Raw score calculator

The formula

x = μ + z·σ

μ is the mean, σ the standard deviation and z the number of standard deviations above (positive) or below (negative) the mean. For a percentile P, first find z = Φ⁻¹(P/100), the inverse of the standard normal cumulative distribution.

A worked example from a z-score

The default uses an IQ-style scale with μ = 100 and σ = 15, and a z-score of 1.5.

x = 100 + 1.5 × 15 = 100 + 22.5 = 122.5

A person 1.5 standard deviations above the mean has a score of 122.5. On a normal distribution 93.3% of scores fall below that, so it is roughly the 93rd percentile.

A worked example from a percentile

Switch the input to a percentile and keep the default of 90. The z-score with 90% of the area to its left is 1.2816, so

x = 100 + 1.2816 × 15 = 119.22

A score of about 119.2 marks the 90th percentile: 90% of people score below it and 10% above. The same method answers questions such as "what score do you need to be in the top 5%?" Enter the 95th percentile, z = 1.6449, and the answer is 100 + 24.7 = 124.7.

Interpreting the result

The raw score is in the units of the original measurement: points, centimetres, seconds. The z-score is unit-free, which is why it is used to compare across tests; converting back is how you report what a standardised result means on a real scale. The chart shades the area below the score, which is the percentile.

The z to raw-score conversion itself is exact for any distribution; it is only a change of scale. The link between z and percentile is where the normal assumption comes in. For test scores, heights and measurement errors it is usually reasonable; for incomes, waiting times and other skewed data it can be badly off in the tails.

Other standard scales

The same formula converts between any standardised scales. T-scores use μ = 50 and σ = 10, so z = 1.5 is a T-score of 65. Stanines use μ = 5 and σ = 2, and many exam boards scale marks to a chosen mean and spread in exactly this way. Convert a score to z on one scale, then use x = μ + zσ with the other scale's mean and standard deviation.

Common mistakes

• Dropping the sign of a negative z-score. z = −1.5 gives 77.5, not 122.5.
• Using the variance instead of the standard deviation. With a variance of 225, σ is 15, not 225.
• Reading a z-table for the area above when you need the area below, which gives the 10th percentile instead of the 90th.
• Applying the percentile method to strongly skewed data.