Z-Score
1.5000
Percentile
93.32%
P(above)
6.68%
Must be greater than 0
The value 85 is 1.50 standard deviations above the mean of 70.
93.32%
6.68%
86.64%
Probability within ±1.50 standard deviations
z = (x - μ) / σCalculates how many standard deviations a value is from the mean. Positive z-scores indicate above-average values, negative z-scores indicate below-average values.
Where:
x= The observed value (data point)μ= The population mean (average)σ= The population standard deviationPercentile = Φ(z) × 100The cumulative distribution function Φ(z) gives the probability that a standard normal random variable is less than or equal to z, which directly translates to the percentile.
Where:
Φ(z)= Standard normal CDF evaluated at zz= The z-score valueInputs
Result
A score of 85 is 1.5 standard deviations above the class mean of 70 (SD = 10), placing it at the 93.32nd percentile. Only 6.68% of students scored higher.
Inputs
Result
A height of 62 inches is 1.33 standard deviations below the mean of 66 inches, placing it at roughly the 9th percentile.
Inputs
Result
An IQ of 130 is exactly 2 standard deviations above the mean of 100. This places the individual at the 97.72nd percentile, meaning they score higher than about 97.7% of the population.
A z-score (standard score) measures how many standard deviations a data point is from the mean. A z-score of 0 means the value equals the mean, positive z-scores are above the mean, and negative z-scores are below. It standardizes values from any normal distribution to the standard normal distribution.
| Z-Score | Percentile | Meaning |
|---|---|---|
| -2.0 | 2.28% | Far below average |
| -1.0 | 15.87% | Below average |
| 0.0 | 50.00% | Exactly average |
| +1.0 | 84.13% | Above average |
| +2.0 | 97.72% | Far above average |
The z-score formula is z = (x - μ) / σ, where x is your observed value, μ is the population mean, and σ is the standard deviation. For example, if a test score is 85 with a mean of 70 and standard deviation of 10, the z-score is (85 - 70) / 10 = 1.5.
| Component | Symbol | Example |
|---|---|---|
| Observed value | x | 85 |
| Population mean | μ | 70 |
| Standard deviation | σ | 10 |
| Z-score | z | 1.5 |
The 68-95-99.7 rule (empirical rule) describes how data is distributed in a normal distribution: approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. This corresponds to z-score ranges of ±1, ±2, and ±3.
| Range | Z-Score | % Within |
|---|---|---|
| ±1 SD | ±1.0 | 68.27% |
| ±2 SD | ±2.0 | 95.45% |
| ±3 SD | ±3.0 | 99.73% |
| ±4 SD | ±4.0 | 99.9937% |
To convert a z-score to a percentile, use the standard normal cumulative distribution function (CDF). A z-score of 0 equals the 50th percentile. Positive z-scores are above the 50th percentile and negative z-scores are below. For example, z = 1.96 corresponds to the 97.5th percentile.
| Z-Score | Percentile | Common Use |
|---|---|---|
| ±1.645 | 5% / 95% | 90% confidence interval |
| ±1.96 | 2.5% / 97.5% | 95% confidence interval |
| ±2.576 | 0.5% / 99.5% | 99% confidence interval |
Z-scores are used in standardized testing (SAT, IQ scores), quality control (Six Sigma), medical diagnostics (growth charts, BMD scores), finance (risk assessment), and scientific research (hypothesis testing). They allow comparison across different scales and units.
| Field | Application | Example |
|---|---|---|
| Education | Test scoring | SAT percentiles |
| Manufacturing | Quality control | Six Sigma |
| Medicine | Growth charts | BMD T-scores |
| Finance | Risk modeling | VaR calculation |
The z-score is one of the most fundamental concepts in statistics. It transforms any value from a normal distribution into a standardized score that tells you exactly how far that value is from the mean, measured in units of standard deviation. This standardization makes it possible to compare values from completely different distributions.
The standard normal distribution has a mean of 0 and a standard deviation of 1. When you convert a raw score to a z-score, you are placing it on this universal scale. A z-score of 1.5 always means 1.5 standard deviations above the mean, whether you are measuring test scores, heights, or temperatures.
Our z-score calculator instantly computes the standard score, corresponding percentile, and probabilities. Enter any value along with the population mean and standard deviation to see where that value falls in the normal distribution.
Last Updated: Mar 9, 2026
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