How Do You Compute a Raw Score?

The first thing we need to understand is that a raw score is associated to a z-score. This is, for a given normal distribution, for a given mean \(\mu\) and standard deviation \(\sigma\), a z-score is uniquely associated with only one raw score.

This unique association between raw-scores and z-scores makes us see them as equivalent scores: This is, they are different numbers but they represent the same thing. Or rather, they contain the same information.

Connection Between z-scores and raw scores

Now that we know about this association, we also know that a z-score is obtained from normalizing the raw score. So, how do we get a raw score from a z-score? Well, let us recall the formula of how we compute a z-score:

\[ Z = \displaystyle \frac{X-\mu}{\sigma}\]

Raw Score to X Score

Then, based on the definition formula of a z-score, we need to solve for (\(X\)) in order to find the corresponding raw score, and we apply the following formula obtained from solving the equation. We get:

\[X = \mu + Z \times \sigma\]

Raw Score Example

Say that you have a normal distribution with mean \(\mu = 4\) and standard deviation \(\sigma = 2\). Say that we have a z-score equal to \(Z = 2.5\). What is the raw score? Using the formula above, we find that

\[X = \mu + Z \times \sigma = 4 + 2.5 \times 2 = 9\]

Z-score Related Calculators

Alternatively, you may be interested in using our z-score calculator with steps , if you are provided with a raw score and need to get the z-score. Or you may be interested in computing probabilities associated to the standard normal distribution, in which case you could use our z score area calculator .