z-scores
Master z-scores for NSW Year 12 Mathematics Standard 2. A z-score standardises a value by measuring how many standard deviations it lies above or below the mean, using \(z=\dfrac{x-\mu}{\sigma}\).
You will learn to calculate a z-score from a value, read its sign and size (above or below the mean, and by how many standard deviations), work backwards to a score with \(x=\mu+z\sigma\), and use that a set of z-scores always has a mean of \(0\) and a standard deviation of \(1\) — a core Standard 2 skill for the normal distribution.
Theory
A z-score is the number of standard deviations a value lies above or below the mean. This Year 12 Standard 2 (NSW) guide shows how to standardise a score with \(z=\dfrac{x-\mu}{\sigma}\), read the sign and size of the z-score, find a score back from its z-score with \(x=\mu+z\sigma\), and use that z-scores have mean \(0\) and standard deviation \(1\).
A z-score (or standardised score) is the number of standard deviations a value lies above or below the mean of its data set. You find it by subtracting the mean and dividing by the standard deviation.
The sign tells you the side of the mean: a positive z-score is above the mean, a negative z-score is below it, and a z-score of \(0\) means the value equals the mean. The size tells you how far: \(z=2\) is two standard deviations away, \(z=-1.5\) is one and a half.
Standardising is reversible with \(x=\mu+z\sigma\), and converting an entire data set to z-scores always produces z-scores with a mean of \(0\) and a standard deviation of \(1\).
Standardise a score with the z-score formula:
Rearrange it to recover the original score from a z-score:
How to calculate and use a z-score
- Identify the value \(x\), the mean \(\mu\) and the standard deviation \(\sigma\).
- Subtract the mean: \(x-\mu\). A positive result is above the mean, a negative result is below.
- Divide by the standard deviation: \(z=\dfrac{x-\mu}{\sigma}\).
- Interpret the sign and size, or go backwards with \(x=\mu+z\sigma\) to find a score from its z-score.
Substitute into \(z=\dfrac{x-\mu}{\sigma}\), then read the sign and size.
| \(z\) | \(=\) | \(\dfrac{x-\mu}{\sigma}\) |
| \(z\) | \(=\) | \(\dfrac{86-70}{8}\) |
| \(z\) | \(=\) | \(\dfrac{16}{8}=2\) |
\(z=2\): the mark is 2 standard deviations above the mean.
The same formula applies; a time under the mean gives a negative z-score.
| \(z\) | \(=\) | \(\dfrac{x-\mu}{\sigma}\) |
| \(z\) | \(=\) | \(\dfrac{31-34}{2}\) |
| \(z\) | \(=\) | \(\dfrac{-3}{2}=-1.5\) |
\(z=-1.5\): the time is 1.5 standard deviations below the mean — a fast swim.
Rearrange to \(x=\mu+z\sigma\) and substitute the z-score.
| \(x\) | \(=\) | \(\mu + z\sigma\) |
| \(x\) | \(=\) | \(250 + (-1.5)(20)\) |
| \(x\) | \(=\) | \(250 - 30 = 220\) |
A z-score of \(-1.5\) is a reaction time of \(220\) ms.
Rearrange to \(\sigma=\dfrac{x-\mu}{z}\), then reuse it to standardise \(18\).
| \(\sigma\) | \(=\) | \(\dfrac{x-\mu}{z}=\dfrac{30-24}{1.5}=4\) |
| \(z_{18}\) | \(=\) | \(\dfrac{18-24}{4}\) |
| \(z_{18}\) | \(=\) | \(\dfrac{-6}{4}=-1.5\) |
(i) \(\sigma=4\). (ii) A mark of \(18\) has \(z=-1.5\), i.e. 1.5 SDs below the mean.
Common pitfalls
Frequently asked questions
What is a z-score?
A z-score, or standardised score, is the number of standard deviations a value is above or below the mean of its data set. A positive z-score is above the mean, a negative one is below, and a z-score of 0 means the value equals the mean.
How do you calculate a z-score?
Subtract the mean from the value, then divide by the standard deviation: z equals (x minus mu) divided by sigma. For example, a mark of 86 in a test with mean 70 and standard deviation 8 has z = (86 - 70) / 8 = 2.
What does a negative z-score mean?
It means the value is below the mean. The size still tells you how far below: a z-score of -1.5 means the value is one and a half standard deviations below the mean.
What are the mean and standard deviation of z-scores?
When a whole data set is converted to z-scores, the z-scores always have a mean of 0 and a standard deviation of 1. That standard scale is what lets you compare values fairly.
How do you find the original score from a z-score?
Rearrange the formula to x = mu + z times sigma. Multiply the z-score by the standard deviation and add the mean. For example, with mean 250 and standard deviation 20, a z-score of -1.5 gives x = 250 + (-1.5)(20) = 220.
Is a higher z-score always better?
Not always. A high positive z-score just means the value is far above the mean. Whether that is good depends on the context: a high z-score is good for a test mark but not for a race time, where a low (negative) z-score means a faster, better result.