The Normal Distribution: Why Finance Uses It and Where It Fails

The normal distribution is one shape with a single hump, fixed by a centre, a spread and a rule for how fast it thins. The economy of three numbers is why finance reaches for it, and that same economy is the trap. A shape built from quiet months and stressed months reports the same centre of 1.00 per cent and the same spread of 5.00 per cent, and carries more than five times the weight far out.

The pieces this rests on are already in place. A distribution is the full list of what can happen with a weight attached to each entry, and a centre and a spread are two compressions of that list rather than the list itself. A warning came attached to those two numbers: they describe a shape, they do not pin one down. The warning becomes concrete on the one shape everybody uses: two shapes can report the identical centre and the identical spread and still disagree about the far end by a factor of more than five.

The monthly change of the traded unitAny single thing with a price that gets quoted again and again, so a change from one quote to the next can be measured. What is being traded does not change how a change from one quote to the next is measured. called the Nakshatra unit, an invented example, carries a centre of 1.00 per cent and a spread of 5.00 per cent, and the second shape below is assembled from those same two numbers rather than measured anywhere.

What is the normal distribution, and what fixes it?

Consider how long a morning commute takes. Most days it lands close to the usual, forty minutes or so. Some days it is thirty five, some days it is forty eight. Very occasionally something goes badly wrong and it is an hour and a quarter. A year of mornings, drawn as how often each length came up, makes a pile with a fat middle and two thin edges: the further from the usual, the less often it happens.

The normal distribution is one particular version of that pile, written down exactly. The shape has a single hump. The curve is symmetricA shape is symmetric when the half to the left of the centre is the mirror image of the half to the right, so an outcome the same distance either side carries the same weight., meaning the left half mirrors the right half. The tails never quite reach zero on either side, so there is no largest possible outcome and no smallest, only outcomes that get less and less likely. And the shape thins away from the centre at a rate it dictates itself. Nobody gets to choose that rate.

Once a centre and a spread are named, the entire shape follows, and there is nothing else left to set. That is a very strong claim and it is worth being uncomfortable about it. Two numbers, and every question about the shape already has an answer: how much weight sits between any two levels, how far out the weight becomes negligible, how the two ends compare. Two numbers in, an unlimited amount of detail out. Nothing in the world owes anybody that. The deal has a price, and the price is paid at the far ends of the shape.

The two numbers do two different jobs and it is worth watching them separately. The centre slides the whole pile left or right without changing its width. The spread stretches or squeezes it without moving where it sits. In the picture below, two of the shapes have the same width and sit in different places, and two of them sit in the same place at different widths. There is no third thing to adjust.

What do the three bands actually claim?

The three bands are the most quoted fact about this shape, and they are usually quoted without being attached to anything. Attach them. The Nakshatra unit has a centre of 1.00 per cent and a spread of 5.00 per cent, so one spread either side of the centre runs from minus 4.00 per cent to 6.00 per cent, and the normal shape puts 68.27 per cent of the weight inside that band. Two spreads either side runs from minus 9.00 per cent to 11.00 per cent, and holds 95.45 per cent. Three spreads runs from minus 14.00 per cent to 16.00 per cent, and holds 99.73 per cent.

In rupees it stops being abstract. The Nakshatra unit is priced at Rs 100/- at the start of a month. The first band says that roughly two months in every three end somewhere between Rs 96/- and Rs 106/-. The third band says a month ending below Rs 86/- or above Rs 116/- is the rare case. Same statement, now with a picture of what a month looks like from the inside.

The third band leaves 0.27 per cent of the weight outside it, and because the two halves are mirror images, that leftover splits into 0.135 per cent below minus 14.00 per cent and 0.135 per cent above 16.00 per cent. The 0.135 per cent is worth holding on to. The figure is small enough to wave away in conversation, and it is exactly where two shapes that agree on everything else part company. A sentence quoting that figure sounds equally confident whether it is right or wrong.

Why does finance keep reaching for this one shape?

Three honest reasons stand behind the habit. Every one of the three is a reason of convenience, and none of them is evidence.

The first is that it travels light. Two numbers are enough to hand the whole shape to somebody else. A risk report can carry a centre and a spread in one line of a table, and the person reading that line can rebuild every reading in the shape without ever seeing a month of the underlying record. Very little else compresses that well.

