# Z-Score Calculator

URL: https://speedycalculator.com/calculators/statistics/z-score/
Last updated: 2026-10-25

> The z-score calculator standardises a value by subtracting the mean and dividing by the standard deviation. It then uses the normal distribution to give the percentile (area below), the area above and the two-tailed p-value for that z-score.

## Inputs

- Value (x)
- Mean (μ)
- Standard deviation (σ)

## Results

- z-score
- Percentile (area below)
- Area above
- Two-tailed p-value

## Formula

```
z = (x − μ) ÷ σ
Percentile = Φ(z)
```

## Explanation

A z-score tells you how many standard deviations a value lies from the mean. It lets you compare results from different scales, such as two exams with different averages.

### How to use it

Enter the value, the mean and the standard deviation. Type a target z into the result to find the raw value that corresponds to it.

### Interpreting z

- About 68% of normal data lies within z = ±1.
- About 95% lies within ±1.96.
- |z| above 3 is unusual.

Percentiles assume the data are approximately normally distributed.

### Inputs explained

FieldWhat to enterValue (x)A numberMean (μ)A numberStandard deviation (σ)A number

### What each result means

- **z-score** = ([Value (x)] - [Mean (μ)]) ÷ [Standard deviation (σ)]
- **Percentile (area below)** = ncdf([z-score]) × 100
- **Area above** = (1 - ncdf([z-score])) × 100
- **Two-tailed p-value** = 2 × (1 - ncdf(abs([z-score])))

### Example values

How the main result changes when **Value** changes and everything else stays at the example values above:

Value (x)z-score41-3.62561.5-1.063821.5102.54.0631236.62516411.75

## Worked example

Score 82 on a test with mean 70 and SD 8.

z = (82 − 70) ÷ 8 = **1.5**. About 93.3% of scores fall below 82.

## Key terms

- **Standard deviation**: Typical distance of values from the mean.
- **Standard normal distribution**: Normal distribution with mean 0 and SD 1.
- **p-value**: Probability of a result at least as extreme as observed, assuming the null hypothesis.

## FAQ

**Can a z-score be negative?**
Yes, it means the value is below the mean.

**Do I need normal data?**
The z-score itself does not; percentiles and p-values do.

**What z is the 95th percentile?**
About 1.645.

**What formula does the Z-Score Calculator use?**
z = (x − μ) ÷ σ.

**Can I work backwards from the result?**
Yes. Type the result you want into z-score and the calculator finds the matching value (x).


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)