Correlation and Causation: The Distinction That Costs Money

Correlation says two series moved together and stops there. Causation says one of them moved the other. Over ten invented months the Nakshatra unit and the Vasant unit correlate at 0.8694. A correlation of 0.8694 is strong by any reading, and the single figure fits four completely different accounts of why. Nothing computable from those ten months chooses between them.

Two words that behave like synonyms in ordinary speech behave nothing alike once a decision rests on them. In conversation the difference rarely matters, so nobody says moved with when they mean caused. In written analysis it is the difference between a sentence the record supports and a sentence the record cannot support at any strength. One gap runs under the whole subject: a correlation is computed from a table, and a cause is a claim about the situation the table came out of, and no amount of arithmetic crosses from the first to the second.

One invented setup carries every figure that follows, and it is worth having in full before any claim is made about it. There is a traded unitSomething with a price that other people trade, so it carries a fresh value at the close of each month. Here it does nothing except supply one figure per month; how it would be held, priced or exchanged never comes into it. called the Nakshatra unit, invented for teaching, and a second one called the Vasant unit, also invented. For each of ten months there is a monthly changeHow far something shifted over a single month, expressed against the level it began at. A reading of plus 6.00 per cent means the month closed six per cent higher than it opened. Nothing else about the month is written down. for each. The Nakshatra unit ran 1.00, 6.00, minus 4.00, 11.00, 1.00, minus 9.00, 6.00, 1.00, minus 4.00 and 1.00 per cent, averaging 1.00 per cent. The Vasant unit ran 3.00, 18.50, 2.50, 17.00, minus 3.00, minus 13.00, 6.50, minus 1.00, minus 7.50 and minus 3.00 per cent, averaging 2.00 per cent.

Two things about that setup carry weight later. The first is that the months are paired observationsTwo measurements that belong to the same occasion and cannot be shuffled independently. Month four of one list goes with month four of the other, and separating them destroys the only thing the pair can be used for.: month four of the Nakshatra unit belongs with month four of the Vasant unit and with no other month. The second is that they are in time orderThe months are listed in the order they happened rather than sorted by size. Sorting them makes a neater picture and throws away the only information that lets a later check ask whether one month followed another., first month first. A later check on the same ten months needs the order intact, so the months stay as they happened rather than sorted into a tidier picture.

What does a correlation actually claim?

A correlation is one number, and it lives on a fixed scale that runs from minus one to plus one. The correlation answers a single question about two series: when one of them was above its own average, how reliably was the other above its own average as well? Plus one is a perfect match, every time. Minus one is a perfect mismatch, every time. Zero is no tendency in either direction. For the Nakshatra unit and the Vasant unit across those ten months it comes to 0.8694.

The number is built so that the sizes of the two series are divided out of it. Dividing the sizes out is what makes the scale fixed. A series that swings by tens and a series that swings by hundredths can both reach 0.8694 with a partner, and the figure means the same thing in both cases. The correlation is a statement about how closely the two moved in step, expressed independently of how far either one moved. A correlation is a claim about movement together, and it contains no direction of influence, no mechanism and no promise about any month outside the record it was computed from.

Sit with what is missing from that. Think about a tea stall outside one office building. On evenings when the office runs late, the stall sells more tea. Somebody who watches for a month has a real correlation in their notebook. But the sentence the correlation supports is only this: late evenings and heavy tea sales turned up together. The sentence is equally true written the other way round, that heavy tea sales and late evenings turned up together, and the notebook contains nothing that prefers one arrangement to the other. The stall owner may be certain which way it runs, and may well be right. The certainty is coming from knowing how offices work, not from the tally in the notebook.

Correlation vs Covariance: why can a covariance not be read on its own?

Underneath the correlation sits a rawer figure called the covariance, and on these ten months it is 50.0000. The covariance measures the same joint movement, taken before the sizes of the two series are divided out. Take each month's distance from the Nakshatra unit's own average, multiply it by that month's distance from the Vasant unit's own average, add the ten products and divide by nine. The result is 50.0000, and it is positive for the same reason the correlation is positive: months above average on one tended to be above average on the other.

Now ask the question a reader always wants to ask. Is 50.0000 large? The question has no answer. Multiplying a distance in per cent by another distance in per cent produces a figure measured in per cent squared. Nothing in daily life is measured in per cent squared, so nobody has an intuition for it. There is no benchmark to hold 50.0000 against and no ceiling it cannot exceed. A covariance carries the units of both series multiplied together, so the figure on its own gives the sign of the joint movement and nothing at all about its strength.

