Height Is Not Weight

Last timeWhy Averages Settle Down

When outcomes form a continuum every single value has weight zero, probability lives in areas rather than points, and a density is a rate whose height can exceed one.

Every lesson so far has had a list of outcomes. Many quantities do not come in a

list. A latency can be any positive number, a temperature any real one, a

position anywhere in a room. The machinery needs one adjustment, and only one,

but the adjustment is the thing people get wrong.

Start with the problem. If every value in an interval carried some positive

weight, and there are uncountably many of them, no total of one is possible. So

on a continuum every single value has weight exactly zero. The chance that a

request takes exactly 12.4 milliseconds, to infinite precision, is zero. That is

not a claim of impossibility; it is a statement that points are the wrong objects

to ask about.

The right objects are intervals.

FIG 1Probability lives in the area

: the density, the height of the curve at the value x, which is not a probability
: the area under the curve between the two ends of the interval
: the probability, which is that area and is always between zero and one
: a window rather than a point, because points carry no weight

This single line is the whole adjustment. The two rules from the first lesson survive unchanged in it: the density is never negative, and the total area under the whole curve is one. Setting a equal to b gives an area of zero, which is the formal statement that a point has no weight.

FIG 2What is being asked when you ask about an interval

The marked point carries no weight however carefully it is specified. The window between the two ends does. Every usable question about a continuous quantity is secretly a question about a window, including the ones that sound like points: asking about a latency of six and a half milliseconds really means asking about a window around it that is as wide as the measurement.

Why the height can be above one

Here is the error worth inoculating against. The height of the curve is not a

probability. It is probability per unit of the quantity, which for a latency

means probability per millisecond. Its units are the reciprocal of the units of

the thing being measured, and it has no upper bound at all.

The reason is the area constraint. The total area must be one. If a quantity is

tightly concentrated, so that the curve is narrow, then the curve must be tall to

enclose that area. Squeeze the width to a quarter and the height rises to four.

Nothing is wrong, and nothing is four hundred percent.

FIG 3Two densities, both enclosing an area of one

spread out, height one at the left edgeconcentrated, height four at the left edge

Both curves enclose exactly the same total area of one. The second is concentrated near zero and therefore has to be four times as tall to do it. If the heights were probabilities the second curve would be making an impossible claim; as a rate per unit it is making a perfectly ordinary one.

The rules survive

Everything from the earlier lessons carries over by replacing the sum with an

integral. Nothing conceptual changes.

The lesson stops here

3 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 plan The 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

Height Is Not Weight

Why the height can be above one

The rules survive

3 more paragraphs to go

This is the reading half

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