Z-Score Calculator (with Normal Probability)
- Free
- No sign-up
- Works on phones
- Shows the working
By Yasir Hafeez · Updated
This z-score calculator tells you how many standard deviations a value lies from the mean, and what that means as a probability. Enter the value x, the mean \( \mu \) and the standard deviation \( \sigma \). You get the z-score, the probability of a value below it, \( P(Z < z) \), the probability above it, the probability within \( \pm z \) of the mean, and the percentile, all computed accurately from the normal distribution, so you don’t need a z-table.
It’s free and needs no sign-up.
How to use it
- Enter the value x you’re interested in.
- Enter the mean \( \mu \) and standard deviation \( \sigma \) of the distribution. For a standard normal variable, use 0 and 1.
- Press Calculate.
Formula
$$z = \frac{x - \mu}{\sigma}$$
A positive z is above the mean, a negative z below it. The probabilities are areas under the standard normal curve:
$$P(Z < z) = \Phi(z) = \int_{-\infty}^{z}\frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt$$
Worked example: a test score
Scores have mean 70 and SD 10. A student scores 85 (as loaded above):
$$z = \frac{85 - 70}{10} = 1.5$$
\( P(Z < 1.5) \approx 0.9332 \), so the student scored better than about 93.3% of test-takers: the 93rd percentile.
Worked example: below the mean
A machine fills bottles with mean 70 ml and SD 5 ml. What fraction contain less than 62 ml? \( z = \frac{62 - 70}{5} = -1.6 \), and \( P(Z < -1.6) \approx 0.0548 \): about 5.5% of bottles.
Worked example: comparing scores
SAT score 1300 (mean 1050, SD 200) gives \( z = 1.25 \). An ACT score of 28 (mean 21, SD 5) gives \( z = 1.4 \). The ACT result is relatively stronger, even though the raw numbers aren’t comparable.
Key values
| z | \( P(Z < z) \) | Meaning |
|---|---|---|
| −2 | 0.0228 | 2.3% below |
| −1 | 0.1587 | 15.9% below |
| 0 | 0.5000 | the mean |
| 1 | 0.8413 | 84th percentile |
| 1.645 | 0.9500 | 95th percentile |
| 1.96 | 0.9750 | middle 95% lies within ±1.96 |
| 2.576 | 0.9950 | middle 99% lies within ±2.576 |
For the full table, see z-score and the z table.
Common mistakes
- Using the variance instead of the SD. \( \sigma \) is the square root of the variance.
- Reading the wrong tail. \( P(Z > z) = 1 - P(Z < z) \).
- Non-normal data. The probabilities assume a bell-shaped (normal) distribution. The z-score itself is still a valid measure of distance from the mean.
The calculus behind it
The normal curve \( \frac{1}{\sqrt{2\pi}}e^{-z^2/2} \) has no elementary antiderivative, so its areas are computed numerically. Its total area is 1, a fact proved with a famous double integral; see the integral of e^(x²) and improper integrals.
Related guides
Further reading
- Standard score (Wikipedia) — the z-score, its uses and interpretation.
- Normal distribution (Wikipedia) — the bell curve and the 68-95-99.7 rule.
FAQ
What is a z-score?
The number of standard deviations a value is from the mean: \( z = \frac{x - \mu}{\sigma} \).
What does a negative z-score mean?
The value is below the mean. \( z = -1 \) is one standard deviation below.
How do I convert a z-score to a percentile?
Find \( P(Z < z) \) and multiply by 100. A z-score of 1 is about the 84th percentile.
What is a good z-score?
It depends on context. Above 2 or below −2 is usually considered unusual, since only about 5% of normal data is that far from the mean.