An everyday version makes the trouble obvious. Suppose the covariance between the number of guests at a wedding and the weight of food ordered is reported as 4,000. Four thousand what? Guest kilograms, whatever those are. Change the caterer's book from kilograms to grams, exactly the same wedding, and the figure becomes 4,000,000. Nothing about the wedding changed. Dividing the sizes out is precisely what a correlation does, so the correlation would sit at the same value in both books.

What happens to both figures when one series is rescaled?

Here is the move that settles the difference between the two figures without any argument about it. Take the Vasant unit and double every single one of its ten monthly changes. Change nothing else. Leave the Nakshatra unit exactly as it is, add no month and remove none. Doubling one column is a change of nothing but scale: the same ten occasions, the same shape, the same order, with the second column written in units twice as large as before.

Three quantities are worth watching through that change. The covariance goes from 50.0000 to 100.0000, which is exactly double. The slopeThe number of points the second series moved for each one point of the first, taken from the straight line fitted through the ten months. How that line is found and how well it fits is covered separately. of the line through the ten months goes from 1.5000 to 3.0000. The slope doubles too. And the correlation goes from 0.8694 to 0.8694. The correlation does not move at all, and not approximately: to every decimal that can be printed, it is the same figure before and after.

The covariance changed and the relationship did not. Any figure that moves when the units change is measuring the units as well as the relationship. Nobody imposed that as a rule. The cancellation falls straight out of the arithmetic: doubling one column doubles every product that goes into the covariance, and it also doubles the spreadA single figure for how widely a set of numbers is scattered about its own middle. Established in earlier notes and taken as read here. Multiply every entry by two and this figure doubles along with them. of that column, and the correlation divides the first by the second, so the doubling cancels exactly.

Which of the two goes to somebody else, then? The correlation, every time, unless the other person knows precisely what both series are measured in. Two people comparing the covariance of a pair measured in per cent against the covariance of a pair measured in rupees are not comparing anything. Both figures sit on the same fixed scale, whatever went into them, so two people comparing 0.8694 against 0.7100 are comparing like with like. A correlation can be carried between pairs and a covariance cannot, and that portability is the entire practical difference between them.

The picture below makes the same point a second way, and it is worth looking at closely because it is easy to misread. The two panels are plots of the same ten paired months, the left as recorded and the right with every Vasant figure doubled. The shape does not change. Every point sits in exactly the same place in its panel, and the fitted line runs at exactly the same angle. Only the numbers written up the side changed, and they doubled. A reader who looks at shapes rather than at axes sees two identical pictures and is exactly right to do so.

What do the same ten months look like written out in full?

The table below is the whole record, in time order, with the doubled column beside it. The first two columns hold the ten paired months behind every figure worked so far. The third holds the rescaled version: the same ten occasions with the second measurement written in units twice as large. The third column contains no new information whatsoever. The correlation computed from it is identical for exactly that reason.

Worked out of the first two columns, the three readings are a covariance of 50.0000 in per cent squared, a slope of 1.5000 and a correlation of 0.8694. Worked out of the first and third columns instead, they are a covariance of 100.0000, a slope of 3.0000 and a correlation of 0.8694. Months 5 and 10 are the identical pair, 1.00 per cent against minus 3.00 per cent, and the plot above shows nine visible points for ten months for that reason. Nothing was reordered to make any of this tidier.

Correlation vs Causation: what would a causal claim actually require?

Causation is a claim of a completely different kind. To say the Nakshatra unit causes the Vasant unit to move is to say that if somebody reached in and changed the Nakshatra unit, the Vasant unit would move in response. The claim is about what would happen under an intervention that nobody performed. The ten months contain no interventions at all. The ten months contain two columns of numbers that were watched.

Three things would have to be in place before that claim could be made, and it is worth being precise about each rather than gesturing at the general idea. The first is something that fixes the direction: a reason, outside the two columns, why influence would run one way and not the other. The second is something that rules out a third thing moving both: a reason to believe there is no further series, absent from the table, that pushed each of these two around independently. The third is a change rather than an observation: some occasion on which the input was actually altered and the outcome was watched afterwards.

All three of those are conditions on the situation the numbers came from, not conditions on the numbers, and no quantity computed from a table can supply any of them. This is the point where most readers want an escape and there is not one. Adding months does not help; a thousand months of watching is still watching. Adding decimal places does not help. Adding a more elaborate method does not help either. Every method on offer takes the same two columns as its input, and no method can extract from a table something that was never recorded in it.