The second is a result covered properly on its own terms elsewhere: when a great many small independentTwo things are independent when knowing how one turned out says nothing about how the other turned out. Whether that actually holds is a separate question from what a shape looks like. pushes are added together, the total tends towards this shape, whatever the individual pushes looked like. The result is real and it is famous. What people do with the result matters more: they reach for the normal shape partly because the result exists, and then treat the shape as settled for a total nobody ever checked was a sum of many small independent pushes at all.

The third is that the arithmetic closes. Questions about this shape have closed formA closed form answer comes from evaluating a formula directly, rather than from running a large number of trials and counting how often something happened. answers, so a formula is evaluated instead of ten million trials being run and counted. When a number has to be produced every evening before a deadline, that is not a small consideration.

Not one of those three reasons is a claim that the shape is true of anything, and reading them as though they were is the most common mistake made with this distribution. Cheap to carry, backed by a famous result about sums, and easy to compute with. All three would be exactly as true of a shape that described the situation badly. In the long runOver a stretch long enough for occasional outcomes to have turned up in something like their proper proportion. How long that has to be depends on how rare the outcome is. the convenience is what survives in memory and the caveat is what gets dropped.

What is a z score, and why does it turn every question into one question?

Here is the move that makes the whole shape workable. How likely a change below minus 9.00 per cent is depends on whichever centre and spread happen to be in hand. So ask a different question instead: how many spreads below the centre does minus 9.00 per cent sit? The count of spreads is the z score, and once the count exists the original units have gone.

Worked on the Nakshatra unit, the level is minus 9.00 per cent. Subtracting the centre of 1.00 gives minus 10.00. Dividing by the spread of 5.00 gives minus 2.00. So a month below minus 9.00 per cent is a month sitting more than two spreads below the centre, and the normal shape says the weight below two spreads down is 2.28 per cent. Roughly one month in forty four.

The count of spreads is why one table of readings answers every question anybody has ever asked of this shape: the table is written in counts of spreads, and every question in the world gets converted into a count of spreads before it is asked. A shop measuring daily takings in rupees, a workshop measuring a part in millimetres and a monthly change measured in per cent all arrive at the same table with their units stripped off at the door. Stripping the units off at the door is the whole trick, and there is nothing else to it.

Where does this shape fail, and by how much?

Everything above is true and none of it is the point. The point is what happens when somebody is handed a centre and a spread, assumes the normal shape, and the thing measured was never that shape. So build the counter example in the open, and make it agree with the normal model on every number anybody normally checks.

Suppose months come in two kinds. Ninety five months in every hundred are quiet, and in a quiet month the change is centred on 1.00 per cent with a spread of 4.00 per cent. Five months in every hundred are stressed, still centred on 1.00 per cent, but with a spread of 14.00 per cent. Nothing else changes. Each month simply arrives in one regimeA stretch of time in which conditions behave one way, before switching to a stretch in which they behave differently. The word says nothing about why the switch happens. or the other, and this is not exotic: a household knows perfectly well that an ordinary month and the month somebody needs an operation are two different kinds of month, not two draws from one calm pile.

Now for what this two-part shape reports as its centre and its spread. The centre is easy. Both kinds of month are centred on 1.00 per cent, so the whole thing is centred on 1.00 per cent. The spread takes one step of care, and the answer does not come from taking a weighted averageA total in which each item is multiplied by its own weight before being added, so items carrying more weight pull the answer further. Building one is covered separately. of the two spreads directly.

Put the numbers in. Squaring the quiet spread gives 16.00 and squaring the stressed one gives 196.00. Weighted, that is 0.95 times 16.00, or 15.20, plus 0.05 times 196.00, or 9.80. The two parts add to exactly 25.00, and the square root of 25.00 is 5.00. So this two-part shape reports a centre of 1.00 per cent and a spread of 5.00 per cent, the identical pair the normal model reports, and any summary line carrying those two numbers cannot tell the two shapes apart.

So both shapes get the same question. Where does each of them put the weight below minus 14.00 per cent, the month that takes Rs 100/- down to Rs 86/-? The normal model says 0.135 per cent, the sliver left outside the third band. The two-part shape says 0.72 per cent. The two-part reading is 5.32 times as much weight in the same place, from two shapes that agree on the centre to the second decimal and agree on the spread to the second decimal.

No summary number anywhere in that comparison distinguished the two shapes, and the disagreement only appeared once somebody asked a question about the far end. The normal shape was not computed wrongly. Every band on it is right, every reading follows from its own rule, and the arithmetic is flawless. The shape was simply assumed, and the assumption did all of its damage in the one region nobody could see from the summary.

