How Do You Calculate a Z-Score (Standardized Value)?

Calculate a z-score by subtracting the mean from the observed value, then dividing by the standard deviation:

$$z=\frac{x-\mu}{\sigma}$$

This measures how many standard deviations the value $x$ lies above or below the mean.

The Method

For a normal random variable $X\sim N(\mu,\sigma^2)$, standardization transforms $X$ into the standard normal variable $Z\sim N(0,1)$. In the formula, is the observed value, is the population mean, and is the population standard deviation.

Use these steps:

1. Calculate the deviation from the mean: $x-\mu$.
2. Divide that deviation by the standard deviation $\sigma$.
3. Interpret the sign and magnitude of the result.

For example, suppose test scores have mean $\mu=70$ and standard deviation $\sigma=4$. For a score of $x=78$:

$$z=\frac{78-70}{4}=\frac{8}{4}=2$$

Therefore, a score of $78$ is two standard deviations above the mean.

The z-score is dimensionless, so values measured in different units can be compared on the same standardized scale. For instance, $z=-1.5$ means the observation is $1.5$ standard deviations below the mean; it does not mean the original value is negative. To reverse the process, rearrange the formula to $x=\mu+z\sigma$. In the example, a z-score of $1.5$ corresponds to . This rearrangement is useful when a standardized value is given and the original observation is required.

A common misconception is to use $\frac{\mu-x}{\sigma}$. This reverses the sign. The numerator must be observed value minus mean, $x-\mu$.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches SL, sub-topic SL 4.12, and the calculation can appear on Paper 1 or Paper 2. For a command term such as calculate, show the substitution and relevant working, not only the final value. On technology-based questions, use the GDC for normal probabilities or inverse-normal calculations, but still write the standardization equation when it is part of the method.