The Quick Answer
A z-score tells you how many standard deviations a value is from the mean. The formula is:
z = (x − μ) / σ
Where x is the value, μ (mu) is the mean, and σ (sigma) is the standard deviation.
A z-score of 0 means the value equals the mean. A z-score of +1.5 means the value is 1.5 standard deviations above the mean. A z-score of −2 means it is 2 standard deviations below.
You can calculate z-scores instantly with our z-score calculator.
What Is a Z-Score?
A z-score (also called a standard score) converts any data point into a common scale. Instead of asking "is 85 a good score?", you can ask "how does 85 compare to everyone else?"
Without context, the number 85 means nothing. On a test where the average is 70 with a standard deviation of 10, a score of 85 is well above average (z = 1.5). On a test where the average is 90 with a standard deviation of 5, a score of 85 is below average (z = −1.0).
Z-scores solve this comparison problem by standardizing values to a single scale centered at 0.
The Z-Score Formula
The formula has three components:
z = (x − μ) / σ
- x = the raw value you want to standardize
- μ = the population mean (average)
- σ = the population standard deviation
If you are working with a sample rather than a full population, use the sample mean (x̄) and sample standard deviation (s) instead. The formula is the same: z = (x − x̄) / s.
Step-by-Step Example
A class of students takes an exam. The mean score is 72 and the standard deviation is 8. A student scored 88. What is their z-score?
- Identify the values: x = 88, μ = 72, σ = 8
- Subtract the mean: 88 − 72 = 16
- Divide by the standard deviation: 16 / 8 = 2.0
- z = 2.0
The student scored exactly 2 standard deviations above the mean. This places them at approximately the 97.7th percentile — better than about 97.7% of students.
Second Example (Negative Z-Score)
Same class. Another student scored 64.
- x = 64, μ = 72, σ = 8
- 64 − 72 = −8
- −8 / 8 = −1.0
- z = −1.0
This student scored 1 standard deviation below the mean, placing them at about the 15.9th percentile.
How to Interpret Z-Scores
The sign and magnitude of a z-score tell you two things:
- Sign (+/−): Whether the value is above (+) or below (−) the mean
- Magnitude: How far from the mean, measured in standard deviations
Quick Interpretation Guide
| Z-Score Range | Meaning | Percentile Range |
|---|---|---|
| −3.0 or below | Extremely low (rare) | Bottom 0.1% |
| −2.0 to −3.0 | Very low | Bottom 2.3% |
| −1.0 to −2.0 | Below average | 2.3% – 15.9% |
| −1.0 to +1.0 | Average range | 15.9% – 84.1% |
| +1.0 to +2.0 | Above average | 84.1% – 97.7% |
| +2.0 to +3.0 | Very high | 97.7% – 99.9% |
| +3.0 or above | Extremely high (rare) | Top 0.1% |
About 68% of values fall within ±1 standard deviation of the mean. About 95% fall within ±2. About 99.7% fall within ±3. This is called the 68-95-99.7 rule (or empirical rule).
Z-Score to Percentile Conversion
A percentile tells you what percentage of values fall below a given z-score, assuming the data follows a normal distribution.
Common Z-Score to Percentile Reference
| Z-Score | Percentile | Meaning |
|---|---|---|
| −3.0 | 0.13th | 99.87% of values are higher |
| −2.5 | 0.62nd | 99.38% are higher |
| −2.0 | 2.28th | 97.72% are higher |
| −1.5 | 6.68th | 93.32% are higher |
| −1.0 | 15.87th | 84.13% are higher |
| −0.5 | 30.85th | 69.15% are higher |
| 0.0 | 50th | Exactly at the median |
| +0.5 | 69.15th | 30.85% are higher |
| +1.0 | 84.13th | 15.87% are higher |
| +1.5 | 93.32th | 6.68% are higher |
| +1.645 | 95th | 5% are higher |
| +1.96 | 97.5th | 2.5% are higher |
| +2.0 | 97.72th | 2.28% are higher |
| +2.5 | 99.38th | 0.62% are higher |
| +2.576 | 99.5th | 0.5% are higher |
| +3.0 | 99.87th | 0.13% are higher |
The z-scores 1.645, 1.96, and 2.576 are used heavily in statistics for confidence intervals at the 90%, 95%, and 99% levels respectively.
Use the z-score calculator to convert any z-score to a percentile instantly.
Reverse Z-Score: From Z-Score to Raw Value
Sometimes you know the z-score and need to find the original value. Rearrange the formula:
x = z × σ + μ
Example
IQ tests have a mean of 100 and a standard deviation of 15. What raw score corresponds to a z-score of +2?
- x = 2 × 15 + 100
- x = 30 + 100
- x = 130
An IQ of 130 is 2 standard deviations above the mean, which places it at the 97.7th percentile.
Real-World Applications
Standardized Testing
Tests like the SAT, GRE, and IQ scales are built on z-score principles. Your raw score is meaningless without knowing the distribution. Z-scores (or derived scales) let you compare performance across different test administrations with different difficulty levels.
SAT scores, for example, use a rescaled z-score system where the mean is 500 and the standard deviation is 100 for each section. A score of 700 corresponds to z = +2.0.
Quality Control
In manufacturing, z-scores identify defective products. If a bolt is supposed to be 10.0 mm with a tolerance of σ = 0.05 mm, a bolt measuring 10.15 mm has z = +3.0 — outside the acceptable range. Six Sigma methodology is named after the goal of keeping defects within ±6 standard deviations.
