Z-Score Calculator
Enter a data value, the mean and the standard deviation to find its z-score — how many standard deviations it sits from the mean.
A z-score tells you how many standard deviations a value is above or below the mean. Subtract the mean from the value and divide by the standard deviation: z = (x − mean) ÷ SD. A test score of 85 in a group with mean 70 and standard deviation 10 has a z-score of (85 − 70) ÷ 10 = 1.5, meaning it is 1.5 standard deviations above average.
How to Use This Calculator
Enter three values: the raw score x you want to standardize, the mean of the distribution, and the standard deviation. The calculator returns the z-score and tells you how many standard deviations, and in which direction, the value lies from the mean.
Use the population mean and standard deviation if you know them; otherwise use the sample estimates. A positive z-score means the value is above the mean, a negative one means below, and zero means it equals the mean. Z-scores let you compare values from different distributions on a common scale — a z of 1.5 is equally unusual whether it came from test scores, heights or reaction times.
How the Calculation Works
Standardizing shifts and rescales a value so it is expressed in standard-deviation units. Subtracting the mean centers the distribution on zero; dividing by the standard deviation rescales it so one unit equals one standard deviation. The result, z = (x − mean) ÷ SD, is dimensionless. Any normal distribution transformed this way becomes the standard normal distribution, which has a mean of 0 and a standard deviation of 1 — the basis for z-tables and probability lookups.
Worked Example
Suppose an exam has a mean of 70 and a standard deviation of 10, and you scored 85. Your z-score = (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5. That means your score is 1.5 standard deviations above the average. Using the standard normal distribution, a z of 1.5 corresponds to roughly the 93rd percentile — you scored higher than about 93% of test-takers. A score of 60 would give z = (60 − 70) ÷ 10 = −1.0, one standard deviation below the mean.
What the Result Means
The z-score places a value on a universal scale of unusualness. Values within about ±1 are typical, values beyond ±2 are uncommon (outside the middle 95% for normal data), and values beyond ±3 are rare. Because z-scores are unit-free, they let you compare across different measurements and are the entry point for finding probabilities and percentiles from a standard normal table. They also underlie outlier detection: a common rule flags any point with |z| greater than 3.
Assumptions and Limitations
The z-score itself is a simple, exact transformation for any data. However, interpreting a z-score as a percentile or probability assumes the underlying distribution is approximately normal — for strongly skewed or heavy-tailed data, the same z-score corresponds to a different percentile. The result is only as good as the mean and standard deviation you supply; using sample estimates from small samples adds uncertainty. Standard deviation must be greater than zero.
When to Use a Z-Score Calculator
Students and analysts use this to standardize a value — to see how many standard deviations it sits from the mean. Run it to compare values from different scales, find percentiles, or flag outliers. Enter the value, mean, and standard deviation to get the z-score.
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Open calculator →Frequently Asked Questions
What is a z-score?
A z-score, or standard score, is the number of standard deviations a value lies from the mean. It is found with z = (x − mean) ÷ standard deviation and is positive above the mean, negative below.
What is a good or normal z-score?
Most values fall between about −2 and +2. A z near 0 is typical, beyond ±2 is uncommon, and beyond ±3 is rare. What counts as 'good' depends on context — for a test, a higher positive z is better.
How do I convert a z-score to a percentile?
Look the z-score up in a standard normal table or use a normal CDF. For example, z = 1.5 corresponds to about the 93rd percentile, meaning the value exceeds roughly 93% of the distribution.
Can a z-score be negative?
Yes. A negative z-score simply means the value is below the mean. A z of −1.0 is one standard deviation below average; the sign shows direction, the magnitude shows distance.
When should I not use a z-score for probability?
When the data is far from normal — strongly skewed or heavy-tailed. The z-score is still valid as a standardized distance, but converting it to a percentile assumes an approximately normal distribution.
What z-score is considered an outlier?
A common rule of thumb flags values with an absolute z-score greater than 3 as potential outliers, since fewer than 0.3% of normal data lies that far from the mean.
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