Z-Score Calculator
Calculate the z-score (standard score) of a data point. The z-score measures how many standard deviations a value is from the mean. This calculator also provides the associated probability, percentile, and a visualization of the standard normal distribution.
What Is a Z-Score?
A z-score (also called a standard score) indicates how many standard deviations a data point is from the mean of a distribution. It is a dimensionless measure that allows comparison of values from different datasets.
z = (x − μ) / σ
Where: x = data point, μ = population mean, σ = population standard deviation
Interpreting Z-Scores
- z = 0: The data point is exactly at the mean.
- z > 0: The data point is above the mean.
- z < 0: The data point is below the mean.
- |z| = 1: The data point is one standard deviation from the mean.
- |z| = 2: The data point is two standard deviations from the mean.
- |z| = 3: The data point is three standard deviations from the mean.
68-95-99.7 Rule (Empirical Rule)
For a normal distribution:
- 68% of data falls within ±1 standard deviation of the mean.
- 95% of data falls within ±2 standard deviations of the mean.
- 99.7% of data falls within ±3 standard deviations of the mean.
Applications of Z-Scores
- Standardizing Test Scores: Compare scores across different exams.
- Quality Control: Identify outliers or defects in manufacturing.
- Finance: Measure risk (Value at Risk, VaR) and volatility.
- Medicine: Compare patient measurements to population norms.
- Psychology: Standardized intelligence and personality tests.
Why Use This Z-Score Calculator?
- Instant Calculation: Get z-score, probability, and percentile in one click.
- Visualization: See the z-score on a standard normal distribution curve.
- Step-by-Step: Understand the calculation process.
- Free & Private: No registration, no data storage.
❓ Z-Score Calculator FAQ
What is a z-score?
A z-score measures how many standard deviations a data point is from the mean. It's calculated as z = (x − μ) / σ.
What does a z-score of 1.5 mean?
A z-score of 1.5 means the data point is 1.5 standard deviations above the mean. Approximately 93.3% of data falls below this value.
What is the difference between z-score and standard deviation?
Standard deviation is a measure of spread in the original data units. A z-score is a standardized measure that tells you how many standard deviations a value is from the mean.
What is a good z-score?
A "good" z-score depends on the context. In hypothesis testing, |z| > 1.96 is often considered statistically significant at the 5% level. In quality control, values with |z| > 3 are often considered outliers.
What is the z-score for 95% confidence?
For a 95% confidence interval, the z-score is approximately 1.96 (two-tailed). This means 95% of data falls within ±1.96 standard deviations from the mean.
What is the z-score for 99% confidence?
For a 99% confidence interval, the z-score is approximately 2.576 (two-tailed).
Can a z-score be negative?
Yes, a negative z-score means the data point is below the mean.
What is the maximum z-score?
There is no theoretical maximum. In practice, z-scores rarely exceed ±6 in normal distributions.
How is the z-score used in hypothesis testing?
In hypothesis testing, the z-score is used as a test statistic to determine whether to reject the null hypothesis. The z-score is compared to a critical value from the standard normal distribution.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.