Normal Distribution Probability: Below, Above and Between
The calculator takes a centre, a spread and one or two thresholds and returns the chance of landing below a threshold, above it, or between two of them under a normal shape. Each threshold is measured as a distance from the centre in spreads. With a centre of 1.00 per cent and a spread of 5.00 per cent, 2.28 per cent sits below minus 9.00 per cent.
The calculator: a centre, a spread and a threshold in, three probabilities out
Figures can be typed in directly, or the threshold dragged so the filled area moves with it. Every step is shown: the two arithmetic moves that turn a threshold into a distance in spreads, the running total that distance lands on, and a check that the parts add back to the whole. The panel opens on the worked case: a centre of 1.00 per cent, a spread of 5.00 per cent and a threshold of minus 9.00 per cent. The threshold sits minus 2.00 spreads out, and 2.28 per cent of the shape sits below it. At that opening setting the between reading runs from the threshold to the far end and reads 81.86 per cent. The 68.27 per cent worked out further down is a different pair of ends, and the one spread band setting moves the threshold to minus 4.00 to reproduce it. The settings for three, four and five spreads out drive the panel out to where the assumed shape supplies the whole of the answer.
Every figure below describes the monthly change of the Nakshatra unit, an invented traded unitAnything that can be bought and sold at a price, so that the price on one date can be compared with the price on another. The kinds that exist and how they are bought are covered separately. written for teaching, treated as sitting around 1.00 per cent with a typical distance from that centre of 5.00 per cent. The normal shape itself, why so much of finance reaches for it and where it breaks are covered separately.
The calculator never works out a centre or a spread. Both arrive from wherever the analyst got them, and everything below is downstream of that choice. Turning observed numbers into a centre and a spread is a different job with its own hazards and it is covered separately. Here they are simply typed in.
What does this calculator compute?
Three readings, from one set of inputs. The chance of landing below a threshold. The chance of landing above it. The chance of landing between two of them. Type a centre, a spread and a threshold, and all three come back at once.
The three readings look like three separate calculations and they are not. All three are the same single quantity read in different directions: the running total of the shape up to a point. Above a threshold is one hundred less below it. Between two thresholds is one below reading with another below reading taken off it. So the machine only ever does one thing, and the three answers are three ways of asking for the result.
Picture the shape as a mound over a horizontal scale, with the whole area under it standing for every outcome there is. A threshold cuts through that mound, and the chance of landing below it is the area to the left of the cut as a share of the whole. A probability here is exactly that: an area, reported as a percentage.
Read the number as a long runA statement about what would happen across a very large number of repetitions, rather than about the next single occasion. The long run says nothing about what any one month will do. claim, not as a forecast of the next month. A reading of 2.28 per cent says that if this shape genuinely described the quantity, then across a very large number of months roughly two in every hundred would land below the threshold. The reading does not say the next one will or will not.
The centre is 1.00 per cent and the spread is 5.00 per cent. How many spreads from the centre is minus 4.00 per cent?
Where does each typed number come from?
Four fields, and the note beside each one says where to find that number rather than what it stands for. A field with a vague label collects a vague number and then reports it to two decimal places, so the distinction matters more on a calculator than anywhere else.
Every input is supplied by the reader, and this calculator produces none of them. The centre and the spread arrive from wherever the reader got them, fully formed; turning observed numbers into a pair like that is a different job with its own hazards and it is covered separately. The threshold is the number the reader walked in with, and it usually comes from outside statistics altogether: a level below which a plan stops working, a figure written into an arrangement, a line somebody drew in a meeting. The fourth field only sets the far end of a between reading and changes nothing else.
One constraint applies, and the panel enforces it visibly. The spread must be above nil. A spread of nil is not a very narrow shape, it is no shape at all, and the panel refuses it with a message rather than printing a nil or an error. Beyond that the calculator has no opinion about any figure typed into it: a calculator that quietly manufactured a centre or a spread would be doing the harder half of the job invisibly, leaving the half that mattered most impossible to audit.
Which of the three inputs does this calculator work out?
How is a threshold turned into a distance from the centre?
One operation, two steps, and it is worth watching once because everything afterwards is a lookup.
Subtract the centre from the threshold. Then divide by the spread. The result is the threshold restated as a number of spreads away from the centre, and that single number is the only thing the shape ever sees.
