ContentsThe library

Chance, Spread and What to Expect

The Number and the Bar Beside It

Last timeWhen the Outcomes Do Not Come in Pieces

A measured proportion is one draw from a distribution, and the only honest way to report it is with the width of that distribution attached to it.

Everything in this course has been about distributions that were given. In

practice nobody gives them. A number is measured from a sample and then has to be

interpreted, and this last lesson is about doing that honestly. It uses every

earlier lesson and introduces nothing new.

Start with the right frame. Two hundred visitors arrived, nine of them signed up,

and the measured rate is four and a half percent. That number is not the sign up

rate. It is one draw from a distribution of numbers that the same experiment could

have produced, centred on the real rate. Run it again next week with two hundred

different visitors and a different number comes out, with no change in the

product at all.

So the question is how wide that distribution is. The answer comes from two

earlier results stuck together.

FIG 1The spread of a measured proportion
the variance of a single yes or no outcome, derived in the spread lesson
the number of observations in the sample
the square root, taking squared units back to the units of the proportion
the typical distance between the measured proportion and the truth
Two earlier results, combined. A yes or no outcome has variance p times one minus p, and an average of n independent draws has that variance divided by n. A measured proportion is exactly such an average, of ones and zeros, so no new derivation is needed.

The numerator has a useful property: it is largest at a half, where it equals a

quarter, and it shrinks towards zero at both ends. So measurement is hardest when

the thing being measured is near fifty percent, and easiest when it is near zero

or one.

FIG 2Where the uncertainty is worst
0.000.070.150.220.300.00.30.50.71.0the true proportion
p times one minus p, peaking at a quarter
The peak sits at one half with a value of one quarter, and the curve falls away symmetrically. This is why a coin flip is the hardest proportion to pin down and why a rare event, despite needing many observations to see at all, has a small spread once seen. Taking the peak value gives the worst case, which is where the rule of thumb comes from.

The rule worth memorising

Put the worst case into the formula. A quarter divided by the sample size, square

rooted, is one over twice the square root of the sample size. Round it up to one

over the square root and you have a rule you can apply in your head to any

proportion anyone quotes at you.

The lesson stops here

5 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the planThe contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

The rest of this course

  1. 01A List, and a Number on Each Item
  2. 02One Number Standing in for a Whole Listopening only
  3. 03The Second Number Every Average Needsopening only
  4. 04A Table Instead of a Listopening only
  5. 05What Is Left After the Newsopening only
  6. 06Four Times the Data for Half the Erroropening only
  7. 07Height Is Not Weightopening only
  8. 08The Number and the Bar Beside Ityou are here

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