Week 4 Session: A Number With a Shape

A national agency publishes its uncertainty on purpose, in a grammar you can borrow — two numbers, a percentage, and a statement of what each one means

15 March 2027

Bring your week 2 chain and your week 3 ledger. You will be rewriting both.

For three weeks you have been asked how wrong a number could be. Today is about saying so out loud, in a form somebody else can use.

Somebody already solved this#

Most institutions publish a single number and let you assume what you like about it. The Bureau of Meteorology does not. Its rainfall forecasts carry their uncertainty on the surface, deliberately, and it explains the grammar in public:

“Rainfall forecasts are probabilistic. This means they provide a possible range of rainfall amounts, and then the chance of receiving at least that amount of rain. This is different to a maximum or minimum temperature forecast, which is represented by one single number.”

Read that last sentence twice. A national agency is telling you that some quantities should not be published as one number, and saying which of its own are which.

The grammar has three parts, and the most common mistake is to weld them together:

• The chance of rain is the probability of at least 0.2 mm — the smallest amount the gauge can measure. It says nothing about how much.
• The two rainfall amounts are a pair with confidences attached. In BOM’s words, “The first number has a high (75%) chance of occurring, while the second number has a low (25%) chance of occurring.” Their own worked example: a forecast showing “a 75% chance of at least 7 mm of rain and a 25% chance of at least 25 mm of rain.”
• The range is chosen, not padded. BOM says it is picked because it “captures the range of rainfall scenarios that are most likely to occur, and the confidence in that forecast.”

Spend the first part of the session with a real forecast open, working out which number is which and what each one is actually claiming. Ask around the room how people had been reading the percentage before today, and whether anyone had been taking it for the chance of the amounts rather than the chance of at least 0.2 mm.

Translate three of your own#

Take three point estimates you already own:

1. your week 2 bus stop number;
2. a published figure you have used recently for anything;
3. a quantity from this week’s news, as reported.

Rewrite each in that grammar: a value you are fairly confident the truth exceeds, a value you think it probably does not, and a plain-language statement of how confident you are. Plain language throughout — “I’d be surprised if it were under X, and quite surprised if it were over Y” is a perfectly good professional sentence.

The discipline is in choosing the width honestly. A range wide enough that nothing could fall outside it has not told anyone anything; you have swapped a confident claim for an empty one.

A range narrowed to look authoritative claims precision you have not earned, which is the week 1 problem again. The width should be an honest report of what you know, and sometimes that is embarrassingly wide.

Commit, then look#

Now one you cannot reason your way around.

You will each be given a BOM weather station and a past date. The rainfall total recorded at that station on that day is already a fact — measured, published, and sitting in the Bureau’s Climate Data Online archive. Nothing about it is undecided. The only uncertainty is in the room.

You are not predicting the weather. You are putting a range around a number that already exists and that you happen not to know.

Write down, before anyone opens a browser:

• a low value you are fairly confident the recorded total is above;
• a high value you are fairly confident it is below;
• how confident, in words;
• and one line: what would have to be true about that day for your range to miss?

Then look it up.

The last line is the one that pays. A range that missed is not a failure worth much on its own; a range that missed for a reason you had already written down is the beginning of knowing something about yourself as an estimator.

Keep the sheet#

A range with a stated confidence records how firmly you held it; a bare number does not. That is what makes today’s sheet worth keeping rather than throwing out, and the lecture takes the argument further.

BOM does the institutional version of this, at a scale no individual sheet reaches: it checks its stated probabilities against outcomes across very large numbers of forecasts, and publishes what that shows.

“when we say there is a 50% chance of getting at least, say, 10mm of rain, that is what should happen 50% of the time. National figures showed that: When the Bureau forecasts a 25% chance of exceeding a rainfall amount for the next day, that result occurred 19% of the time. Similarly, for the 50% chance amount, it occurred 49% of the time and for the 75% chance amount it occurred 80% of the time.”

Today’s ranges are the first entries in a record you will keep all semester and audit at the end of it. Nothing is due, and there is nothing to submit yet. Keep the sheet, keep committing before you look, and keep the line about what would make you miss — that line is the one that turns out to be worth something in week 12.