Work the default all the way through. The threshold is minus 9.00 per cent and the centre is 1.00 per cent. The first step is minus 9.00 less 1.00, and that gives minus 10.00. The spread is 5.00 per cent. The second step is minus 10.00 divided by 5.00, and that gives minus 2.00. The threshold sits two spreads to the left of the centre, and 2.28 per cent of the shape sits below that point.
Notice what the two steps did. Subtracting the centre moved the origin so that the centre now reads as nil. Dividing by the spread changed the unit so that distances are counted in spreads rather than in per cent. After both steps the answer no longer depends on the original units at all. One shape can therefore serve every centre and every spread anybody types.
A tool that shows only its final answer cannot be checked by the person using it, so the panel above sets out both of those steps. A subtraction and a division are within reach of anybody with a pencil, and a reader who recomputes the distance once has verified the machine rather than trusted it. The panel runs a second check of its own on every change: it takes the distance it produced, multiplies by the spread, adds the centre back, and prints what comes out. If that does not return the threshold as typed, the two steps did not undo each other and nothing below them can be relied on.
Before returning to the panel, a prediction is worth committing to. As the threshold is dragged further to the left, what happens to the below reading and to the filled area?
How is a below, an above and a between answer read?
Take the worked case and read all three off it. Below minus 9.00 per cent reads 2.28 per cent. Above minus 9.00 per cent reads 97.72 per cent. Between minus 4.00 per cent and 6.00 per cent reads 68.27 per cent. Three numbers, three questions, one shape.
Below and above are not two facts. They are one fact stated twice, and they must add to 100.00 per cent exactly. Every outcome is either below the threshold or above it, with nothing left over, so one of them gives both. If a quoted pair does not sum to a hundred, something is wrong upstream of the arithmetic and no amount of recomputing will fix it.
The between reading is one running totalThe share of a shape that has accumulated by a given point, counted from the far left. A running total rises from nil to the whole and never falls back. with another taken off it. Below 6.00 per cent is 84.13 per cent and below minus 4.00 per cent is 15.87 per cent, and 84.13 less 15.87 gives 68.27 per cent. The arithmetic is complete at that point. Those two thresholds sit exactly one spread either side of the centre, and that is why the answer lands on a figure many readers will recognise.
The between reading also carries a check inside it. The between reading is the below reading at its right hand end with something taken away, so it can never be larger. The same holds against the above reading at its left hand end. Both comparisons are free and both catch a subtraction done in the wrong direction.
One more property explains a great deal of the panel's behaviour. The normal shape is symmetricA shape that looks the same reflected about its middle, so equal distances left and right of the centre carry equal weight. Not every shape has this property. about its centre, so a threshold sitting one spread below the centre and one sitting one spread above it enclose equal areas on their outer sides. Symmetry is why 15.87 per cent turns up below minus 4.00 per cent and 15.87 per cent turns up above 6.00 per cent.
A below reading comes back as 2.28 per cent. What does the above reading on the same threshold say, and why?
A between reading comes back larger than the below reading at its own right hand threshold. What must have gone wrong?
The same curve answers every question that can be asked of it, so all the thresholds can be seen at once rather than one at a time. Read from left to right, the running total climbs from nil to the whole, passing 2.28 per cent at minus 9.00 and 84.13 per cent at 6.00. Every below reading this calculator can produce is a height on that one curve.
What does every answer assume?
One thing, and it is doing more work than the arithmetic is. Every reading here assumes the quantity follows a normal shape. Not that it is roughly like one. The assumption is that it is one.
Why that shape is reached for, and where it lets people down, is covered under the normal distribution itself. One uncomfortable fact carries forward into this calculator. A different shape can agree with this one on the centre and on the spread, to the decimal, and still return a very different answer at the same threshold.
The comparison worth keeping is the one already worked. Take a mixtureA shape built by combining two shapes in stated proportions, so that most occasions are drawn from a narrow one and a few from a wide one. Built and argued separately. of quiet months and stressed months whose centre is 1.00 per cent and whose spread is 5.00 per cent, exactly matching the inputs typed into this calculator. At minus 14.00 per cent this calculator returns 0.13 per cent. The mixture returns 0.72 per cent. Both shapes agree on both inputs, and they disagree by more than five times on that one question.
