# Free online z-score and normal distribution calculator

> A z-score measures how many standard deviations a value is above or below the mean. This calculator converts between raw scores, z-scores, and percentiles for any normal.

Calculate z-scores from raw values, convert z-scores to percentiles, or find the z-score for any percentile using the standard normal distribution.

**Interactive version:** https://onlinecalculator.me/math/z-score/
**Category:** Math

## How to use

1. Select a mode — convert a raw value to a z-score, look up a z-score's percentile, or find the z-score for a given percentile.
2. Fill in the required fields.
3. Click **Calculate** to see the full output.

## Formulas

**Raw value to z-score:**
```
z = (x − μ) / σ
```
where x is the raw value, μ is the mean, σ is the standard deviation.

**Cumulative probability (CDF):**
```
P(Z ≤ z) = 0.5 × (1 + erf(z / √2))
```

**Error function approximation (Hart):**
```
erf(x) ≈ 1 − (a₁t + a₂t² + a₃t³) × exp(−x²)
where t = 1 / (1 + 0.47047|x|)
a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556
```

**Two-tailed p-value:**
```
p = 2 × min(P(Z ≤ z), 1 − P(Z ≤ z))
```

## Worked example

A student scores **75** on a test with mean **70** and standard deviation **10**:
```
z = (75 − 70) / 10 = 0.50
P(Z ≤ 0.50) ≈ 0.6915 → 69.15th percentile
two-tailed p = 2 × 0.3085 ≈ 0.617
```

Standard critical values:
```
z = 1.645 → 95th percentile (one-tailed 5%)
z = 1.960 → 97.5th percentile (two-tailed 5%)
z = 2.576 → 99.5th percentile (two-tailed 1%)
```

## Frequently asked questions

### What is a z-score?

A z-score (also called a standard score) is a signed number of standard deviations a data point is from the population mean. Formula: z = (x − μ) / σ.

### How is the percentile calculated from a z-score?

The percentile is the cumulative probability P(Z ≤ z) from the standard normal distribution, expressed as a percentage. The calculator uses Hart's rational approximation to the error function (erf), which is accurate to about 4 decimal places.

### What is a p-value?

The two-tailed p-value is 2 × min(P(Z ≤ z), P(Z > z)). It represents the probability of observing a z-score at least as extreme as the one computed, in either tail of the distribution.

### What does "percentile to z-score" mode do?

Given a percentile (0–100), it returns the z-score that corresponds to that cumulative probability. For example, the 97.5th percentile corresponds to z ≈ 1.96, the critical value used in a two-tailed 95% confidence interval.

### Is this calculator accurate for extreme z-scores?

The approximation used is accurate to roughly 4 decimal places for z in the range −5 to +5. At extreme values (|z| > 5), the probabilities are extremely close to 0 or 1 and small absolute errors matter little.

### How do I share my calculation?

Click Share with my numbers to copy a URL that restores your selected mode and inputs.

## Sources

- [Standard score (z-score)](https://en.wikipedia.org/wiki/Standard_score) — Wikipedia
- [Normal distribution](https://en.wikipedia.org/wiki/Normal_distribution) — Wikipedia

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