Raw Score Calculator
Enter a mean, standard deviation, and a z-score to find the corresponding raw score, or enter a raw score to find the z-score. Results include the percentile rank and the two-tailed p-value so you get the full statistical picture from one tool.
What is a raw score in statistics?
A raw score is any value measured directly on its original scale, before any transformation. Test marks, blood pressure readings, reaction times, and salary figures are all raw scores. On their own, raw scores can be difficult to interpret because you need context: is 78 points out of 100 on a hard exam better than 78 points on an easy one? Converting to a z-score or percentile places the value on a common scale so you can compare across different distributions.
The raw score formula and the z-score formula
The z-score formula standardises a raw score by expressing how many standard deviations it is from the mean: z = (x - mu) / sigma. Rearranging gives the raw score formula: x = mu + z times sigma. Both formulas use the same three quantities: the population mean (mu), the population standard deviation (sigma), and either the raw score or the z-score. This calculator works in both directions. Set "Solve for" to "Raw score" and enter a z-score to get the matching data value. Set it to "Z-score" and enter a raw score to standardise it.
Percentile rank and p-values
The percentile rank is the proportion of the distribution that falls at or below your score, expressed as a percentage. It comes directly from the standard normal cumulative distribution function (CDF), often written as Phi(z). A z-score of 0 gives the 50th percentile; a z-score of +1.96 gives about the 97.5th percentile. The two-tailed p-value answers a different but related question: how probable is it to observe a z-score at least as far from zero as this one? That is 2 times (1 - Phi(|z|)). A p-value below 0.05 means the result would appear by chance less than 5% of the time under the null hypothesis that the true value equals the mean.
Population vs sample standard deviation
The raw score formula always uses the population standard deviation (sigma), not the sample standard deviation (s). When you are working with a complete population, sigma is calculated by dividing the sum of squared deviations by n. When you only have a sample and need to estimate the population SD, use the sample SD with n-1 in the denominator, but be aware this introduces a small uncertainty into the z-score. For large samples (roughly n greater than 30) the difference is negligible. For small samples, consider using a t-score with the appropriate degrees of freedom instead.
Common z-scores and their percentiles
| Z-score | Percentile rank | Two-tailed p-value | Interpretation |
|---|---|---|---|
| -3.00 | 0.13% | 0.0027 | Extreme low (beyond 3 SDs) |
| -2.58 | 0.49% | 0.0099 | 99% CI lower bound |
| -2.33 | 0.99% | 0.0198 | 98% CI lower bound |
| -1.96 | 2.50% | 0.0500 | 95% CI lower bound |
| -1.65 | 5.05% | 0.1003 | 90% CI lower bound |
| 0.00 | 50.00% | 1.0000 | Exactly at the mean |
| +1.00 | 84.13% | 0.3173 | 1 SD above the mean |
| +1.65 | 94.95% | 0.1003 | 90% CI upper bound |
| +1.96 | 97.50% | 0.0500 | 95% CI upper bound |
| +2.33 | 99.01% | 0.0198 | 98% CI upper bound |
| +2.58 | 99.51% | 0.0099 | 99% CI upper bound |
| +3.00 | 99.87% | 0.0027 | Extreme high (beyond 3 SDs) |
Frequently referenced z-score benchmarks under the standard normal distribution (mean = 0, SD = 1).
Frequently asked questions
What is the raw score formula?
The raw score formula is x = mu + z times sigma, where x is the raw score, mu is the population mean, z is the z-score, and sigma is the population standard deviation. It is simply the z-score formula rearranged to isolate x.
How do I convert a z-score to a raw score?
Multiply the z-score by the standard deviation and then add the mean: raw score = mean + (z-score times standard deviation). For example, with a mean of 100, a standard deviation of 15, and a z-score of 2, the raw score is 100 + (2 times 15) = 130.
How do I convert a raw score to a z-score?
Subtract the mean from the raw score and divide by the standard deviation: z = (raw score - mean) / standard deviation. A positive z-score means the raw score is above the mean; a negative z-score means it is below.
What does a negative z-score mean?
A negative z-score means the raw score is below the population mean. For example, z = -1.5 means the value is 1.5 standard deviations below the mean. The corresponding percentile rank would be about 6.7%, meaning roughly 93.3% of the distribution falls above that point.
What is the difference between a z-score and a percentile?
A z-score measures distance from the mean in standard deviation units: it can be any real number, negative or positive, and it has the same meaning regardless of the original scale. A percentile rank measures position relative to the whole distribution on a scale from 0 to 100. The two are related by the standard normal CDF: a z-score of +1 corresponds to the 84th percentile; a z-score of +2 to roughly the 97.7th percentile.
Does this formula assume a normal distribution?
The raw score and z-score formulas themselves work for any distribution: they simply rescale the data. However, the percentile and p-value outputs shown here assume the distribution is approximately normal. If your data come from a heavily skewed or non-normal distribution, the percentile estimates will be less accurate.