Academic Grading
Some courses grade on a curve using z-scores. A student's raw score is converted to a z-score based on class performance, then mapped to a letter grade. This adjusts for differences in exam difficulty across sections or semesters.
Finance
In portfolio analysis, z-scores measure how unusual a return is relative to historical performance. A daily return with z = −3 is a rare event — it happens roughly 0.13% of the time under normal assumptions. (In practice, financial returns have "fat tails," so extreme events happen more often than the normal distribution predicts.)
Medical and Health Sciences
Growth charts for children use z-scores to compare a child's height or weight against the reference population. A height z-score of −2 means the child is 2 standard deviations below the median height for their age and sex.
Lab results often include reference ranges based on population z-scores to flag abnormal values.
Z-Scores From a Dataset
If you have raw data instead of pre-calculated statistics, compute the mean and standard deviation first, then apply the formula.
Example With a Small Dataset
Dataset: 4, 7, 8, 10, 11
- Mean: (4 + 7 + 8 + 10 + 11) / 5 = 40 / 5 = 8.0
- Deviations from mean: −4, −1, 0, +2, +3
- Squared deviations: 16, 1, 0, 4, 9
- Variance (sample): (16 + 1 + 0 + 4 + 9) / (5 − 1) = 30 / 4 = 7.5
- Standard deviation: √7.5 ≈ 2.739
- Z-score for value 11: (11 − 8) / 2.739 ≈ 1.095
The value 11 is about 1.1 standard deviations above the mean of this dataset.
Note: For samples, divide by (n − 1) when calculating variance (Bessel's correction). For populations, divide by n.
Common Mistakes
Confusing Population and Sample Standard Deviation
Population standard deviation (σ) divides by n. Sample standard deviation (s) divides by (n − 1). Using the wrong one changes your z-score. For small datasets, the difference matters. For large datasets (n > 30), it is negligible.
Assuming a Normal Distribution
Z-scores can be calculated for any dataset, but the percentile interpretation (using z-tables) only works reliably when the data is approximately normally distributed. For heavily skewed data, a z-score of 2 might not correspond to the 97.7th percentile.
Ignoring Context
A z-score of +3 in exam scores might mean "excellent student." A z-score of +3 in blood pressure readings might indicate a medical concern. The z-score tells you "how unusual," but you must decide whether unusual is good or bad based on context.
Comparing Z-Scores Across Different Distributions
Comparing z-scores is valid when both come from normal (or approximately normal) distributions. Comparing a z-score from a uniform distribution to one from a normal distribution is misleading.
Using Z-Scores for Very Small Samples
With fewer than about 10 data points, z-scores are unreliable because the sample mean and standard deviation are poor estimates of the true population parameters. For small samples, consider using the t-distribution instead.
Z-Score vs. T-Score
Both measure distance from the mean in standard deviations, but they are used in different situations:
- Z-score: Use when you know the population standard deviation, or when the sample size is large (n > 30). Based on the normal distribution.
- T-score: Use when the population standard deviation is unknown and the sample size is small. Based on the t-distribution, which has heavier tails to account for uncertainty.
As sample size increases, the t-distribution approaches the normal distribution, and t-scores converge to z-scores.
Frequently Asked Questions
What is a z-score in simple terms?
A z-score tells you how far a value is from the average, measured in standard deviations. A z-score of 0 is exactly average. A z-score of +1 means one standard deviation above average. A z-score of −2 means two standard deviations below average.
What is a "good" z-score?
It depends on context. In testing, a positive z-score (above average) is usually desirable. In quality control, a z-score close to 0 (on target) is ideal. There is no universally "good" z-score — it depends on what you are measuring and what outcome you want.
Can z-scores be negative?
Yes. A negative z-score means the value is below the mean. A z-score of −1.5 means the value is 1.5 standard deviations below the mean.
Can z-scores be greater than 3 or less than −3?
Yes, but values beyond ±3 are rare in normally distributed data (about 0.3% of all values). In datasets with outliers or non-normal distributions, extreme z-scores occur more frequently.
What z-score corresponds to the 95th percentile?
A z-score of approximately 1.645 corresponds to the 95th percentile (one-tailed). This means 95% of values in a normal distribution fall below z = 1.645. For two-tailed 95% confidence intervals, the critical z-score is 1.96.
How do I use a z-table?
A z-table lists cumulative probabilities for z-scores. Find the row matching the ones and tenths digit of your z-score, then find the column matching the hundredths digit. The cell value is the probability that a random value falls below that z-score. For example, z = 1.96 gives 0.9750, meaning 97.5% of values fall below.
What is the difference between a z-score and a percentile?
A z-score measures distance from the mean in standard deviations. A percentile tells you what percentage of values fall below a given point. They are related: you can convert z-scores to percentiles using a z-table or calculator. A z-score of +1 corresponds to approximately the 84th percentile.
Do I need a normal distribution to use z-scores?
You can calculate z-scores for any numerical data. However, the standard percentile interpretations (z = 1 ≈ 84th percentile) are only accurate when the data follows an approximately normal distribution. For non-normal data, z-scores still indicate relative position but the percentile mapping changes.
What is the z-score of the mean?
The z-score of the mean is always 0, because z = (μ − μ) / σ = 0 / σ = 0. The mean is, by definition, zero standard deviations from itself.
How are z-scores used in hypothesis testing?
In hypothesis testing, z-scores measure how far a sample statistic is from the hypothesized population parameter. If the z-score exceeds a critical value (e.g., ±1.96 for a 95% confidence level), you reject the null hypothesis. This is the basis of the z-test.