Move the slider up from the left end and two things happen at once. The far reading climbs the whole way, from 0.13 per cent when no month is stressed to 1.70 per cent when twelve months in a hundred are. And the middle gets sharper rather than flatter. The quiet spread is being crushed downward to make room: at twelve stressed months in a hundred the quiet months are running on a spread of just 1.30 per cent. The shape is being pulled inward in the middle and outward at the ends, and the two summary numbers absorb both movements and report nothing.

The near reading does not behave like the far one, and the panel is the easiest place to catch it. The weight below minus 9.00 per cent starts at 2.28 per cent, falls to a low of about 1.78 per cent at around five stressed months in a hundred, and then climbs back up to 2.61 per cent at eleven and 2.85 per cent at twelve. The near reading falls, and then it rises. Anybody who says a heavy tail simply means more weight everywhere out from the centre has not looked at a reading like that one.

Is the shape with the heavier far tail worse everywhere?

Almost everybody assumes it is, and the panel says otherwise. Ask both shapes about minus 9.00 per cent, two spreads below the centre. The normal model says 2.28 per cent. The two-part shape says 1.78 per cent. The shape with the heavier far tail reports a smaller chance of the moderately bad month, not a larger one.

The smaller near reading is not an error and it is not a rounding artefact. Each shape has to account for every month that could possibly happen, so each shape carries exactly one unit of weight in total. So if the two-part shape has extra weight sitting out at minus 16.00 and minus 20.00 per cent, that weight had to be taken from somewhere, and where it came from is the region a little way out from the centre. The quiet months, running on a spread of 4.00 rather than 5.00 per cent, simply do not reach minus 9.00 per cent as often.

A heavy far tail means weight was moved, never that weight was added, and that is why the two curves have to cross somewhere rather than one of them sitting above the other throughout. Work out where. Below minus 10.07 per cent the two-part shape starts reporting more than the normal model; above it, less. At the crossing itself both shapes report exactly 1.34 per cent, and a reader who compared them only at that level would find perfect agreement and conclude that the choice of shape made no difference at all.

The picture below plots, for every level along the bottom, the running totalThe weight collected sweeping across a shape from one end, so at any level it is the total weight lying to that side of it. Building one is covered separately. of weight lying to the left of it, which is exactly the quantity both shapes were asked for above. Follow the two lines from left to right and watch them swap places.

Does a third summary number close the gap?

The obvious repair is to add another number. There is a standard measure of how much weight sits far from the centre, and it separates these two shapes cleanly: it reads 3.00 for any normal shape at all, whatever its centre and spread, and 10.39 for the two-part shape built above. Two shapes that agreed on the first two numbers to the second decimal disagree on the third by a factor of nearly three and a half.

The third number is a genuine improvement and it is worth having. A summary line carrying three numbers rather than two would have caught the difference at issue here, and a reader seeing 10.39 sitting beside a centre and a spread would know immediately not to reach for the bands.

The third number is another summary, so it narrows the gap rather than closing it, and the argument that beat two numbers beats three. A third number compresses the shape into one more reading, and other shapes can be built that match all three and still disagree about where the weight actually sits. Each number added makes the counter example harder to construct, and none of them makes it impossible. The honest position is that summaries are compressions, compressions lose things, and the thing they lose most reliably is the far end.

What is this shape safe for, and what is it not safe for?

None of this makes the normal shape useless, and treating it as useless is a worse mistake than the one being corrected. The working rule splits by which part of the shape a question lands on.

An analyst putting two quantities on a common footing is on solid ground. Turning a level into a count of spreads is the single most useful operation in the toolkit, and it does not depend on the shape being right at the ends. The middle is exactly the region where the two shapes above agreed with each other, so a lender describing what a typical month looks like for a borrower is on solid ground too. So is anybody who needs the arithmetic to close before a deadline, provided they say out loud that they assumed a shape in order to get there.

Now the other side, and it does not get softened. The worst case sits precisely where the two summary numbers stopped carrying information, so sizing a worst case with this shape is not safe. The month being sized is out past three spreads, and out past three spreads the answer stopped being a fact about the record and became a fact about the shape somebody assumed. A household deciding how many months of expenses to keep aside is doing exactly this, sizing the bad case. The honest version of that decision keeps more aside than the tidy arithmetic suggests, not because the arithmetic is wrong, but because the shape it ran on was picked for convenience.