Z-Score Calculator: Calculate Standard Scores Online
Calculate Z-scores, percentiles, and probabilities instantly with our free online Z-score calculator. A Z-score (standard score) measures how many standard deviations a value is from the mean, making it essential for statistical analysis, standardized testing, and data comparison.
What is a Z-Score?
A Z-score (also called a standard score) is a statistical measurement that describes a value's relationship to the mean of a group of values. It's measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean. A Z-score of 1.0 indicates a value that is one standard deviation from the mean.
Z-Score Formula
The Z-score formula is: Z = (X - μ) / σ
- Z = Z-score (standard score)
- X = Raw score (the value you're evaluating)
- μ = Population mean
- σ = Population standard deviation
Interpretation Guide
- Z > 3: Very High (Exceptional) - 99.9%+ percentile
- Z = 2 to 3: High (Above Average) - 95-99.9% percentile
- Z = 1 to 2: Above Average - 84-95% percentile
- Z = -1 to 1: Average (Normal Range) - 16-84% percentile
- Z = -2 to -1: Below Average - 5-16% percentile
- Z = -3 to -2: Low - 0.1-5% percentile
- Z < -3: Very Low (Exceptional) - <0.1% percentile
Common Uses of Z-Scores
- Standardized Testing: SAT, ACT, GRE, IQ tests use Z-scores to compare performance across different test versions
- Research & Data Analysis: Compare data from different scales or distributions in social sciences, psychology, and medical research
- Quality Control: Identify outliers and defects in manufacturing and production processes
- Finance: Assess investment risk and identify unusual market movements
- Sports Analytics: Compare athlete performance across different metrics and seasons
- Medical Diagnosis: Evaluate test results and identify abnormal values
How to Use This Calculator
- Choose Calculation Mode: Select whether you want to calculate Z-score, raw score, or percentile
- Enter Known Values: Input your raw score, mean, and standard deviation (or percentile depending on mode)
- View Results: Get instant calculations including Z-score, percentile, and probability
- Interpret: Use the interpretation guide to understand what your Z-score means
Understanding Percentiles
A percentile indicates the percentage of scores that fall below a particular value. For example, if your Z-score corresponds to the 84th percentile, it means your score is higher than 84% of all scores in the distribution. The 50th percentile represents the median (mean in a normal distribution).
Normal Distribution
Z-scores are most commonly used with normal (Gaussian) distributions. In a normal distribution:
- 68% of values fall within ±1 standard deviation (Z = -1 to 1)
- 95% of values fall within ±2 standard deviations (Z = -2 to 2)
- 99.7% of values fall within ±3 standard deviations (Z = -3 to 3)
Frequently Asked Questions
What does a negative Z-score mean?
A negative Z-score indicates that the raw score is below the mean. For example, a Z-score of -1.5 means the value is 1.5 standard deviations below the mean.
What is a good Z-score?
"Good" depends on context. For standardized tests, a Z-score above 1 (84th percentile) is generally considered above average. For quality control, values beyond ±2 or ±3 may indicate problems.
Can Z-scores be used for non-normal distributions?
While Z-scores can be calculated for any distribution, their interpretation (especially percentiles) is most accurate for normal distributions. For non-normal data, consider other standardization methods.
How do I convert between Z-score and percentile?
Use a Z-table or calculator (like this one) to convert. For example, Z = 1.0 corresponds to approximately the 84th percentile, and Z = -1.0 corresponds to the 16th percentile.