See exactly where a value falls in the bell curve.
Enter a value, the mean, and the standard deviation. Get the z-score, the percentile, and the probability of landing above or below, plotted on the standard normal curve.
The numbers
The individual data point you want to evaluate.
The center of the distribution, the population or sample average.
The typical spread of values around the mean. Must be greater than 0.
What this z-score is telling you
Where the value sits on the curve
Standard normal distribution
Result scorecard
Z-score to percentile reference
Cumulative probability P(X < x) for z-scores around your result. Your value is highlighted.
| Z-score | Percentile | P(X < x) | P(X > x) | Two-tailed |
|---|
Z-scores, explained
What Is a Z-Score?
A z-score tells you how many standard deviations a data point is from the mean. The formula is z = (x - μ) / σ. A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative z-scores are below. It standardizes values from different distributions for comparison.
Interpreting Z-Scores
In a normal distribution, a z-score of 1.0 means the value is one standard deviation above the mean (84th percentile). A z-score of 2.0 corresponds to the 97.7th percentile. Values with z-scores beyond ±3 are considered outliers, occurring less than 0.3% of the time.
Z-Scores and Probability
Z-scores connect directly to probability through the standard normal distribution table. The cumulative probability tells you what percentage of the population falls below that z-score. This is essential for hypothesis testing, quality control, and determining statistical significance.
Applications of Z-Scores
Z-scores are used in standardized testing (SAT, IQ scores), financial analysis (identifying unusual stock returns), quality control (Six Sigma), medical research (comparing patient measurements to reference populations), and any field that requires comparing values from different scales.
Common questions
What does a negative z-score mean?
A negative z-score means the value is below the mean. For example, a z-score of -1.5 means the value is 1.5 standard deviations below average.
What z-score corresponds to the 95th percentile?
A z-score of approximately 1.645 corresponds to the 95th percentile (one-tailed), and ±1.96 captures the middle 95% of the distribution (two-tailed).
Can I use z-scores for non-normal distributions?
You can calculate z-scores for any distribution, but the probability interpretations are only accurate for approximately normal (bell-shaped) distributions. For skewed data, consider other methods.
What is a good z-score?
It depends on context. In testing, higher is usually better. In quality control, closer to 0 means closer to target. Generally, z-scores between -2 and +2 are considered typical.
Probability interpretations assume the data is approximately normal (bell-shaped). For skewed or heavy-tailed distributions the percentile is only a rough guide. Estimates for learning and planning only.