Find a z-score from a value, mean and standard deviation — or go the other way — plus its percentile on the standard normal distribution.
Enter x, mean and standard deviation to find the z-score
Enter a z-score, mean and standard deviation to find x
This free z-score calculator — also a standard score calculator and standard normal calculator — converts a raw value into a z-score and percentile using the mean and standard deviation of its distribution, or runs the process in reverse to recover a raw value from a known z-score. A z-score tells you how many standard deviations a value sits from the mean, which is the single most useful trick for comparing measurements from completely different scales, whether that's test scores, heights, stock returns, or manufacturing tolerances.
A z-score standardizes a value by measuring its distance from the mean in units of standard deviation, using z = (x − μ) ÷ σ. A z-score of 0 means the value equals the mean exactly. Positive z-scores sit above the mean; negative z-scores sit below it. Because the scale is standard deviations rather than raw units, z-scores from entirely different measurements — a test score and a person's height, for example — can be directly compared.
This tool is built for students learning standard scores in a statistics course, teachers building practice problems, data scientists and analysts standardizing features or flagging anomalies, quality-control engineers checking whether a measurement falls within an acceptable range, and researchers comparing values across differently-scaled variables. No advanced statistics background is required — just a value, a mean, and a standard deviation.
Raw numbers alone rarely tell you whether a value is typical or unusual — a score of 85 could be excellent or mediocre depending on the exam. Z-scores solve this by re-expressing any measurement on the same standardized scale, letting you instantly compare it against the rest of its population and convert it into a percentile, the intuitive "higher than X% of values" statement most people find easier to interpret than a raw z-score.
Standardized testing programs convert raw scores into z-scores and percentiles to rank test-takers fairly across different test forms. Quality-control teams use z-scores to flag measurements that stray too far from a manufacturing target. Data scientists standardize features before feeding them into machine-learning models and use z-score thresholds to detect outliers or fraud. Growth charts convert a child's height and weight into age-adjusted z-scores and percentiles for pediatric screening.
How this z-score calculator converts a raw value into a standardized score and a percentile
Positive z = above the mean; negative z = below the mean. The larger |z| is, the further the value is from typical.
About 68% of values fall within z = ±1, 95% within z = ±2, and 99.7% within z = ±3 for normally distributed data.
A 90th percentile z-score means the value is higher than 90% of the distribution — not that it scored 90% on some test.
From picking a direction to interpreting the result
Choose "Find Z-Score" to convert a raw value into a z-score, or "Find x from Z-Score" to convert a z-score back into a raw value.
For Find Z-Score, enter x, the mean (μ), and the standard deviation (σ). For Find x, enter the z-score, mean, and standard deviation instead.
The calculator applies z = (x − μ) ÷ σ, or its inverse x = μ + z·σ, and evaluates the standard normal CDF for the percentile.
The result includes the computed number, the percentile, and a plain-language line explaining how far above or below the mean the value sits.
A positive z-score is above the mean, a negative one is below. Compare the magnitude against the empirical rule (±1, ±2, ±3) for a quick sense of how unusual the value is.
Plug your result back into the other formula — z = (x − μ) ÷ σ or x = μ + z·σ — to confirm it returns your original input.
A test score of 85, on an exam with mean 75 and standard deviation 10
A student scores 85 on a test where the class mean is 75 and the standard deviation is 10. How many standard deviations above average is this score, and what percentile does it correspond to?
Explanation: A z-score of +1.0 places this score solidly in the "above average" range without being extreme — per the empirical rule, about 68% of scores fall within z = ±1, so this result sits right at the upper edge of the typical range. Had the same student instead scored 95 (z = 2.0), the percentile would jump to roughly 97.7%, illustrating how nonlinear the tail of the normal distribution is as z increases.
What the sign, magnitude, and percentile actually mean
| Z-Score Range | General Interpretation | Approx. Percentile |
|---|---|---|
| z ≤ −2 | Far below average — unusual on the low side | Below ~2.3rd percentile |
| −2 < z ≤ −1 | Below average, but within the typical range | ~2.3rd – 15.9th percentile |
| −1 < z < 1 | Close to average — the most common range for normal data | ~15.9th – 84.1st percentile |
| 1 ≤ z < 2 | Above average, but within the typical range | ~84.1st – 97.7th percentile |
| z ≥ 2 | Far above average — unusual on the high side | Above ~97.7th percentile |
Reading the sign: a positive z-score always means the value is above the mean; negative always means below. A z-score of exactly 0 means the value equals the mean and sits at the 50th percentile.