The arithmetic in this tool is exact and the assumption underneath it is a choice. The further into the tail the threshold sits, the more of the answer is the choice rather than the arithmetic. Near the centre almost any sensible shape returns almost the same number, because that is where the bulk sits and the bulk is hard to get wrong. Out in the tail there is very little to be wrong about and the shape supplies almost all of it. A fat tailA shape that puts more weight on far outcomes than the normal shape does, at the same centre and the same spread. The causes of a fat tail, and the ways to spot one, are covered separately. changes the answer most at precisely that distance.
At minus 14.00 per cent this calculator returns 0.13 per cent. What did the shape with the same centre and the same spread return at that threshold?
Two answers are quoted to two decimal places. One threshold sits close to the centre and the other sits far out in the tail. In which one is more of the answer coming from the assumption rather than the arithmetic?
How rare does this shape say a far move is?
Percentages stop carrying meaning once they get small. Two decimal places will happily print 0.00 for two quantities that differ by a factor of a hundred, and a reader who sees 0.00 twice has been told nothing. Stating the reading as a frequency instead restores the meaning. One month in so many is a sentence a person can picture, and the panel above prints it for whatever threshold is set.
Walk the ladder outward from the centre on the same two inputs, a centre of 1.00 per cent and a spread of 5.00 per cent. Two spreads below the centre, at minus 9.00 per cent, the reading is 2.28 per cent, or about one month in 44. Three spreads out, at minus 14.00 per cent, it is 0.13 per cent, about one month in 741, or one month in 62 years of monthly readings. Four spreads out, at minus 19.00 per cent, it is 0.00317 per cent: about one month in 31,574, or one month in 2,631 years. Five spreads out, at minus 24.00 per cent, it is 0.0000287 per cent, about one month in 34,88,556, or one month in 2,90,713 years.
By four spreads out this shape has stopped saying unlikely and started saying never, and it says so on the strength of a choice nobody tested. Nothing was measured to produce 2,631 years. Two numbers were typed in, a shape was assumed, and the arithmetic did the rest. Out there the shape is all there is, so the shape supplies the whole of that answer.
The two part shape agrees on both typed inputs to the decimal. Set the same four thresholds against it. At two spreads it says one month in 56, slightly rarer than the shape assumed here says: 1.78 per cent against 2.28 per cent, lower rather than higher. At three spreads it says one month in 139 against one month in 741. At four spreads it says one month in 261 against one month in 31,574, and at five it says one month in 539 against one month in 34,88,556. So the assumption does not push answers one way. Somewhere between two and three spreads the two shapes cross, and past that crossing they run apart without limit.
Read the last comparison slowly. One shape calls a move at minus 24.00 per cent something to expect in a working lifetime of monthly readings. The other calls it something to expect once in a few hundred thousand years. Both shapes were handed identical inputs.
Both shapes here were written for teaching, and every figure came out of arithmetic done on them. Records of traded prices have repeatedly produced single moves that a normal shape fitted to the very same record treats as effectively never happening. The normal shape is therefore best held as a description with known limits rather than as a law. It is convenient, it is easy to explain, and it is at its weakest at exactly the distance where a far threshold puts the question. The panel can be driven out to where the mechanism stops being trustworthy.
Set the panel four spreads below the centre and it returns about one month in 31,574, or one month in 2,631 years. What was measured to produce that figure?
How is a quoted probability checked?
Quoted probabilities arrive from other people far more often than they are produced first hand. A note says the chance of falling below some level is such and such per cent, quoted to two decimals, with nothing in the note about where it came from. Four checks take about a minute between them, and none of them needs anything the reader does not already have.
Recompute the distance. Ask for the centre and the spread, take the centre off the threshold, divide by the spread, and see whether the number of spreads makes sense against the probability quoted. Two spreads out on the low side is roughly two in a hundred; three spreads out is roughly one in a thousand. If somebody quotes four per cent at three spreads out, the arithmetic and the shape are not talking to each other.
Check the pair. If a below reading and an above reading are both given, add them. The two must come to a hundred. The check costs nothing and it catches a threshold quietly changed between the two lines of a table.
Check the between. A between reading must be smaller than the below reading at its upper end and smaller than the above reading at its lower end. Either comparison catches a subtraction done backwards, and a backwards subtraction is the commonest arithmetic fault in this whole operation.
