Z-Score Calculator – Standard Score and Probability in 2 Easy Steps
Standard Score (Z-Score) Calculator
Determine how many standard deviations a raw score is from the population mean, along with probability mapping.
| Z-Score Range | Data Coverage | Interpretation |
|---|---|---|
| Between -1 and +1 | ~68.27% of Data | The value is very common and falls within one standard deviation of the mean. |
| Between -2 and +2 | ~95.45% of Data | The value is relatively common but lies on the outer edges of the bell curve. |
| Between -3 and +3 | ~99.73% of Data | The value is an outlier or extreme observation in a normal distribution. |
Use this free z-score calculator to see how far a value is from the mean in standard deviations, and how unusual it is.
Quick answer
A z-score (standard score) tells you how many standard deviations a value X is above or below the mean: z = (X − μ) / σ. Example: a test score of 85 with mean 70 and standard deviation 10 gives z = 1.5. Under a normal curve, P(Z < 1.5) = 0.93319, so the score beats about 93% of results, and P(Z > 1.5) = 0.06681. The calculator updates the z-score and both probabilities as you type.
Table of Contents
Z-scores put different measurements on the same scale. They let you compare a maths score with an English score, spot outliers in a dataset, convert values to percentiles and work out p-values for z-tests.
This z-score calculator gives you the standard score and both tail probabilities at once, along with a sentence that explains the result, so you do not need a z-table.
Z-Score Formula
z = (X − μ) / σ
X = raw score
μ = mean
σ = standard deviationA positive z-score is above the mean, a negative one is below it, and z = 0 is exactly the mean. To go back from a z-score to a raw score, use X = μ + z × σ.
Worked Examples
Example 1: exam score
X = 85, μ = 70, σ = 10. z = (85 − 70) / 10 = 1.5. P(Z < 1.5) = 0.93319, so the student scored higher than about 93.3% of the class.
Example 2: below-average value
X = 62, μ = 70, σ = 10. z = −0.8. P(Z < −0.8) = 0.21186, so about 21% of values are lower.
Example 3: comparing two tests
Ali scored 85 in maths (μ = 70, σ = 10, z = 1.5) and 180 in a fitness test (μ = 170, σ = 8, z = 1.25). Although the raw numbers are very different, the z-scores show the maths result is relatively stronger.
How to Use the Z-Score Calculator
Step 1: Enter X, the Mean and the Standard Deviation
Type the X Raw Score, the μ Population Mean and the σ Standard Deviation. Results appear as soon as all three are filled in. Click Clear Values to start again.
Step 2: Read the Z-Score and Probabilities
The Statistical Results panel shows the calculated z-score, the left-tailed probability P(Z < z), the right-tailed probability P(Z > z) and a summary such as “1.50 standard deviations above the mean”. Click Copy Report to copy the result.
Common Z-Scores and Percentiles
| z-score | P(Z < z) | Percentile |
|---|---|---|
| −2.0 | 0.0228 | 2nd |
| −1.0 | 0.1587 | 16th |
| 0.0 | 0.5000 | 50th |
| 1.0 | 0.8413 | 84th |
| 1.645 | 0.9500 | 95th |
| 1.96 | 0.9750 | 97.5th |
| 2.0 | 0.9772 | 98th |
| 3.0 | 0.9987 | 99.9th |
How to Interpret a Z-Score
| |z| | Meaning |
|---|---|
| Below 1 | Common, within one standard deviation (about 68% of data) |
| 1 to 2 | Less common but normal |
| 2 to 3 | Unusual; only about 5% of values are beyond ±2 |
| Above 3 | Very rare; often treated as an outlier |
Using Z-Scores for Hypothesis Tests
For a one-sample z-test, calculate the test statistic and read the tail probability. Use the right-tailed value for a “greater than” test, the left-tailed value for a “less than” test, and double the smaller tail for a two-tailed test. At α = 0.05, |z| > 1.96 is significant for a two-tailed test.
Common Mistakes
- Using the variance instead of the standard deviation.
- Reading P(Z < z) when the question asks for “greater than”.
- Applying normal probabilities to data that are clearly skewed.
- Forgetting the sign: a negative z means below the mean.
Frequently Asked Questions
What does a z-score calculator do?
It converts a raw value into a z-score using the mean and standard deviation, and shows the probability of a value below and above it on a normal curve.
What is a good z-score?
It depends on context. Above 0 is above average, above 1 is better than about 84% of values, and above 2 is better than about 98%.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean.
How do I turn a z-score into a percentile?
Use the left-tailed probability P(Z < z) and multiply by 100. z = 1.5 gives about the 93rd percentile.
What z-score is significant at 0.05?
For a two-tailed test, |z| greater than 1.96. For a one-tailed test, z greater than 1.645.
Is the z-score calculator free?
Yes. It is free, runs in your browser and needs no sign-up.