What z actually measures, and the backwards direction that carries the marks.
Ask which is the better performance, 70 in Maths or 102.5 in History.
They will pick the bigger number. Both are z = 1.25 and both beat 89.4% of their cohort, which is the argument for standardising made in one line instead of a paragraph.
Then sweep z negative. Watch for anyone who reads a negative z as a negative mark; it is a surprisingly common reading and it is better to catch it here than in an examination.
Mark to z is the easy way and gets all the practice. The marks sit in the reverse: given a probability, find μ or σ.
Convert the probability to a left-hand area first, every time, in writing. "P(X > 75) = 0.1" becomes "0.9 to the left" on its own line before the calculator is touched. That single habit removes most of the errors in this sub-topic.
| 1. z for 48 in N(60, 8²) | (48 − 60)/8 = −1.5. |
| 2. μ when σ=10 and P(X<85)=0.9 | z = 1.2816, so μ = 85 − 12.816 = 72.2. |
| 3. z = 1.4 against z = 1.1 | B. She did better relative to her cohort in Maths. |
1 markA sketch with the region shaded.
1 markConverting to a left-hand area before using the inverse normal.
1 markSolving the resulting equation for the unknown parameter, with σ not σ².
| They give | What it means |
|---|---|
| +1.5 (Q1) | Lost the sign. 48 is below the mean, so z must be negative, and the sketch would have caught it. |
| −0.1875 (Q1) | Divided by 64. N(60, 8²) has σ = 8; the squared notation is deliberately testing this. |
| 97.8 (Q2) | Added rather than subtracted. If μ were 97.8 then 85 would be below the mean and P(X < 85) could not be 0.9. |
| 85 (Q2) | Gave the mark back. Worth asking what P(X < 85) would be if μ were 85. |
| A (Q3) | Read z as a mark, or assumed the bigger z means the bigger raw score. The two figures are on screen showing otherwise. |
"Can z be negative?" Yes, for anything below the mean, and it is not a bad score. Half of every normal distribution has a negative z.
"Why standardise at all, my calculator does it directly?" Because z is how you compare across different distributions, and because the reverse questions are solved in z and nowhere else.
"What does z = 3 mean?" About one in 740 above it. Worth having, because it makes "a three sigma event" meaningful rather than just impressive-sounding.
| What is happening | |
|---|---|
| 1 | 70 or 102.5? Reveal that they are the same z. |
| 2 | The definition, with a negative example done deliberately. |
| 3 | Forward questions on the calculator, with a sketch every time. |
| 4 | The reverse direction, finding μ and then σ, with the left-area line written first. |
| 5 | A question with both unknown, which needs two equations and separates the group. |
Do not call z a probability, even loosely. It is a position, and the confusion is hard to undo once it sets.
Do not skip the sketch because the calculator is quicker. It is a mark and it is the cheapest error check available.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.