To standardise, subtract the mean and divide by the standard deviation: Z = (X − μ) ÷ σ. The result says how many standard deviations a value sits above or below the mean.
This lesson is part of probability distributions. The z-score is then used in reading normal-distribution tail probabilities.
What does a z-score mean?
A z-score of 2 means the value is 2 standard deviations above the mean. A z-score of −1 means 1 standard deviation below it.
Because every normal distribution becomes the same standard normal Z after standardising, one table serves all of them.
Worked example: the same N(50, 16) done two ways
Here is an original example. The masses X of fictional parcels are normally distributed as X ~ N(50, 16), in kg. Find the z-score for a parcel of mass 58 kg.
Step 1: read the notation. N(50, 16) means μ = 50 and σ² = 16. So the standard deviation is σ = √16 = 4.
Step 2: standardise. z = (58 − 50) ÷ 4 = 8 ÷ 4 = 2.
A parcel of 58 kg is 2 standard deviations above the mean.
The wrong route. A student divides by 16 instead: z = 8 ÷ 16 = 0.5. The working looks correct, but the z-score is four times too small, and the probability read from the table will be wrong as well.
| Step | Wrong | Right |
|---|---|---|
| Value of σ | 16 (the variance) | 4 (√16) |
| Calculation | 8 ÷ 16 | 8 ÷ 4 |
| z-score | 0.5 | 2 |
A value below the mean
Use the same parcels. Find z for a parcel of mass 45 kg.
z = (45 − 50) ÷ 4 = −5 ÷ 4 = −1.25.
The negative sign is correct and should be kept. It is used later when you draw the shaded region and choose between a left tail and a right tail.
Going back from z to X
Sometimes the question gives a z-score and asks for the value. Rearrange the formula: X = μ + zσ.
For the parcels, find the mass with z = 0.75.
X = 50 + 0.75 × 4 = 50 + 3 = 53 kg.
Check by standardising again: (53 − 50) ÷ 4 = 0.75. The check takes five seconds and catches a wrong sign.
The mistake that costs marks
The common slip is to treat the second number in N(μ, σ²) as σ. Some questions give the standard deviation in words and others give the variance in the notation, so check which one you have.
- If the question says “standard deviation 4”, use 4.
- If the question says N(50, 16), use σ = 4 because 16 is the variance.
- If the question says N(50, 4²), use 4 directly, since the square is shown.
Check yourself
X ~ N(30, 25). (a) Find the z-score for X = 22.5. (b) Find the value of X that has z = 0.8.
Answer
The variance is 25, so σ = 5.
(a) z = (22.5 − 30) ÷ 5 = −7.5 ÷ 5 = −1.5.
(b) X = 30 + 0.8 × 5 = 30 + 4 = 34.
Check (b): (34 − 30) ÷ 5 = 0.8.
What to study next
With z-scores secure, move to reading normal-distribution tail probabilities, where the z-score becomes a probability. Later, finding unknown normal-distribution parameters runs the formula backwards.
If you want a teacher to watch you standardise on fresh numbers, see online one-to-one Additional Mathematics tuition.