Why students who can find z still cannot answer the question, and the inverse direction they always meet cold.
Drag the boundaries and ask what the number in the panel is.
Some will say it is the z value. It is the shaded area, and the distinction matters because z is a position and the probability is a quantity of area.
Then drag to make the shaded region the tails rather than the middle. Watch who can still predict the number. The ones who can have understood the area; the ones who cannot were memorising a procedure.
Going from a value to a probability is what gets practised. Going from a probability back to a value, which is the “top 10% start at what score” question, is what gets examined just as often and rehearsed far less.
The trap is which side the quoted percentage is on. “Top 10%” means 0.9 to the left, and every calculator wants the left-hand area. Make them write which side before they type.
| 1. P(X < 115) | 115 is one standard deviation above the mean, so it is 0.5 plus half of 68%: 0.8413. |
| 2. P(X > 130) | 130 is two standard deviations up, and about 95% lies within two, so each tail is about 2.5%: 0.0228. |
| 3. Top 10% start at | 0.9 to the LEFT gives z = 1.2816, so 100 + 1.2816 × 15 = 119.2. |
1 markA sketch with the region shaded. It is worth a mark on its own and it prevents the complement error.
1 markStandardising correctly, or using the calculator's normal function with μ and σ the right way round.
1 markOn an inverse question, using the left-hand area that matches the wording.
| They give | What it means |
|---|---|
| 0.1587 (Q1) | The complement. They found the tail rather than the body, which a sketch would have caught. |
| 0.6827 (Q1) | Gave the central 68% without the half-and-add. They have remembered the empirical rule and not used it. |
| 1 (Q1) | Rounded 0.8413 to 1, or typed a bound the wrong way. Either way, a probability of exactly 1 should prompt a check. |
| 0.9772 (Q2) | The complement again, and the same fix. |
| 0.0455 (Q2) | Both tails. They answered a two-tailed question that was not asked. |
| 80.8 (Q3) | Used 0.1 to the left, so they found where the BOTTOM 10% ends. This is the side error, and it is the most common single mistake on the sub-topic. |
| 1.2816 (Q3) | Gave the z value and stopped. They did the hard part and skipped the easy one. |
"Do I still need z tables?" No, your calculator does it directly, and the examination expects that. Teach z anyway, because it is what makes 115 and 130 interpretable without a calculator at all.
"What if it is not normal?" Then none of this applies, and saying so is sometimes the mark. Skewed data, or a count, needs something else.
"Why 68, 95, 99.7?" They are just the areas within one, two and three standard deviations. Worth memorising because they make every answer sanity-checkable in your head.
| What is happening | |
|---|---|
| 1 | Dragging, with the “what is this number” question asked twice. |
| 2 | The empirical rule, used to predict Q1 and Q2 before any calculator comes out. |
| 3 | Calculator method for both directions. |
| 4 | The inverse question, with which side written down first, every single time. |
| 5 | A question where σ is the unknown, which forces z to be used properly. |
Do not skip the sketch to save time. It is a mark and it is the cheapest error check in the whole topic.
Do not say “the normal distribution is everywhere”. Heights are, incomes are not, and overclaiming here sets up the central limit theorem confusion at Higher Level.
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.