What a z-score tells you
A z-score (standard score) says how many standard deviations a value is from the mean. A z of 0 is exactly average, +1 is one standard deviation above, and −2 is two below. For data that follow a normal distribution, the z-score converts directly into a probability.
Example: test scores average 70 with a standard deviation of 10. A score of 85 has z = (85 − 70) ÷ 10 = 1.5. Φ(1.5) = 0.9332, so 93.32% of scores fall below 85: the 93rd percentile. Between 60 and 80 (z from −1 to 1) lies 68.27% of scores.
The 68-95-99.7 rule
| Within | Share of values |
|---|---|
| ±1 standard deviation | 68.27% |
| ±2 standard deviations | 95.45% |
| ±3 standard deviations | 99.73% |
Common critical values: z = 1.645 marks the 95th percentile (a one-tailed 5% test), and z = ±1.96 bounds the middle 95% (a two-tailed 5% test or a 95% confidence interval).
To get the mean and standard deviation from your own data, use the standard deviation calculator.
Frequently asked questions
What is a good z-score?
It depends on what you're measuring. On a test, a z of +1 beats about 84% of scores and +2 beats about 98%. In quality control, values beyond ±3 are usually flagged as unusual.
Can a z-score be negative?
Yes. A negative z-score just means the value is below the mean. z = −1 is at about the 16th percentile.
How do I find a p-value from a z-score?
For a one-tailed test, the p-value is the area beyond your z: the "probability above" for a positive z. For a two-tailed test, double it. z = 1.96 gives a two-tailed p-value of 0.05.
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