Typical ranges (the empirical rule): for normally distributed data, about 68% of values fall within z = ±1, 95% within z = ±2, and 99.7% within z = ±3. Values beyond z = ±3 are rare enough to often be treated as outliers.
Manual verification: the fastest sanity check is to plug your z-score back into x = μ + z·σ and confirm you recover the original raw value — as shown in the worked example above.
A fractional z-score works exactly like a whole one — it simply means the value sits that many standard deviations from the mean. z = 0.65 is 0.65 standard deviations above average, which lands at about the 74th percentile, while z = 1.20 falls near the 88.5th percentile. Because this calculator evaluates the standard normal CDF at your exact z value, you don't have to round to the nearest table entry or interpolate by hand.
Since the standard normal CDF is fixed, the same percentiles always map to the same z-scores: the 95th percentile is z ≈ 1.645, the 97.5th is z ≈ 1.96, and the 99th is z ≈ 2.326. These values show up constantly in statistics because they are the thresholds behind common confidence intervals and significance levels — 1.645 for a one-tailed 5% cutoff, 1.96 for a two-tailed 5% level, and 2.326 for a one-tailed 1% cutoff.
This is a frequent mix-up. The empirical rule says about 68% of values fall within z = ±1, meaning between 1 standard deviation below and 1 above the mean. A single value at z = 1 sits at the upper edge of that band, which is the 84.13th percentile. The 68% describes the area between −1 and +1, not the area below +1 alone.
Enter the standard deviation that describes the population your value belongs to. If a test score is being compared against an entire exam cohort, use the population σ. If your numbers are just a sample, the sample standard deviation with its n − 1 divisor is the more honest estimate of that population spread. Entering the wrong one shifts every z-score and percentile this calculator returns.
Both modes evaluate the exact same standard normal CDF at the same z-score. Find Z-Score computes z from your value and then reports Φ(z), while Find x from Z-Score uses the z you typed and reports Φ(z) directly. Because the percentile depends only on z — never on the raw units — converting forward and then backward returns the same percentile and the same original value every time.
Switch to the Find x from Z-Score mode and enter the z-score for the 90th percentile, which is z ≈ 1.2816. With your mean and standard deviation in place, the calculator returns x = μ + 1.2816·σ — the value above which roughly 10% of the distribution falls. For the bottom 10%, enter the same z-score with a minus sign.
Where standardizing a value into a z-score and percentile is genuinely useful
Convert a raw test score into a percentile rank comparable across different exam forms.
Flag data points with an unusually high or low z-score as potential outliers or anomalies.
Assess how unusual a return or price move is relative to its historical volatility.
Flag manufacturing measurements that fall outside an acceptable number of standard deviations.
Standardize variables measured on different scales so they can be directly compared.
Compare a child's height or weight against age-based population norms using standardized scores.
Compare an athlete's performance metric against the league or team average.
Flag a temperature or rainfall reading as unusual relative to historical norms.
Verify whether a component's measurement falls within an acceptable standardized range.
Standardize performance scores across teams or review cycles for fairer comparison.
Convert raw exam scores into standardized grades relative to the class distribution.
Compare a biological measurement against a reference population distribution.
Standardize input features to a common scale before training a model.
What this z-score calculator does well, and where care is still needed
Three related but distinct ways to describe the same value
| Term | Definition | Example |
|---|---|---|
| Raw Score (x) | The original, unstandardized measurement in its native units | 85 points on a test |
| Z-Score | The standardized distance from the mean, in standard deviation units | z = (85−75)/10 = 1.0 |
| Percentile | The share of the distribution falling at or below the value | Φ(1.0) × 100 ≈ 84.13rd percentile |
Summary: This z-score calculator gives you an instant, free way to standardize a raw value into a z-score and percentile, or reverse the process to recover a raw value from a known z-score. Pair it with related tools like the Normal Distribution Calculator and Standard Deviation Calculator for a fuller statistical picture of your data.
Common questions about z-scores
Trusted educational references to go deeper on standardized scores
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