# What is the z-score?

Computes the z-score of a value from the mean and standard deviation, or any one of the four from the other three, with the percentile and tail probabilities of the standard normal distribution.

- Page: https://www.acalculator.org/statistics/z-score-calculator
- JSON spec: https://www.acalculator.org/statistics/z-score-calculator.json
- Version: 3db15853a718

## Default answer

Example with the default inputs (Value (x) 85, Mean (μ) 70, Standard deviation (σ) 10): A value of 85 with mean 70 and standard deviation 10 has a z-score of 1.5; 93.32% of a normal population is below it.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| x | Value (x) | The raw score or measurement. |
| mean | Mean (μ) | The mean of the population. |
| sd | Standard deviation (σ) | The standard deviation of the population, more than 0. |
| z | Z-score | How many standard deviations the value is above (+) or below (−) the mean. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| x | Value (x) | The raw score or measurement. |
| mu | Mean (μ) | The mean of the population. |
| sigma | Standard deviation (σ) | The standard deviation of the population, more than 0. |
| z | Z-score | How many standard deviations the value is above (+) or below (−) the mean: (x − μ) ÷ σ. |
| percentile | Percentile | The percent of a normal population below this z-score: 100 × Φ(z). |
| pBelow | P(Z < z), area to the left | The probability that a standard normal value is below z: Φ(z). |
| pAbove | P(Z > z), area to the right | The probability that a standard normal value is above z: 1 − Φ(z). |
| pBetween | P(−\|z\| < Z < \|z\|), area between | The probability that a standard normal value lies within \|z\| of 0. |
| pOutside | P(\|Z\| > \|z\|), two-tailed | The probability that a standard normal value lies further than \|z\| from 0, in either tail. |

## Method

z = (x − μ) ÷ σ; Φ(z) is the standard normal cumulative distribution function.

## Assumptions

- The mean and standard deviation are those of the population the value comes from.
- The percentile and the probabilities assume the values follow a normal distribution. The z-score itself does not.
- The standard deviation must be more than 0.

## Worked examples

1. x = 85, mean = 70, sd = 10 gives z = 1.5, pBelow = 0.933193, pAbove = 0.066807, percentile = 93.31928%, pOutside = 0.133614. Source: hand calculation in content.mdx: (85 − 70) ÷ 10 = 1.5; NIST e-Handbook 1.3.6.7.1 table, 0.5 + 0.43319 = 0.93319; Python NormalDist().cdf(1.5).
2. z = -2, mean = 100, sd = 15 gives x = 70, pBelow = 0.02275, pBetween = 0.9545. Source: hand calculation in content.mdx: 100 + (−2) × 15 = 70; NIST e-Handbook 1.3.6.7.1 table, 0.5 − 0.47725 = 0.02275; Python NormalDist().cdf(-2).
3. x = 130, mean = 100, z = 2 gives sd = 15. Source: hand calculation in content.mdx: (130 − 100) ÷ 2 = 15.
4. x = 60, z = -1, sd = 5 gives mean = 65, pBetween = 0.682689. Source: hand calculation in content.mdx: 60 − (−1) × 5 = 65; NIST e-Handbook 1.3.6.7.1 table, 2 × 0.34134 = 0.68268; Python 1 − 2 × NormalDist().cdf(-1).
5. x = 70, mean = 70, sd = 12 gives z = 0, pBelow = 0.5, pAbove = 0.5, pOutside = 1, pBetween = 0. Source: hand calculation in content.mdx: (70 − 70) ÷ 12 = 0, and Φ(0) = 0.5 by symmetry.

## FAQ

### How do I calculate a z-score?

Subtract the mean from the value, then divide by the standard deviation: z = (x − μ) ÷ σ. A test score of 85 in a class with mean 70 and standard deviation 10 has z = (85 − 70) ÷ 10 = 1.5.

### What does a z-score tell me?

It tells you how many standard deviations a value is from the mean. A positive z is above the mean, a negative z is below it, and z = 0 is exactly the mean. Because it has no units, you can compare values from different scales, such as an SAT score and an ACT score.

### How do I turn a z-score into a percentile?

Look up Φ(z), the area to the left of z under the standard normal curve, and multiply by 100. For z = 1.5, NIST’s table gives an area of 0.43319 from 0 to 1.5, so Φ(1.5) = 0.5 + 0.43319 = 0.93319, the 93rd percentile. This assumes the values follow a normal distribution.

### What is a good or unusual z-score?

No z-score is good or bad on its own. In a normal distribution, about 68.3% of values lie within 1 standard deviation of the mean, 95.4% within 2, and 99.7% within 3. So a z-score beyond ±2 is uncommon (about 1 value in 22), and beyond ±3 is rare (about 1 in 370).

### Can a z-score be negative?

Yes. A negative z-score means the value is below the mean. A z-score of −2 is two standard deviations below the mean, and about 2.3% of a normal population is lower still.

### How do I find the value for a given z-score?

Rearrange the formula: x = μ + z × σ. With a mean of 100 and a standard deviation of 15, a z-score of −2 is the value 100 − 2 × 15 = 70. Enter the mean, standard deviation, and z-score, and leave the value empty.

### Should I use the population or the sample standard deviation?

The formula uses the population mean and standard deviation. If you only have a sample, the sample mean and sample standard deviation are the usual estimates, and the result is an estimate of the z-score.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1, Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
- NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7.1, Cumulative Distribution Function of the Standard Normal Distribution (table). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm