Last updated: July 16, 2026
Z-Score Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
A z-score (standard score) tells you how many standard deviations a data point is above or below the population mean, using Z = (X − μ) / σ. This calculator supports four modes: computing a z-score from a data value, finding a data value from a known z-score, recovering the mean, or solving for the standard deviation. It also computes the corresponding left-tail and right-tail p-values and percentile using the standard normal cumulative distribution function.
A z-score measures how far a data point is from the mean in units of standard deviations. Calculate it with Z equals X minus mu divided by sigma. A z-score of zero means the value is average; positive z-scores are above average and negative z-scores are below average.
Key Takeaways
- A z-score measures how many standard deviations a data point is from the population mean, using the formula Z = (X − μ) / σ.
- Z-scores allow comparison of values from different distributions by standardizing them to a common scale with mean 0 and standard deviation 1.
- The sign of the z-score indicates direction: positive means above the mean, negative means below the mean.
- Under a normal distribution, 68% of values have |z| < 1, 95% have |z| < 1.96, and 99.7% have |z| < 3.
- Z-scores directly translate to percentiles and p-values through the standard normal cumulative distribution function (CDF).
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Z = (X - μ) / σ
Where:
- Z=Z-Score (Standard Score)(dimensionless)
- X=Data Point Value(same as data)
- μ=Population Mean(same as data)
- σ=Population Standard Deviation(same as data)
Worked Examples
Student Test Score
A student scores 62 on an exam where the class mean is 58.75 and the standard deviation is 7.854. Calculate the z-score.
- 1Identify the data point: X = 62
- 2Identify the population mean: μ = 58.75
- 3Identify the standard deviation: σ = 7.854
- 4Apply the formula: Z = (X − μ) / σ = (62 − 58.75) / 7.854
- 5Calculate numerator: 62 − 58.75 = 3.25
- 6Divide by σ: Z = 3.25 / 7.854 ≈ 0.4138
- 7The student is about 0.41 standard deviations above the class mean
IQ Score from Z-Score
Given a z-score of −1.5 on a standard IQ scale (μ = 100, σ = 15), find the corresponding IQ value.
- 1Known z-score: Z = −1.5
- 2Population mean: μ = 100
- 3Standard deviation: σ = 15
- 4Rearrange the formula: X = Z × σ + μ
- 5Substitute: X = (−1.5) × 15 + 100
- 6Calculate: X = −22.5 + 100 = 77.5
- 7An IQ of 77.5 corresponds to a z-score of −1.5
Find the Population Mean
A data point of 85, with z-score 1.0 and standard deviation 10, is used to find the population mean.
- 1Known data point: X = 85
- 2Known z-score: Z = 1.0
- 3Known standard deviation: σ = 10
- 4Rearrange: μ = X − Z × σ
- 5Substitute: μ = 85 − 1.0 × 10 = 85 − 10 = 75
- 6The population mean is μ = 75
Introduction
The z-score (also called a standard score or normal score) measures how many standard deviations a data point lies above or below the population mean. It is one of the most fundamental tools in statistics, enabling comparison of values from different distributions, identification of outliers, and calculation of probabilities under the normal distribution. A z-score of 0 means the value equals the mean; positive z-scores indicate values above the mean, and negative z-scores indicate values below it. Our calculator supports four modes: computing a z-score, finding a data value from a known z-score, recovering the mean, or solving for the standard deviation.
What Is a Z-Score?
A z-score quantifies how far and in what direction a particular data point deviates from the mean, expressed in units of standard deviations. The formula Z = (X − μ) / σ transforms any normally distributed variable into a standard normal distribution with mean 0 and standard deviation 1. This standardization allows direct comparison between measurements on completely different scales — for example, comparing a student's performance on two exams with different means and variances. Z-scores are dimensionless, making them a universal yardstick in statistics. For a deeper understanding of the underlying spread measure, see our Standard Deviation Calculator.
Z-Scores and the Normal Distribution
Z-scores are most powerful when data follows a normal (bell-shaped) distribution. The Normal Distribution Calculator can help you verify normality and compute probabilities. Under a standard normal distribution, approximately 68% of values fall within ±1σ (z between −1 and 1), 95% within ±2σ, and 99.7% within ±3σ — the celebrated empirical rule. The cumulative distribution function (CDF) evaluated at a z-score gives the left-tail probability (p-value), which directly maps to a percentile. Our calculator uses the Abramowitz & Stegun approximation for fast, accurate CDF computation. For more on the theoretical foundation, see the NIST/SEMATECH e-Handbook of Statistical Methods.
Four Calculation Modes
Beyond the classic Z = (X − μ) / σ, our calculator lets you solve for any missing variable: Calculate Z-Score: Given X, μ, and σ, computes the z-score and associated probabilities. Find Data Value (X): Given Z, μ, and σ, rearranges to X = Z·σ + μ. Useful for finding what raw score corresponds to a desired percentile. Find Mean (μ): Given X, Z, and σ, uses μ = X − Z·σ. Helpful when you know the standardized position but need to recover the original mean. Find Standard Deviation (σ): Given X, μ, and Z, computes σ = (X − μ) / Z. Requires Z ≠ 0.