Then ask a fourth question, and it decides how much to trust the answer. Is this a middle question or a tail question? A threshold near the centre is mostly arithmetic and a threshold far out in the tail is mostly assumption, so the same two decimal places mean very different things in the two cases. A household deciding whether a monthly figure usually lands in a comfortable band is asking a middle question and can lean on the answer. Somebody sizing the worst month they should prepare for is asking a tail question, and there the shape is doing most of the talking.
A tail probability arrives quoted to two decimal places, with no shape named anywhere. What should be asked for?
What does the whole worked case look like in one place?
The quantity is the monthly change of the Nakshatra unit, invented for teaching. The centre is 1.00 per cent and the spread is 5.00 per cent, both typed in rather than worked out. The shape is assumed to be normal.
| The question asked | Distance in spreads | The answer |
|---|---|---|
| Below minus 9.00 per cent | minus 2.00 | 2.28 per cent |
| Above minus 9.00 per cent | minus 2.00 | 97.72 per cent |
| Below minus 4.00 per cent | minus 1.00 | 15.87 per cent |
| Below 6.00 per cent | 1.00 | 84.13 per cent |
| Between minus 4.00 and 6.00 per cent | minus 1.00 to 1.00 | 68.27 per cent |
| Below nil per cent | minus 0.20 | 42.07 per cent |
| Below minus 14.00 per cent | minus 3.00 | 0.13 per cent |
| Below minus 19.00 per cent | minus 4.00 | 0.00317 per cent |
| Below minus 24.00 per cent | minus 5.00 | 0.0000287 per cent |
| That same minus 14.00 per cent threshold under a shape agreeing on both inputs | minus 3.00 | 0.72 per cent |
The minus 14.00 per cent row and the ruled row beneath that repeats it repay slow reading. Same threshold. Same centre. Same spread. Two answers, and the gap between them is more than five times. Everything above that rule is the calculator working correctly, and the rule itself is where the arithmetic stops and the assumption starts.
These numbers were made up. The monthly change used throughout was chosen so the arithmetic lands on round distances and stays checkable by hand.
The two decimal places that read as knowledge
A reader takes a far tail answer to two decimal places and treats the decimals as something that was measured. At minus 14.00 per cent this calculator returns 0.13 per cent, cleanly and correctly. A shape agreeing with both of its inputs returns 0.72 per cent, just as cleanly and just as correctly. The precision is entirely real and it belongs entirely to the arithmetic. The answer belongs to the assumption.
The cost is a number that reads as measured when it was chosen. The number travels into a note, then into a summary, then into a decision, and by the third stop nobody remembers that a shape was picked at the start. The decimals survive the whole journey and the choice does not, and that is exactly the wrong half to lose.
The fix is a habit rather than a calculation. A tail answer is quoted with the shape it assumed attached to it, in the same sentence, and the two then cannot be separated by anybody repeating it. Not in a footnote and not in an appendix, because those get dropped. In the sentence itself, where 0.13 per cent under an assumed normal shape is a very different statement from 0.13 per cent flat, and a reader can see the difference without being told to look for it.
The normal shape itself, why so much of finance reaches for it, and where it lets people down are covered under the normal distribution, along with the reference setThe full collection of occasions a probability is being counted over. Change what is being counted and the probability changes with it, even when nothing else moves. a probability is counted over. The skewed shape used for prices is set out under the shape used for prices themselves. Working a centre or a spread out from observed numbers, and everything that goes wrong in the attempt, is covered separately.
Which outside source stands behind these numbers?
Not one. Every quantity printed above came out of arithmetic done on a monthly change that was written for teaching, and the arithmetic is displayed at each step so it can be redone with a pencil and disagreed with. There is no maintained record underneath any of it, so there is no document to name and no date on which anything was read. A genuine quantity, for a real thing over a real period, has to be fetched from outside and brought here as a typed centre and a typed spread. The calculator will then compute cleanly on it and will still be assuming the shape.
| Source | Document | Where |
|---|---|---|
| None used | The centre, the spread and both thresholds are supplied by the reader, so no external record is involved at any point | not applicable |
| Method | Shown as arithmetic in the one operation block above, and reproducible with two subtractions and a division | not applicable |
The Nakshatra unit is invented.
Educational material. Not advice on any investment, tax, budget or market position.