Raw Score Calculator

How to Calculate a Raw Score from a Z-Score

To calculate a raw score from a z-score, multiply the z-score by the standard deviation and add the mean. The formula is x = μ + zσ, where x is the raw score, μ is the mean, z is the z-score and σ is the standard deviation. If you already know how to calculate a z-score, this is the same relationship worked backwards.

1. Find the mean (μ) and the standard deviation (σ) of the distribution.
2. Take the z-score (z) you want to convert.
3. Multiply the z-score by the standard deviation: z × σ.
4. Add the result to the mean: x = μ + zσ.
5. Find the percentile for the z-score, so you can see where the raw score ranks.

Worked example: with a mean of 100, a standard deviation of 15 and a z-score of +2.00, the raw score is x = 100 + (2.00 × 15) = 130. That score sits at the 97.72nd percentile, so about 97.72% of scores fall at or below it. The calculator at the top of this page shows every step, so you can check your working line by line.

To go the other way, use the reverse direction on this calculator or the z-score calculator: subtract the mean from the raw score and divide by the standard deviation, z = (x − μ) ÷ σ. A raw score of 115 with a mean of 100 and a standard deviation of 15 gives z = +1.00.

Raw Score, Z-Score and Percentile: How the Three Are Connected

A raw score, a z-score and a percentile are three ways of describing the same result. The raw score is the value on its original scale, the z-score says how many standard deviations that value sits from the mean, and the percentile says what share of scores fall at or below it. Once you know any one of them, together with the mean and standard deviation, you can find the other two, as long as the scores follow a normal distribution. The normal distribution calculator shows the same relationship for probabilities and areas under the bell curve.

Notice the pattern: a z-score of 0 always gives the mean as the raw score, and the percentile is always 50. Positive z-scores give raw scores above the mean, and negative z-scores give raw scores below it. If you need a cut-off value rather than a score, the critical z-value calculator finds the z-score for a chosen level of significance, and the Empirical Rule percentile calculator turns a percentile back into a score.

When to Use a Raw Score, a Z-Score or a Percentile

Use a raw score when the original units matter, for example a test mark, a height in centimetres or a time in seconds. Use a z-score when you need to compare scores from different tests or different scales, because it puts every result on the same standard scale. Use a percentile when you need to explain a result to someone else, because “97th percentile” is easier to picture than a z-score of +2.00.

Test scores are the most common use. Two exams with different means and spreads cannot be compared mark for mark, but their z-scores can. IQ scores work the same way: they are raw scores on a scale with a mean of 100 and a standard deviation of 15, as the IQ bell curve calculator shows, so an IQ of 115 is exactly one standard deviation above average. Teachers also use the three measures together when grading on a curve, and researchers use z-scores to work out what a p-value is for a test statistic. The normal distribution behind all of this is defined in the NIST/SEMATECH e-Handbook of Statistical Methods, which describes the standard normal distribution and standardising scores. You can browse the full set in the statistics tools hub.

Frequently Asked Questions About Raw Scores

How do you calculate a raw score from a z-score?

Multiply the z-score by the standard deviation and add the mean: x = mean + z times standard deviation. For example, with a mean of 100, a standard deviation of 15 and a z-score of +2.00, the raw score is 100 + (2.00 times 15) = 130.

What is a raw score?

A raw score is the original, unconverted value of a measurement or test result, such as 85 marks out of 100 or an IQ test score of 115. It is the value before it is turned into a z-score or a percentile.

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 = (x - mean) / standard deviation. For example, a raw score of 115 with a mean of 100 and a standard deviation of 15 gives z = (115 - 100) / 15 = +1.00.

What percentile is a raw score of 130 with a mean of 100 and a standard deviation of 15?

The 97.72nd percentile. A raw score of 130 has a z-score of +2.00, and 97.72 percent of scores in a normal distribution fall at or below that value.

What is the difference between a raw score and a z-score?

A raw score is the original value on its own scale, while a z-score is that value expressed as a number of standard deviations from the mean. A raw score of 130 and a z-score of +2.00 describe the same result when the mean is 100 and the standard deviation is 15.

What is a good z-score?

There is no single good z-score: it depends on the test. A z-score of 0 is exactly average, +1.00 is around the 84th percentile, and +2.00 is around the 98th percentile. In most school and psychological tests, z-scores between -1.00 and +1.00 are in the typical range.

Can I convert several z-scores to raw scores at once?

Yes. Use the Batch mode on this calculator, enter the z-scores separated by commas or one per line, and you will get a table of raw scores with a percentile for each one.

How are raw scores used for IQ scores and test grades?

IQ scores are raw scores on a scale with a mean of 100 and a standard deviation of 15, so an IQ of 115 is one standard deviation above average. Teachers also use raw scores, z-scores and percentiles to compare results fairly when tests have different means or different spreads.

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