Raw Score Calculator

A raw score is the original, unstandardized value in a dataset — for example, the number of points a student earned on a test. It can be converted to a z-score by subtracting the mean and dividing by the standard deviation, or recovered from a z-score using the formula x = μ + z·σ.

The formula is x = μ + z·σ, where x is the raw score, μ is the mean, z is the z-score, and σ is the standard deviation. This is simply the rearrangement of the standard z-score formula z = (x − μ) / σ.

A z-score tells you how many standard deviations a value lies above or below the mean. A z-score of 0 means the value equals the mean; positive z-scores are above average, and negative z-scores are below average.

Yes. Negative z-scores simply indicate that the raw score falls below the mean of the distribution. For example, a z-score of −1.5 with a mean of 75 and SD of 10 gives a raw score of 60.

The percentile rank tells you what percentage of values in a standard normal distribution fall below your z-score. It is calculated using the cumulative distribution function (CDF) of the normal distribution, Φ(z). A z-score of 1.5 corresponds to approximately the 93.32nd percentile.

A standard deviation of zero is mathematically invalid for this formula, as it would mean all values in the dataset are identical and no conversion is meaningful. The calculator requires σ > 0 to produce a result.

A z-score calculator converts a raw score into a z-score using z = (x − μ) / σ. This raw score calculator works in reverse — you provide the z-score, mean, and standard deviation, and it recovers the original value x using x = μ + z·σ.

Raw score to z-score conversion (and the reverse) is widely used in education for standardized tests, in psychology for IQ and personality scales, in finance for risk modeling, and in any field that uses the normal distribution to compare results across different datasets or measurement scales.