Probability Distributions: The Shape of Uncertainty

A probability distribution is the whole account of an uncertain quantity: every value it can take, and how much weight sits on each one. A distribution answers how likely each value is, never which one will happen. A centre and a spread compress that account and lose most of it. Two shapes can share a centre of 1.00 per cent and a spread of 5.00 per cent and still disagree completely about their worst months.

Two things arrive already built and are used below without rework. The earlier material on probability established what a probability claims and what the reference setThe collection of situations a probability is a statement about. Once the months, the companies or the draws are named, the number has something to be a share of. behind it is. Another built the step from an uncertain outcome to a number, and with it the invented Nakshatra unit: a traded unitSomething that can be bought and sold at a price, where the price moves. The word says nothing about what it is made of, only that it has a price that changes., priced monthly, whose monthly change takes five values, minus 9.00, minus 4.00, 1.00, 6.00 and 11.00 per cent, carrying weights of 0.08, 0.18, 0.48, 0.18 and 0.08. Its expected valueThe average of what a quantity can be, with each possible value counted according to how likely it is. Established earlier and used here without rework. is 1.00 per cent. The shape of that unit is a different question from its average. Not what the unit averages to, but what its whole shape is, and what gets thrown away every time that shape is described in fewer numbers than it actually has.

Before anything else, one warning, and it decides how every number below should be read. Every weight below is stated, not measured. Nobody watched the Nakshatra unit for a hundred months and counted. The five values and their weights were written down as the definition of the thing, in the same way a fair die is written down as having six faces before anybody rolls it. Working the other way round, from a stack of observed months back to a shape, is a genuinely different job and is covered separately.

What is a probability distribution?

A probability distribution is the complete list of what an uncertain quantity can be, with a weight on each entry, and the weights adding to exactly 1.00. The definition stops there. A distribution is not a formula, not a curve and not a summary. A distribution is a list.

Consider ten shops along one corridor of a mall. The whole account of their trade is ten lines: this shop took this much, that shop took that much, all ten of them. Somebody who reports the average takings across the ten has said something true and has also thrown away nine tenths of what was in front of them. There is no longer any way to tell whether the ten are close together or whether one shop is carrying the corridor. Nor whether the worst shop is quietly failing. The average is a sentence about the list; the list is the thing itself.

A distribution sits in exactly that relationship to every summary number there is. The Nakshatra generator has five lines, and those five lines are the distribution. Everything else here, the running total, the centre, the spread, the three centres, is worked out from those five lines and holds less than they do. Reading down the figure below, each step keeps less than the one above it, and the last step keeps so little that two completely different shapes can produce the identical answer.

What are the two pictures of one distribution, and what is each one for?

The same distribution can be drawn two ways, and the two answer different questions. The first puts the weight on each value: how likely is this exact value? The second puts the running total up to each value: how likely is this value or anything worse than it?

Both hold the identical five numbers. Nothing is added and nothing is lost going from one to the other. But the second is almost always the one actually wanted. Nobody asks how likely a month at exactly minus 4.00 per cent is; they ask how likely a month that bad or worse is, and only the running total answers that without arithmetic.

The falls read straight off the third column. A fall is any month below zero, and the two values below zero are minus 9.00 and minus 4.00 per cent. The running total has already added them: it stands at 0.26 by the time it has passed minus 4.00 per cent and has not yet reached 1.00 per cent. So the Nakshatra unit falls in 26.00 per cent of months, and no arithmetic was needed to find that out. The unit is priced at Rs 100/- at the start of a month, so those two fall months would leave it at Rs 91/- and Rs 96/- respectively.

What changes when the quantity can land anywhere in a range?

So far the Nakshatra unit has been allowed five values and nothing else. Suppose instead the monthly change can land anywhere between two limits, at any depth of decimal. Something surprising happens to the weights, and it is not a technicality.

