# Raw Score Calculator

URL: https://speedycalculator.com/calculators/statistics/raw-score/
Last updated: 2026-10-30

> The raw score calculator converts a z-score back into an original data value with X = μ + z·σ. Enter the mean, the standard deviation and either a z-score or a percentile, and it returns the raw score, its distance from the mean and the share of values below and above it.

## Inputs

- I know the
- Mean (μ)
- Standard deviation (σ)
- z-score
- Percentile (%)

## Results

- Raw score (X)
- z-score
- Distance from the mean (z·σ)
- Share of values below X
- Share of values above X
- Mirror value on the other side of the mean

## Formula

```
X = μ + z × σ · from a percentile P: z = Φ⁻¹(P), X = μ + Φ⁻¹(P) × σ
```

## Explanation

A raw score is a value in its original units – points on a test, centimetres of height, seconds on a stopwatch – before any standardisation. Statistics often works with z-scores, which describe a value by how many standard deviations it lies from the mean, but at the end you usually need to translate back: which test score marks the top 10%? What height is two standard deviations above average? What fill weight sits 1.5 standard deviations below the target?

The conversion simply reverses the z-score formula. Starting from z = (X − μ)/σ, multiplying by σ and adding μ gives X = μ + z·σ. A positive z gives a raw score above the mean, a negative z one below it, and z = 0 returns the mean itself. All three inputs must describe the same distribution – the mean and standard deviation of the very group the z-score refers to.

If you know a percentile rather than a z-score, the calculator first finds the matching z with the inverse normal distribution (the 90th percentile is z = 1.2816) and then converts it, which assumes the scores are roughly normally distributed.

### How to use the Raw Score Calculator

- **I know the**
- **Mean (μ)**
- **Standard deviation (σ)**
- **z-score**
- **Percentile**
- Want to work backwards? Type a value into **Raw score (X)** and **z-score** is calculated for you.

Results update as you type, and everything is calculated in your browser.

### How it works

For a z-score, multiply it by the standard deviation and add the mean. For a percentile P, take z = Φ⁻¹(P/100) from the inverse standard normal distribution first. The area below the raw score is Φ(z), the area above 1 − Φ(z).

### Worked example

An aptitude test is scaled to mean 500 and standard deviation 100. A candidate with z = 1.25 has a raw score of **X = 500 + 1.25 × 100 = 625**, higher than about 89.4% of test takers. To find the cut-off for the top 10%, use the 90th percentile: z = 1.2816 and X = 500 + 128.16 = 628.2.

### Percentiles and their z-scores

PercentilezX for μ = 500, σ = 10010th−1.2816371.825th−0.6745432.650th050075th0.6745567.490th1.2816628.295th1.6449664.599th2.3263732.6

### Inputs explained

FieldWhat to enterI know theChoose one: z-score, PercentileMean (μ)A numberStandard deviation (σ)A numberz-scoreA numberPercentileIn %

### What each result means

- **Raw score (X)** = [Mean (μ)] + [z-score used]×[Standard deviation (σ)]
- **z-score** = [z-score used]
- **Distance from the mean (z·σ)** = [z-score used]×[Standard deviation (σ)]
- **Share of values below X** = ncdf([z-score used])×100
- **Share of values above X** = (1 - ncdf([z-score used]))×100
- **Mirror value on the other side of the mean** = [Mean (μ)] - [z-score used]×[Standard deviation (σ)]

## Worked example

Inputs — I know the: z-score; Mean (μ): 500; Standard deviation (σ): 100; z-score: 1.25.

Results — **Raw score (X) = 625**; Distance from the mean (z·σ) = 125; Share of values below X = 89.44 %; Share of values above X = 10.56 %; Mirror value on the other side of the mean = 375.

## Key terms

- **Raw score**: A value in its original units, before standardisation.
- **z-score**: Number of standard deviations a value lies from the mean.
- **Inverse normal Φ⁻¹**: Function that turns a cumulative probability into a z-score.

## FAQ

**How do you find a raw score from a z-score?**
X = μ + z × σ. With μ = 500, σ = 100 and z = 1.25: X = 500 + 125 = 625.

**What raw score corresponds to the 90th percentile?**
z = Φ⁻¹(0.90) = 1.2816, so X = μ + 1.2816σ – 628.2 for a mean of 500 and SD of 100.

**What does a negative z-score give?**
A raw score below the mean. With z = −4, σ = 3 and μ = 60, X = 60 − 12 = 48.

**Can I find the standard deviation from a raw score?**
Yes: σ = (X − μ) ÷ z, as long as z is not zero.

**Do I need normal data?**
Not for X = μ + zσ. Converting a percentile to a z-score does assume a normal distribution.

**Why do the mean and SD have to come from the same data set?**
The z-score is defined relative to one distribution; mixing parameters from different groups gives a meaningless raw score.

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


## Sources

- [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e)
- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)
- [Khan Academy – Statistics and probability](https://www.khanacademy.org/math/statistics-probability)