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Z-score Calculator
Calculate the standard Z-score of a raw score given the mean and standard deviation, and find the left and right tail normal probabilities.
Input Score & Parameters
Results
| Z-score | — |
|---|---|
| Percentile / Left-tail Probability — P(Z < z) | — |
| Right-tail Probability — P(Z > z) | — |
What is a Z-score?
A **Z-score** (also known as a standard score) measures how many standard deviations a raw score ($x$) lies above or below the mean ($\mu$) of a population.
By converting a value into a Z-score, you standardise it. This allows you to compare values from different normal distributions. For example, you can compare a score on an SAT test to a score on an ACT test.
The Z-score Formula
The formula to calculate a Z-score is:
$$\text{Z} = \frac{x - \mu}{\sigma}$$
Where:
- $x$ is the raw value.
- $\mu$ is the population mean.
- $\sigma$ is the population standard deviation.
Interpreting Z-scores
- A Z-score of **0** means the value is exactly equal to the mean.
- A **positive** Z-score indicates the value is above the mean (e.g., $1.5$ means 1.5 standard deviations above).
- A **negative** Z-score indicates the value is below the mean (e.g., $-2.0$ means 2 standard deviations below).
Normal Probabilities
Under a standard normal distribution (a normal distribution with a mean of 0 and a standard deviation of 1):
- **Left-tail Probability — P(Z < z):** The probability of a score being less than the given value. This corresponds to the percentile rank of the score.
- **Right-tail Probability — P(Z > z):** The probability of a score being greater than the given value.