Free Z-Score Calculator
Calculate the standard score (z-score) of any value: how many standard deviations it sits above or below the mean, computed instantly in your browser.
Enter a value, the mean, and the standard deviation to get the standard score z = (x − μ) / σ.
- Exact value
- 1.5
- x − μ (deviation)
- 15
The value lies 1.5 standard deviations above the mean.
A positive z is above the mean, a negative z is below it, and z = 0 sits exactly on the mean. Values with |z| greater than 2–3 are unusually far from average.
Quick answer
A z-score is the number of standard deviations a value lies from its mean, found with z = (x − μ) / σ, where x is the raw value, μ is the population mean, and σ is the standard deviation. For example, a score of 85 in a distribution with mean 70 and standard deviation 10 has a z-score of (85 − 70) / 10 = 1.5, meaning it is 1.5 standard deviations above average. A positive z is above the mean, a negative z is below it, and z = 0 sits exactly on the mean.
Formula & method
Standard score (z-score)
z = (x − μ) / σ
- z — Standard score (number of standard deviations from the mean)
- x — Raw value being standardized
- μ — Population mean of the distribution
- σ — Population standard deviation (must be greater than 0)
Subtract the mean from the value, then divide by the standard deviation. A positive z is above the mean, a negative z is below it, and z = 0 is exactly on the mean. The formula is undefined when σ = 0.
Examples
- Input
- x = 85, μ = 70, σ = 10
- Result
- z = 1.5000
- Why
- z = (85 − 70) / 10 = 15 / 10 = 1.5. The score is 1.5 standard deviations above the class average.
- Input
- x = 78, μ = 85, σ = 4
- Result
- z = −1.7500
- Why
- z = (78 − 85) / 4 = −7 / 4 = −1.75. The value is 1.75 standard deviations below the mean, so the z-score is negative.
- Input
- x = 185, μ = 175, σ = 6
- Result
- z = 1.6667
- Why
- z = (185 − 175) / 6 = 10 / 6 = 1.66666…, which rounds to 1.6667 at 4 decimals.
- Input
- x = 130, μ = 100, σ = 15
- Result
- z = 2.0000
- Why
- z = (130 − 100) / 15 = 30 / 15 = 2. An IQ of 130 is exactly 2 standard deviations above the mean of 100.
When to use this tool
- Comparing values measured on different scales — e.g. an SAT score against an ACT score — by converting both to a common standard-score scale.
- Spotting outliers, where values with |z| greater than about 2 or 3 are unusually far from the mean.
- Looking up probabilities or percentiles in a standard normal (z) table after standardizing a normally distributed value.
- Standardizing features before feeding them into statistics or machine-learning models so each variable is centered and scaled.
Common mistakes
- Swapping x and μ in the numerator. The formula is value minus mean, (x − μ); reversing it flips the sign of every z-score.
- Using the variance instead of the standard deviation in the denominator. You must divide by σ, not σ². If you only have the variance, take its square root first.
- Entering σ = 0. A standard deviation of zero means every value is identical and the z-score is undefined (division by zero), so this tool returns no result.
- Mixing sample and population statistics. The classic z-score uses the population mean μ and population standard deviation σ; if you only have a sample estimate, your z is an approximation, not an exact standard score.
Frequently asked questions
What is a z-score?
A z-score, or standard score, tells you how many standard deviations a value is above or below the mean of its distribution. It is calculated as z = (x − μ) / σ. A z of +1 means the value is one standard deviation above the mean; a z of −2 means it is two standard deviations below the mean.
What does a negative z-score mean?
A negative z-score simply means the value is below the mean. For example, z = −1.75 says the value sits 1.75 standard deviations beneath the average. The sign carries no judgment about good or bad — it only indicates direction relative to the mean.
Is a high z-score good or bad?
It depends on context. A high positive z is desirable for things you want to maximize, like test scores or income, but undesirable for things you want low, like cholesterol or error rates. The z-score only measures distance from the mean; whether that is good depends on what is being measured.
What z-score is considered an outlier?
A common rule of thumb flags values with an absolute z-score greater than 3 as outliers, since under a normal distribution only about 0.3% of data falls beyond ±3σ. Some analysts use a stricter ±2 threshold (about 5% of data) depending on how sensitive the application is.
How do I convert a z-score to a percentile?
Look the z-score up in a standard normal (z) table or use a normal cumulative distribution function. The table gives the proportion of values below that z. For instance, z = 0 corresponds to the 50th percentile, z = 1.0 to roughly the 84th percentile, and z = 2.0 to about the 97.7th percentile.
What is the difference between z-score and standard deviation?
Standard deviation (σ) is a fixed property of a dataset describing its typical spread. A z-score is computed per value and expresses that value's distance from the mean in units of standard deviation. In short, σ is the ruler and the z-score is the measurement read off that ruler.
Sources & references
External references open in a new tab. We are independent and not affiliated with these organizations.
- ✓ Free to use
- ✓ No sign-up required
- ✓ Runs entirely in your browser — nothing is uploaded.
- ✓ Formula and method shown above
Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.
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