P-Values and Percentiles
Once a z-score is computed, it maps directly to probabilities and percentiles through the standard normal CDF. The left-tail p-value P(Z ≤ z) gives the proportion of values in the distribution that fall at or below the observed value — this equals the percentile divided by 100. The right-tail p-value P(Z > z) = 1 − P(Z ≤ z) represents the proportion of values exceeding the data point. For example, z = 1.645 corresponds to the 95th percentile: 95% of observations fall below this value. Use the Percentile Calculator to explore percentile rankings in empirical datasets, and see Khan Academy: Normal distributions and the empirical rule for an accessible introduction.
Real-World Applications
Z-scores appear across virtually every quantitative field: Education: Standardized test scores (SAT, IQ, GRE) are often reported as or derived from z-scores to allow fair comparisons across different test editions. Finance: The Altman Z-Score predicts corporate bankruptcy risk. In trading, z-scores flag statistically unusual price movements. Medicine: Lab reference ranges are defined partly by z-scores (e.g., bone density T-scores in DEXA scans for osteoporosis). Quality Control: Six Sigma uses z-scores to measure process capability; a process running at 6σ produces fewer than 3.4 defects per million opportunities. Research: Hypothesis testing relies on converting test statistics to z-scores to determine statistical significance. For comparing groups, also explore our Confidence Interval Calculator.
Common Z-Score Reference Values
Several z-score values appear so frequently in statistics that they are worth memorizing: | Z-Score | Percentile | Two-Tailed Coverage | |---------|-----------|---------------------| | ±1.000 | 84.13% / 15.87% | 68.27% | | ±1.645 | 95.00% / 5.00% | 90.00% | | ±1.960 | 97.50% / 2.50% | 95.00% | | ±2.326 | 99.00% / 1.00% | 98.00% | | ±2.576 | 99.50% / 0.50% | 99.00% | | ±3.000 | 99.87% / 0.13% | 99.73% | These critical values are the foundation of confidence intervals and hypothesis tests. For a comprehensive overview of how variance relates to z-scores, see our Variance Calculator and Casella & Berger's *Statistical Inference* (2002), available at major academic libraries.
Quick Reference Card
Z-Score Quick Reference
Quick reference • Z-Score Calculator
Z = (X − μ) / σValid range: Theoretically −∞ to +∞; practically |z| > 3 indicates a rare extreme value
Common Values
⚠ Watch Out
- •Z-scores assume normally distributed data; probability interpretations are invalid for heavily skewed distributions.
- •Use z-scores only with the population standard deviation (σ); use t-scores when only the sample standard deviation (s) is known.
- •A zero standard deviation makes z-scores undefined — every data point would equal the mean.
- •Z-scores identify outliers but do not explain their cause; always investigate extreme z-scores in context.
Pro Tips
- →To convert a z-score back to the original scale: X = Z × σ + μ.
- →Use z = ±1.96 as the critical value for two-tailed hypothesis tests at the 5% significance level.
- →For sample means (not individual values), use the standard error σ/√n instead of σ.
- →Chebyshev's theorem guarantees at least 75% of values lie within |z| ≤ 2, regardless of distribution shape.
FAQs
What does a z-score of 0 mean?
A z-score of exactly 0 means the data point equals the population mean. It is neither above nor below average and sits at the 50th percentile of the normal distribution.
Can a z-score be negative?
Yes. A negative z-score indicates the data point is below the population mean. For example, a z-score of −2 means the value is two standard deviations below the mean, placing it at approximately the 2.3rd percentile.
What is a 'good' or 'bad' z-score?
Context determines this entirely. In academics, a high positive z-score is often good (above-average performance). In risk assessment or medical testing, extreme z-scores (|z| > 2 or 3) signal unusual values that warrant attention. There is no universally good or bad z-score.
Do z-scores require normally distributed data?
Z-scores can be computed for any dataset with a mean and standard deviation, but the probability interpretations (percentiles, p-values) are only strictly valid when the data follows a normal distribution. For non-normal data, Chebyshev's inequality provides a weaker but distribution-free bound on probabilities.
What is the difference between a z-score and a t-score?
A z-score uses the known population standard deviation (σ), while a t-score uses the sample standard deviation (s) and is appropriate when the population σ is unknown and the sample is small. The t-distribution has heavier tails than the normal distribution; as sample size grows, the t-distribution approaches the normal distribution.
How is a z-score used in hypothesis testing?
In a z-test, you compute the test statistic z = (x̄ − μ₀) / (σ / √n), where x̄ is the sample mean, μ₀ is the hypothesized population mean, and n is the sample size. You then compare the z-statistic to critical values (e.g., ±1.96 for a two-tailed α = 0.05 test) or compute the p-value using the normal CDF to decide whether to reject the null hypothesis.