fill weights) as z-scores and flag anything beyond ±3 as a defect signal — under the 99.7% rule, such values should almost never occur naturally.
• Admissions and screening. Schools and employers convert applicant scores to z-scores or percentiles so candidates from different tests, years, or pools can be compared on one scale — many employee screening reports are built on exactly this standardization.
• Sports analytics. Analysts compare athletes across eras and leagues with z-scores: a batting average or sprint time means little across decades of changing conditions, but standard deviations above the league average travel well.

Frequently asked questions

Related calculators

One toolkit for every z-score task — each calculator below handles one job so you never convert the wrong way round: