Z-Score Calculator
Calculate the Z-score (standard score) for a data point. Find the probability and percentile from a standard normal distribution.
Z-Score
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percentile
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P(X ≤ x)
What is Z-Score Calculator?
A Z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. A positive Z-score means the value is above the mean; a negative Z-score means it is below. Z-scores are used in statistics to compare values from different distributions and to find probabilities using the standard normal distribution.
How to use
- 1 Enter the data point value (x) you want to standardize.
- 2 Enter the mean (mu) of the population or distribution.
- 3 Enter the standard deviation (sigma) — it must be greater than zero.
- 4 The result shows the Z-score along with the formula used.
- 5 The percentile and probability cards show where this value falls in the standard normal distribution.
Formula
Example calculation
A student scores 85 on a test with a class mean of 70 and a standard deviation of 10. Z = (85 - 70) / 10 = 1.50. This score is 1.5 standard deviations above the mean, placing the student at approximately the 93rd percentile.
Frequently asked questions
What is a good Z-score?
That depends on context. In quality control, values with |Z| greater than 3 are often treated as outliers. In academic testing, a Z-score of 1.0 means you scored better than about 84% of the population.
What does a negative Z-score mean?
A negative Z-score means the data point is below the mean. For example, Z = -1.5 means the value is 1.5 standard deviations below the mean, placing it at approximately the 7th percentile.
What is the difference between Z-score and percentile?
The Z-score is a raw measure of distance from the mean in standard deviation units. The percentile converts that Z-score into the proportion of values that fall below it, assuming a normal distribution.
Can Z-scores be used for non-normal distributions?
You can always calculate a Z-score, but interpreting it as a percentile only works reliably for normally distributed data. For skewed or non-normal data, the percentile conversion will be inaccurate.
What is the Z-score for the 95th percentile?
A Z-score of approximately 1.645 corresponds to the 95th percentile in a standard normal distribution. This value is widely used in hypothesis testing as the threshold for a one-tailed test at the 5% significance level.