Take the five values and split each one into four nearby values, each carrying a quarter of the original weight. The middle value of 1.00 per cent becomes four values around it, each carrying 0.12 instead of 0.48. Split those into four again and each carries 0.03. Keep splitting. A fixed amount of weight is being shared among a growing crowd of values, so the weight on any one exact value shrinks toward nothing. Once the quantity is free to settle anywhere in a range, the weight on any single exact value is zero, and only an intervalA stretch of the number line between two limits, such as everything from minus 1.50 to 3.50 per cent. A single point is not an interval. carries any weight at all.

Watch what does not change while that happens. Look at the slice from minus 1.50 to 3.50 per cent in the figure below. The value of 1.00 per cent sits inside that slice, so at five values the slice holds 0.48. At twenty values all four splinters of that value stay inside the slice and no splinter of any other value wanders in, so the slice still holds 0.48. Keep splitting forever and it still holds 0.48. The weight did not disappear. The weight stopped belonging to a point and started belonging to a stretch.

A continuous shape is therefore drawn as a curve whose area is the probability, rather than as a set of bars whose heights are. The height of the curve is not a weight and cannot be read as one. Asked how likely a change of exactly 1.0000 per cent is, the honest answer is zero. The answer sounds absurd until it becomes clear that an exact value was never the question wanted. The question wanted was the chance of landing somewhere near 1.00 per cent, or below minus 9.00 per cent, or anywhere between two numbers that matter. Questions with two limits have answers. With a continuous quantity every honest question names two limits, never one value.

How is a whole shape squeezed into a centre and a spread?

Two numbers do most of the compressing that gets done to distributions, and it is worth knowing exactly what each one is made of before trusting either.

The centre is the expected value, already built, and it is a weighted averageAn average where some entries count for more than others. Each value is multiplied by how much it counts before anything is added up.: each value multiplied by its weight, and the five products added. For the Nakshatra unit that comes to 1.00 per cent. The centre is a balance point. With the five weights laid along a plank at their five values, the plank would balance at 1.00 per cent.

The spread is the harder one, and the arithmetic explains itself when it is followed in order. The spread starts with the distance of each value from the centre. Minus 9.00 per cent is 10.00 below. Minus 4.00 per cent is 5.00 below. The value of 1.00 per cent sits exactly on it. And 6.00 and 11.00 per cent are 5.00 and 10.00 above. Added up as they stand, those distances come to nothing at all. The ones below cancel the ones above. The cancelling is not a nuisance but a requirement: the centre is the balance point, so the distances either side of it must come out even.

A squared number is never below zero, and squaring every distance before weighting it is what stops the two sides cancelling. So square each distance, weight each square by how likely that value is, and add the five results. The five results add to 25.00. Squared per cent is not a unit anybody can feel, so take the square rootThe number that returns the original when multiplied by itself. The square root of 25.00 is 5.00, because 5.00 times 5.00 is 25.00. at the end and the figure is back in per cent: 5.00. The spread of the Nakshatra unit is that 5.00 per cent.

Notice two things in that build that the finished number of 5.00 per cent hides completely. Its distance from the centre is zero and zero squared is still zero, so the value sitting on the centre contributes nothing whatever to the spread. And the two outermost values happen only 16.00 per cent of the time between them, yet contribute 16.00 of the 25.00 total, very nearly two thirds of the whole thing. The spread is mostly made of the rare values, and says correspondingly little about the common ones.

The centre and the spread move independently of one another, and that is easier to see than to argue. Take weight out of the middle of the Nakshatra shape and push it onto the two end values in equal amounts. The shape flattens and widens. The spread climbs from 3.00 per cent to 7.00 per cent. Every unit of weight that goes to minus 9.00 per cent is matched by an equal unit going to 11.00 per cent, and those two values sit the same distance either side of 1.00. So the centre does not budge from 1.00 per cent, not by a hundredth.

Where do the mode, the median and the mean land on this shape?

There are three different answers to the question what is a typical value, and they are not the same question dressed differently. The mode is the heaviest value, the one carrying the most weight. The median is the value at which the running total first reaches a half. The mean is the balance point computed above.

