Standardising and z-values

Two tests. Maths is N(60, 8²) and History is N(90, 10²). Slide z and watch both marks move together while the shaded area stays identical.

z = 1.25

70.0Maths mark

102.5History mark

89.4%below, on both

One z, two very different raw marks, one standing.

What z is

z = x − μσ, how many standard deviations x sits above the mean. It is a position, not a probability. Negative means below the mean, and z = 0 is the mean exactly.

Maths, x = 70→ z = 70 − 608 = 1.25 History, x = 102.5→ z = 102.5 − 9010 = 1.25 P(Z < 1.25)= 0.8944 the same for both x = 52 in Maths→ z = −1, and P(Z < −1) = 0.1587

The direction that is examined

Going from a mark to a z is the easy way round. The questions that carry the marks go backwards: you are told a probability and asked for μ or σ.

Worked: μ = 60 and P(X > 75) = 0.1. Find σ.

0.1 above→ 0.9 below, so z = 1.2816 so→ 1.2816 = 75 − 60σ σ= 151.2816 = 11.7

Convert the probability to a left-hand area first, every time. Every calculator and every table wants the area below. “P(X > 75) = 0.1” means 0.9 to the left, and writing that line down before you touch the calculator is what stops the sign error.

Your turn

1. X ~ N(60, 8²). Find the z-value of 48.

2. σ = 10 and P(X < 85) = 0.9. Find μ to 1 decimal place.

3. Ana scores z = 1.4 in Maths and z = 1.1 in History. This tells you:

She scored a higher mark in Maths. She did better relative to her cohort in Maths. Her Maths mark was 1.4 and her History mark 1.1. Nothing, since the tests are different.

Where the marks go

A sketch with the region shaded, on every normal question. It is often a mark and it prevents the complement error.

Converting to a left-hand area before using the inverse function.

Keeping σ and σ² apart. N(60, 8²) has standard deviation 8, not 64, and the notation is deliberately testing that.

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