Why adding a constant changes the mean and not the variance, shown rather than asserted.
Slide b and ask them to watch the width bar, not the curve.
The curve slides across the screen and the plus-or-minus one standard deviation bar never changes length. The faint ghost of the original is left in place so the shift is unmistakable.
Then slide a. Now the bar changes, and the mean moves too. Two controls, two different behaviours, and the rule E(aX + b) = aE(X) + b with Var(aX + b) = a²Var(X) stops being two formulas to confuse and becomes a description of what they just watched.
Variance is built from squared distances from the mean. Adding b moves every value and the mean by the same amount, so every distance is unchanged, so the variance is unchanged.
Multiplying by a scales every distance by a, and the squares by a². The square is on the variance and not on the standard deviation, which is why sd(aX + b) is |a|sd(X) with no square at all. The absolute value matters: a negative a flips the distribution and a spread cannot be negative.
| 1. E(3X + 5) when E(X) = 4 | 3 × 4 + 5 = 17. |
| 2. Var(3X + 5) when Var(X) = 2 | 9 × 2 = 18. The 5 contributes nothing. |
| 3. sd(−2X) | |−2| × sd(X), so the spread doubles and stays positive. |
1 markApplying a to the mean and adding b, in that order.
1 markSquaring a for the variance and dropping b entirely.
1 markTaking the absolute value for a standard deviation, which is the step a negative a is testing.
| They give | What it means |
|---|---|
| Variance including b | They applied the mean rule to the variance. The width bar in the widget is the answer; make them watch it again. |
| Variance with a not squared | Confused the variance rule with the standard deviation rule. Ask which one has units of X and which has units of X squared. |
| A negative standard deviation | They took −2 straight through. A spread cannot be negative, and that alone should stop them. |
| Mean unchanged by b | Rare, but it means they have over-generalised “b does not matter” from the variance rule to everything. |
"Does this work for any distribution?" Yes. These two rules need nothing about shape, which makes them unusually powerful and worth saying explicitly.
"What about X + Y?" Different rule, and it needs independence. Flag it now and do it properly next lesson; conflating the two is a real risk here.
"Why would anyone do this?" Unit changes. Celsius to Fahrenheit is exactly aX + b, and asking what happens to the standard deviation of a set of temperatures makes the rule concrete in one sentence.
| What is happening | |
|---|---|
| 1 | Slide b with the instruction to watch only the width bar. |
| 2 | Slide a. Name both rules from what they saw. |
| 3 | The squared-distance argument for why b vanishes. |
| 4 | The three questions, including a negative a. |
| 5 | A unit conversion question in context, then set up sums of random variables for next lesson. |
Do not teach the two rules as a pair of formulas to memorise. They will be swapped under pressure, and the widget prevents it in two minutes.
Do not skip the negative case. It is where the absolute value lives and it is examined.
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.