On the Nakshatra shape all three land on 1.00 per cent. The mode is 1.00 per cent because its weight of 0.48 beats every other. The median is 1.00 per cent because the running total is 0.26 just below it and 0.74 at it, so the halfway mark is crossed exactly there. The mean is 1.00 per cent from the weighted arithmetic. Three questions, one answer.

The three coincide here because this shape matches itself either side of the centre, and that is a property of this particular generator rather than a fact about shapes. Look at the pairs. Minus 9.00 and 11.00 per cent sit 10.00 either side of 1.00 and carry the same weight of 0.08. Minus 4.00 and 6.00 per cent sit 5.00 either side and carry the same 0.18. Fold the picture along 1.00 per cent and the two halves land on each other exactly. Folding is what forces the balance point, the halfway point and the heaviest point into the same place.

Take the folding away and the three separate, sometimes by a lot, and then which of the three is meant stops being pedantic and starts mattering. Shapes that lean to one side are what prices are usually described with, and they are covered separately along with what the gap between the three does there.

Do a centre and a spread pin a shape down?

No. A centre of 1.00 per cent and a spread of 5.00 per cent are compatible with more than one shape, and those shapes can disagree violently about the thing most worth knowing.

Set the Nakshatra generator beside a smooth shape that matches itself either side of its centre, carrying that same centre of 1.00 per cent and that same spread of 5.00 per cent. Two numbers each, and the two numbers are identical. Now ask both of them one question. How often does a month at minus 9.00 per cent or worse turn up?

Minus 9.00 per cent is the worst value of the generator and carries a weight of 0.08, so the generator answers 8.00 per cent straight off its running total. The smooth shape answers 2.28 per cent. The gap is not a rounding difference and not a disagreement about the centre or the spread, on which the two agree exactly. A difference of more than three times in how often the bad month arrives comes out of two shapes that report the same pair of summary numbers.

Notice which of them says nothing wrong. Neither. The generator is telling the truth about the generator, and the smooth shape is telling the truth about the smooth shape. The centre and the spread are also both true, of both. The question itself has gone missing. A centre says where the weight balances. A spread says how far the weight sits from that balance on average, after squaring. Neither of them was ever asked about the far end of the shape, and neither of them answers about it. The summary is not lying; it simply does not contain the answer being squeezed out of it.

The smooth shape itself, why finance reaches for it so often, how its tailThe far end of a shape, out where the extreme values sit and the weight has thinned to very little. Every shape has two of them. behaves and where the 2.28 per cent comes from, is covered under the normal distribution.

What is worth asking about any shape before using it?

The mechanism is complete. Four questions turn it into a check for the moment a shape lands on somebody's desk with two numbers attached and no shape supplied. Asked in that order, the whole check runs to about a minute.

First, what values can this thing take at all, and is there a floor or a ceiling? A shape with a hard floor and a shape that runs on forever downward behave completely differently at the bad end, and no summary number says which of the two is in hand. For the Nakshatra generator the answer is blunt: five values, nothing else, floored at minus 9.00 per cent and capped at 11.00 per cent. By definition, nothing outside those five can happen at all.

Second, does it match itself either side of the centre? If it does, the mode, the median and the mean agree, and a typical value can be named without anybody having to ask which of the three was meant. If it does not, those three separate and the word typical needs a footnote.

Third, what does the running total say at the value that actually matters? Not at the centre, already known, but at the number that would ruin the month. The running total at that value is the one question a summary cannot reach, and it is worth asking out loud. Somebody who has the shape can read it off in a second. Somebody who has only the summary has to guess.

Fourth, and this is the one that separates a reader who has understood the material from one who has not: what would change if the shape were wrong while the centre and the spread stayed exactly where they are? In the worked case above the answer was a factor of more than three at one threshold. Anyone who cannot say what a wrong shape would cost them is relying on a shape they have not checked.

A household budget is the same test in plain clothes. Knowing that a household spends an average of Rs 42,000/- a month with a typical swing of Rs 6,000/- says nothing about the month the boiler goes. The average does not contain that month, and neither does the swing. The boiler month sits out in the shape, and the only way to know how often it turns up is to look at the shape.