Raw Score Calculator - Convert Z-Score to Original Value

What is a raw score?

A raw score is the original, unstandardized value in a dataset, such as a test score in points or a measured length in millimeters. A raw score calculator converts a standardized z-score back into that original value, using the dataset's mean and standard deviation. This page explains the formula, walks through worked examples, and answers common questions about raw scores.

Raw score formula

The raw score $x$ is calculated from a z-score with this formula:

$$x = \mu + z \times \sigma$$

Where:

• $x$ = raw score (the original value)
• $\mu$ = mean of the dataset
• $\sigma$ = standard deviation of the dataset
• $z$ = z-score (how many standard deviations $x$ is from the mean)

This formula is the inverse of the standard z-score formula, $z = (x - \mu) / \sigma$. Instead of turning a raw value into a z-score, it turns a z-score back into a raw value.

How to calculate a raw score from a z-score

1. Find the mean ($\mu$) of the dataset.
2. Find the standard deviation ($\sigma$), which measures how spread out the data is.
3. Take the z-score ($z$), the number of standard deviations the value sits from the mean. A positive z-score is above the mean; a negative one is below it.
4. Multiply the z-score by the standard deviation, then add the mean: $x = \mu + z \times \sigma$.

Worked examples

Example 1: Test score

A class takes a test with a mean score of 80 and a standard deviation of 5. A student scored a z-score of 1.2. What was the student's raw score?

$$x = 80 + 1.2 \times 5 = 86$$

The student's raw score was 86.

Example 2: Manufacturing measurement

A factory measures a part with mean length 150 mm and standard deviation 2 mm. One part has a z-score of −1.5. What is its actual length?

$$x = 150 + (-1.5) \times 2 = 147$$

The part measures 147 mm.

Valid input ranges

The calculator on this page requires:

• Standard deviation greater than 0 (a standard deviation of zero or less is not mathematically valid).
• Standard deviation no larger than 1,000,000.
• Mean between −1,000,000 and 1,000,000.
• Z-score between −10 and 10.

Most real-world z-scores fall between −3 and 3, since that range covers about 99.7% of values in a normal distribution. A z-score outside that range marks a rare, extreme value.

Where raw scores are used

• Education. Standardized tests report z-scores or scaled scores; converting back to a raw score shows the actual number of points or questions answered correctly.
• Psychology. IQ tests and other assessments are standardized, then converted to raw values to track a person's results over time.
• Manufacturing. Quality control measures parts against a target mean and flags ones with an unusual z-score, then reports the raw measurement for inspection.
• Finance. Standardized risk or performance metrics get converted back to real currency amounts or return percentages for reporting.

Related statistical scores

• Percentile shows a value's rank in a dataset on a 0–100 scale.
• T-score is standardized with a mean of 50 and standard deviation of 10, common in psychology.
• Stanine is a nine-point scale used in education.

Frequently asked questions

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

A raw score is the original value in its natural units, such as points or millimeters. A z-score is that value restated as a distance from the mean, measured in standard deviations. The formula $x = \mu + z \times \sigma$ converts a z-score back to a raw score.

How do I find a raw score from a percentile?

First convert the percentile to a z-score using a standard normal table. Then apply the raw score formula. For example, the 84th percentile is close to a z-score of 1.0.

Can a raw score be negative?

Yes. If the underlying data includes negative values, such as temperature changes or profit and loss, the raw score can be negative too.

What raw score corresponds to a z-score of 0?

A z-score of 0 always gives a raw score equal to the mean, since $x = \mu + 0 \times \sigma = \mu$.

Does the raw score formula work for non-normal data?

The formula itself works for any dataset with a defined mean and standard deviation. But the usual interpretation of a z-score, such as "about 68% of values fall within one standard deviation of the mean," assumes the data follows a normal distribution. For strongly skewed data, that interpretation does not hold even though the arithmetic still does.

How precise is the result?

The calculation is a simple multiplication and addition, so the result is exact for the numbers entered. Precision is limited only by how precisely the mean, standard deviation, and z-score are known.