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
| Percentile | z | X for μ = 500, σ = 100 |
|---|---|---|
| 10th | −1.2816 | 371.8 |
| 25th | −0.6745 | 432.6 |
| 50th | 0 | 500 |
| 75th | 0.6745 | 567.4 |
| 90th | 1.2816 | 628.2 |
| 95th | 1.6449 | 664.5 |
| 99th | 2.3263 | 732.6 |
Inputs explained
| Field | What to enter |
|---|---|
| I know the | Choose one: z-score, Percentile |
| Mean (μ) | A number |
| Standard deviation (σ) | A number |
| z-score | A number |
| Percentile | In % |
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)]
Formula
X = μ + z × σ · from a percentile P: z = Φ⁻¹(P), X = μ + Φ⁻¹(P) × 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.
Frequently asked questions
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.