Z-Score Calculator
Find a z-score from a value, mean and standard deviation, with the percentile and full steps.
Results
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Steps
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Quick Answer
A z-score shows how many standard deviations a value is from the mean: z = (x − μ) ÷ σ. A test score of 85, with a mean of 70 and a standard deviation of 10, has a z-score of 1.5. That is about the 93rd percentile in a normal distribution.
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Table of Contents
How This Z-Score Calculator Works
A z-score tells you how far a value sits from the average, measured in standard deviations. A positive score is above the mean, and a negative score is below it. The calculator also shows the share of values below yours, assuming a normal distribution.
Quick explanation: z = (x − μ) ÷ σ.
- x: your value.
- μ: the mean.
- σ: the standard deviation.
Core Formulas
- Z-score: z = (x − μ) ÷ σ
- Value from a z-score: x = μ + z × σ
- Percentile: the area under the standard normal curve to the left of z
Worked Example
A value is 60, the mean is 75, and the standard deviation is 5.
- 60 − 75 = −15
- −15 ÷ 5 = −3.00
- Only about 0.13% of values are below this in a normal distribution
Reading Your Result
- Near 0: close to the average.
- About ±1: around 68% of normal data falls within this range.
- About ±2: around 95% falls within this range.
- Beyond ±3: very unusual in a normal distribution.
Frequently Asked Questions
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) ÷ σ.
It depends on the context. For a test score, higher is better. A z-score of 1 is one standard deviation above the mean.
Yes. A negative z-score means the value is below the mean.
Use the area under the normal curve to the left of z. This calculator does that for you.
It means the value is equal to the mean.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Standard score | Definition and formula of the z-score. | https://en.wikipedia.org/wiki/Standard_score |
| Wikipedia, 68-95-99.7 rule | Share of normal data within each range. | https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule |
| Wikipedia, Normal distribution | Normal curve and percentiles. | https://en.wikipedia.org/wiki/Normal_distribution |