An everyday case shows how ordinary this is. A landlord notices that in the months when a household pays rent late, the household also orders fewer deliveries. The two go together reliably. Does paying late cause fewer deliveries? Almost certainly not: a thinner month causes both, and the thinner month is nowhere in the landlord's records. The landlord did not need statistics to work that out. The landlord needed to know something about how households run. Knowledge of that kind is knowledge about the situation and not about the tally.

What are the four things one strong correlation can mean?

Take the 0.8694 and lay out everything it is consistent with. The arithmetic presents four readings at equal weight, and the honest way to set them out is at equal weight too.

The first reading is that the Nakshatra unit moved the Vasant unit. The second is that the Vasant unit moved the Nakshatra unit. The same story with the arrow reversed fits the ten months exactly as well. The third is that some third thing, not in the table at all, moved both of them, and neither one ever touched the other. The fourth is that nothing moved anything: ten paired numbers fell out this way and there is no relationship of any kind behind them.

The correlation is 0.8694 under all four readings, identically. A reader who reaches for the first one has chosen it rather than discovered it. That is a hard sentence and it is worth being clear about what it does not say. The sentence does not say the first reading is wrong. The first reading may well be the true one. The claim is narrower: the choosing was done somewhere other than in the arithmetic, and an analysis that does not say where has a hole in it.

The four are not equally plausible in every situation, and nothing requires pretending they are. In the tea stall case, the fourth reading is hard to take seriously after a month of watching, and the second is odd on its face. The ranking comes from knowing about offices and tea. Ranking the four readings is a legitimate and necessary act of judgement, and the moment it is presented as a finding rather than a judgement, the analysis has misrepresented where its conclusion came from.

Which of the four does the arithmetic settle, and which does it leave open?

A correlation is not useless, and it is worth being exact about what the ten months do settle. The ten months settle the strength of the joint movement, at 0.8694. The same ten months settle the direction of movement, and the direction is positive: months above average on one tended to be above average on the other. A third settled figure is the size of the association, at 1.5000 points of the Vasant unit for each point of the Nakshatra unit. And they settle how much of the Vasant unit's variation the straight line through them accounts for, leaving an unexplained shareThe portion of the second series that the fitted line does not account for, here 24.41 per cent. The make-up of that leftover, and what it can be used to check, is covered separately. of 24.41 per cent that the line does not account for at all.

The ten months do not settle which of the four readings holds, and the temptation to add something to the table becomes strongest here. Adding never works. The limit here is not a limit of these particular ten months but a limit of what a correlation is, so no larger record, no finer measurement and no additional column removes it.

One consequence catches careful people, and it is worth stating flatly. A higher correlation is not stronger evidence of cause. A higher correlation is stronger evidence of moving together, and moving together is a different claim. A pair of series with no connection whatsoever can post a correlation far above 0.8694 across ten months. Strength and mechanism are separate axes, and reading one off the other is the single commonest way an analysis goes wrong without anybody noticing.

How does an analyst report a relationship they cannot explain?

The honest position is not silence, and the practical question is what an analyst says instead. An analyst at a research desk, a lender sizing an exposure, an investor reading a note and a household deciding whether last year's pattern will hold again all face the same problem: a relationship that is real in the record and unexplained in the world. There are five sentences always available, and each one is defensible in front of somebody who checks.

First comes the strength and the record it came from: the Nakshatra unit and the Vasant unit correlate at 0.8694 across ten months. Second comes the direction the arithmetic runs in, together with the fact that the arithmetic does not fix that direction: the Vasant unit moved 1.5000 points for each point of the Nakshatra unit, and that figure would read the same if the influence ran the other way. Third comes the third thing worth looking for: the series that might be moving both, named even where the record does not contain it. Fourth comes what would have to be true for the causal reading to hold. Fifth comes the leftover: 24.41 per cent of the Vasant unit's variation is not accounted for by the line, and that share is the honest measure of how much of the story is missing.

Every one of those five sentences survives being checked, and surviving a check is the only test that matters. Saying less than the record supports is the only version that survives it. Very little is given up. The reader of that note learns the strength, the size, the direction of movement, the limit and the thing to go and find next. The one thing they do not get is a claim about cause dressed as a finding, and no table ever entitled them to that.

There is one habit that catches the mistake before it reaches paper. The finding is written twice, once with the word moved and once with the word caused, and both are read against the decision that rests on it. If the decision needs the second version and the record only supports the first, the gap has been found before somebody else finds it.