# Z-Score Calculator

- **URL:** https://codeasystem.com/calculators/statistics/z-score-calculator/
- **Category:** statistics
- **Description:** Convert a raw score into a z-score and see its percentile on the standard normal curve instantly.
- **Primary output:** Z-score: 1.5

## Inputs
- Raw score (x) (name: `x`, type: number, example: 85)
- Population mean (μ) (name: `mean`, type: number, example: 70)
- Population SD (σ) (name: `sd`, type: number, example: 10)

## Outputs
- Z-score: 1.5
- Percentile (P(X ≤ x)): 93.32%
- Share above this value: 6.68%
- Distance from the mean: 1.5 standard deviations above

## Formula / methodology
```
z = (x − μ) / σ   Percentile = Φ(z)
```

A z-score re-expresses a raw value as the number of standard deviations it sits above or below the mean. This strips away units and scale, letting you compare a SAT score against a height against a lab measurement on one common ruler.

The percentile comes from the standard normal CDF Φ(z), evaluated numerically with the Abramowitz–Stegun approximation (error below 7.5×10⁻⁸); accurate enough for any reporting purpose.

## Assumptions & limitations
- You supply the true population mean and SD, not estimates.
- Percentiles assume the underlying data is approximately normally distributed.
- σ must be greater than zero.

## How to use
1. **Enter your raw score**; The individual value you want to place in context.
2. **Add the population parameters**; Use published or historical μ and σ for whatever reference group applies.
3. **Read z and percentile**; Positive z means above average; the percentile converts it into “beats X% of the group”.

## Example
Scoring 85 on a test with μ = 70 and σ = 10 gives z = 1.5; about the 93.32nd percentile, meaning roughly 93% of test takers scored lower.

Result for these inputs:

```
Z-score: 1.5
```

## About this calculator
### Why standardize?

Raw scores are only meaningful relative to their distribution. Being +1.5 SD above the mean on height and +0.2 SD above on a fitness test tells you immediately where someone truly stands; something raw units could never reveal across different measures.

### Rules of thumb worth memorizing

- |z| < 2 covers ~95% of normal data.
- z > 2 or z < −2 is conventionally “unusual”.
- Percentile 84 ≈ z = 1; percentile 97.7 ≈ z = 2.

## FAQs
### Can a z-score be negative?

Yes; negative z simply means the value sits below the mean. The percentile row handles interpretation automatically.

### What if I only have sample mean and SD?

Results are still informative but technically a t-statistic; with samples under ~30, use t-distribution percentiles instead of the normal.

### How accurate is the percentile?

The numerical approximation errs by less than 7.5×10⁻⁸ in CDF terms; effectively exact to the displayed decimals.

### Does the percentile assume normality?

Yes. For heavily skewed data, report the empirical percentile from raw data instead of the normal-based one.

## Related calculators
- [Standard Deviation Calculator](https://codeasystem.com/calculators/statistics/standard-deviation-calculator/)
- [Confidence Interval Calculator](https://codeasystem.com/calculators/statistics/confidence-interval-calculator/)
- [Mean, Median & Mode Calculator](https://codeasystem.com/calculators/statistics/mean-median-mode-calculator/)

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Last updated: 2026-08-23 · Version: 1.0.0 · [HTML version](https://codeasystem.com/calculators/statistics/z-score-calculator/)