Z-Score Calculator
Calculate z-scores, percentiles, and probabilities for normal distributions
Formula:
z = (x - μ) / σ
x = μ + (z × σ)
Z-Score
Percentile
%
Better than % of values
Corresponding Raw Score
Z-Score Scale (Normal Distribution)
Probability (P)
P(X ≤ x)
Right Tail
P(X > x)
Two-Tailed
P(|Z| > |z|)
Step-by-Step Calculation
Common Z-Scores Reference
| Z-Score | Percentile | Interpretation |
|---|---|---|
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About Z-Score Calculator
What is a Z-Score?
A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. It allows you to compare values from different normal distributions and find probabilities.
Z-Score Formula:
z = (x - μ) / σ
Where:
- x = individual value (raw score)
- μ (mu) = population mean
- σ (sigma) = population standard deviation
- z = z-score (standard score)
How to Use This Calculator
Mode 1: Calculate Z-Score
- Enter the raw score (x)
- Enter the population mean (μ)
- Enter the standard deviation (σ)
- Get the z-score and corresponding percentile
Mode 2: Find Raw Score from Z-Score
- Enter the z-score
- Enter the population mean (μ)
- Enter the standard deviation (σ)
- Get the corresponding raw score
Understanding Z-Scores
Interpretation
- z = 0: The value equals the mean
- z > 0: The value is above the mean
- z < 0: The value is below the mean
- |z| = 1: The value is 1 standard deviation from the mean
- |z| = 2: The value is 2 standard deviations from the mean
Common Z-Score Values
| Z-Score | Percentile | Interpretation |
|---|---|---|
| -3.0 | 0.13% | Extremely below average |
| -2.0 | 2.28% | Far below average |
| -1.0 | 15.87% | Below average |
| 0.0 | 50.00% | Average |
| +1.0 | 84.13% | Above average |
| +2.0 | 97.72% | Far above average |
| +3.0 | 99.87% | Extremely above average |
The Empirical Rule (68-95-99.7)
For normally distributed data:
- 68% of values fall within 1 standard deviation (z between -1 and +1)
- 95% of values fall within 2 standard deviations (z between -2 and +2)
- 99.7% of values fall within 3 standard deviations (z between -3 and +3)
Real-World Applications
Education
- Standardized Tests: SAT, ACT, GRE, and IQ tests use z-scores
- Grading on a Curve: Teachers use z-scores to normalize test scores
Statistics & Research
- Hypothesis Testing: Z-tests compare sample means to population means
- Outlier Detection: Values with |z| > 3 are often considered outliers
Quality Control
- Six Sigma: Uses z-scores to measure process capability (6σ = 3.4 defects per million)
- Manufacturing: Identify products outside acceptable tolerances
Finance
- Risk Assessment: Measure how unusual a return is compared to historical data
- Credit Scoring: Some credit models use z-score normalization
Frequently Asked Questions
What is a good z-score?
It depends on context. In statistics, z-scores between -2 and +2 are typical (95% of data). In quality control, higher is better. For test scores, above 0 means above average.
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.
What's the difference between z-score and percentile?
A z-score tells you how many standard deviations from the mean. A percentile tells you what percentage of the population scored below that value. They're related through the normal distribution.
When should I use z-scores?
Use z-scores when:
- Comparing values from different distributions
- Finding how unusual a value is
- Standardizing data for analysis
- Conducting hypothesis tests
What if my data isn't normally distributed?
Z-scores assume normal distribution. For non-normal data, consider using percentile ranks or transforming the data first. The Central Limit Theorem helps when working with sample means.