Z-Score Calculator
Find how many standard deviations a value lies above or below the mean of a distribution. The z-score is fundamental to hypothesis testing, confidence intervals, and comparing values across different scales.
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Formula
z = (x − μ) / σ
Subtract the population mean (μ) from your data point (x), then divide by the standard deviation (σ). The result is the z-score — a dimensionless number expressing how many standard deviations x is from the mean. A positive z-score means x is above the mean; negative means below. The percentile shown is an approximation using an exponential formula and should be treated as an estimate.
How to use the Z-Score Calculator
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Enter your data point (x)
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Enter your mean (μ)
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Enter your standard deviation (σ)
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Read your results instantly
Results update in real time as you type.
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What a z-score means
A z-score of 0 means the value equals the mean exactly. A z-score of +1 means the value is one standard deviation above the mean. A z-score of −2 means the value is two standard deviations below the mean. For a normal distribution, about 68% of values have a z-score between −1 and +1, and about 95% fall between −2 and +2.
For example, if a student scores 85 on a test where the mean is 75 and the standard deviation is 10, their z-score is +1.0. This means they scored better than approximately 84% of test-takers — one standard deviation above average.
Using z-scores to compare across different scales
Z-scores let you compare values measured on different scales. If a student scores 680 on the SAT (mean 500, SD 110) and 28 on the ACT (mean 21, SD 5), which is relatively better? Their SAT z-score is (680−500)/110 = +1.64, and their ACT z-score is (28−21)/5 = +1.40. The SAT score is relatively stronger.
This technique is widely used in sports analytics, finance, psychology, and any field that needs to normalize measurements before combining or comparing them.
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Limitations of the percentile approximation
Converting a z-score to a percentile exactly requires looking up the cumulative normal distribution table (or using the error function). This calculator uses an exponential approximation that is reasonably accurate in the range z = −2 to +2 but less reliable in the tails.
For precise percentile lookups — especially in academic or clinical work — use a z-table or statistical software. The approximation here is best used for quick, informal estimates. The z-score itself is always exact given your inputs.
Tips & Insights
Z-scores work best for normal distributions
Z-scores are most meaningful when data is roughly bell-shaped. For highly skewed distributions, a z-score of +2 may not correspond to the 97.7th percentile — the tails behave differently.
Use sample mean and SD for small datasets
If you calculated your mean and standard deviation from a small sample (fewer than ~30 values), use a t-distribution instead of z-scores for inference. Z-scores assume you know the true population parameters.
Negative z-scores are normal
Half of all values in a symmetric distribution have negative z-scores. A z-score of −0.5 is not bad — it simply means the value is half a standard deviation below the mean.
Worked Examples
Standardized test performance
Z-score: +1.00. This student scored one standard deviation above the mean, placing them at approximately the 84th percentile.
Blood pressure reading
Z-score: +1.00. The reading is one standard deviation above the population mean, suggesting mildly elevated pressure.
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Frequently Asked Questions
What is a z-score?
A z-score (or standard score) measures how many standard deviations a value is from the mean. It standardizes values so you can compare data points from different distributions on the same scale.
What is a good or bad z-score?
There is no universally good or bad z-score. Context determines whether being above or below the mean is desirable. A z-score of +2 is great for an exam grade but concerning for a blood pressure reading.
Can a z-score be negative?
Yes. A negative z-score simply means the value is below the mean. For example, a z-score of −1.5 means the value is 1.5 standard deviations below the mean.
What z-score corresponds to the 95th percentile?
A z-score of approximately +1.645 corresponds to the 95th percentile in a standard normal distribution. This is the critical value used in 90% confidence intervals (one-tailed).
What is the difference between a z-score and a t-score?
A z-score uses the known population standard deviation. A t-score is used when the population standard deviation is unknown and must be estimated from a sample — especially important with small sample sizes.
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