Regression, Correlation and Fit: One Calculator, Ten Months

Two columns of paired readings typed into the table at the top of this calculator return seventeen rows: the fitted line, the correlation, the covariance and four error measures, each step shown as it is built. The ten invented months it opens on give a line of 0.50 plus 1.50 times the input, a correlation of 0.8694, an R squared of 0.7559 and a lag one reading of 0.4862 on the leftover misses.

The two ten month records were built so that the arithmetic lands exactly rather than nearly. Nothing was lifted out of a market, a filing or a database, so no as-of date exists to be quoted.

Paired readings typed in, and the six steps as they build

The table below is the instrument. The table opens on the ten month record this calculator is written around, so the figures it prints the moment it loads are the same ones printed as ordinary text further down. Any cell can be overtyped with fresh readings, rows added and rows dropped, and the six steps, the reconciliation and the chart all recompute as the typing happens. The reconciliation is the part worth watching. The reconciliation puts the input average back into the line the six steps just built and shows the answer landing on the outcome average, with a gap of nothing between them, on every set of pairs that can be typed. The fitted line runs through the crossing of the two averages by construction, never by luck, and that holds on the unrounded figures rather than on the four decimal places shown.

Nothing else is consulted, and no figure below arrives from any source other than the pairs in that table. The published ten describe ten months of an invented traded unitSomething whose price buyers and sellers keep agreeing on, so a record of what it changed by each month can exist at all. Both units named here were invented to be typed into a table. called the Nakshatra unit and a second invented one called the Vasant unit. The meaning of each returned figure is covered separately. The calculator adds the arithmetic behind each one, printed in full with the tool switched off, and a second panel further down that moves a single month so that every row can be watched responding.

What does this calculator take, and what does it hand back?

The calculator takes up to ten paired observationsTwo readings lifted off one and the same occasion and kept side by side. Slip one column against the other by even a single row and every figure underneath describes something that never happened., in the order they actually happened, and it hands back seventeen rows. Three of those seventeen are not really results at all: the count of pairs, and the average of each column. The count and the two averages are handed straight back so that the tool can be confirmed to have read what was intended. The other fourteen are worked out, and every one of them is worked out from the same two columns.

A fit report has no second source of facts standing behind it, and that single origin is the whole reason for gathering fourteen figures a reader usually runs into one at a time into a single table. Met separately, the slope looks like one kind of thing and the correlation looks like another, and it is easy to carry away the impression that a fit report is assembled from several different investigations. It is not. A fit report is one small table of numbers, run through sums, and the sums are then divided by different things.

The everyday version needs no market at all. A household notebook carries two columns for twelve months: what the electricity bill came to, and how many days the air cooler ran. Everything anyone can honestly say about how those two travel together comes off that single sheet of the notebook. There is no thirteenth column arriving from elsewhere to confirm it. If the notebook is wrong, every conclusion is wrong, and no amount of arithmetic afterwards will say so.

How does it get the line out of the pairs?

Two sums do the work. The first asks how much the input column moves about on its own: take each input away from its average of 1.00, square the result so the direction stops mattering, and add the ten squares. The ten squares come to 300. The second asks how much the two columns move together: take each input away from 1.00, take each outcome away from 2.00, multiply the two distances for that month, and add the ten products. The ten products come to 450.

The slope is the second divided by the first. Four hundred and fifty over three hundred is 1.5000. The fitted line always passes through the pair of averages, so the intercept is whatever number puts it there: 2.00 less 1.50 times 1.00 is 0.5000. So the line reads 0.50 plus 1.50 times the input, and the calculator has finished its main job in two divisions and one subtraction.

Both of those figures are exact on the ten default months, not rounded, so a reader who works them out on paper will match the calculator to the last decimal. The ten months were built that way on purpose, leaving no rounding artefact to be explained away. Readings typed in from elsewhere will almost certainly arrive as 1.4997 and 0.5013 instead. A record that was not designed looks exactly like that.

Where do the correlation and the covariance come from?

From the same 450. Reusing that one sum is the part worth slowing down for. The covariance is the movement together divided by nine, one less than the count of pairs, giving 50.0000. Two percentage figures were multiplied together to build the covariance and nothing has taken the square root since, so its units are per cent squared. The correlation is the covariance divided by the two spreadsHow far a column of numbers usually sits from its own centre, written as one number. Covered separately, and used here only as something to divide by., 5.7735 per cent for the input and 9.9610 per cent for the outcome. The answer is 0.8694, and it carries no units at all.

The slope, the covariance and the correlation are one quantity read three times under three different divisors. Swapping any of them for another breaks the reading. 450 divided by the input's own movement of 300 gives the slope. The same 450 divided by nine gives the covariance. Divided by nine and then by both spreads, it gives the correlation. One numerator, three denominators, three answers that mean three different things and are quoted in three different sets of units.

Every one of those denominators is a whole numberA figure that stops at the decimal point, such as ten months or the nine gaps between them. Every divisor named here is one, even where the thing being divided is not. or a length built from the data itself, and not any of them was chosen by preference. The nine is the count of pairs less one. The 300 is a sum that can be added up by hand in a minute.

How is R squared built out of the misses?

Not from a formula that has to be trusted. From a chain of three numbers that can be added up by hand. The worst honest attempt available is the place to start: the same figure guessed for every month, and that figure the outcome column's own average of 2.00 per cent. Each month taken away from 2.00, squared, and the ten squares added, comes to 893. A miss of 893 is how badly the flattest possible answer does.

The line is then fitted and the identical exercise run against it. Each month taken away from what the line predicted for that month, squared, and the ten squares added, comes to 218. The line missed by 218 where the flat guess missed by 893, so the line accounted for the remaining 675. R squared is that difference as a share of the starting point: 675 over 893 is 0.7559.

Adjusted R squared then charges a fee. Fitting a line used up two of the ten pairs, in the sense that two numbers were pulled out of the data to build the line itself, so the adjusted figure divides the leftover 218 by eight rather than by ten and divides the starting 893 by nine rather than by ten. The adjusted figure works out at 0.7254. The gap between the two is 0.0305 on these ten months, and it widens with every extra input added. Widening is the entire reason the second figure exists. On one input the fee is small. The fee stops being small quickly.

Which error measure is read against the outcome?

Four of them come back and they are not interchangeable. The nuisance eases once only two things turn out to separate them: what gets added up, and what it gets divided by. The mean squared error adds the ten squared misses to 218 and divides by ten, giving 21.8000. Its units are per cent squared, so putting it next to a monthly change of 6.50 per cent is comparing an area with a length. The mean squared error is a fine thing to minimise and a poor thing to quote.

The root mean squared error takes the square root of that same 21.8000, giving 4.6690 per cent. A figure in per cent can sit beside a monthly change without embarrassment. The mean absolute error skips the squaring entirely: add the ten miss sizes to 36, divide by ten, and get 3.6000 per cent. The mean absolute error comes in below the root mean squared error because it does not let the two big months of 9.00 and 8.00 dominate. Letting a bad month dominate or not is a choice, never a neutral default.

The fourth is the one that causes the trouble. Two degrees of freedomHow many readings are left to vary after some have already gone into producing a figure. Building the line here uses up two of the ten before any counting starts. were spent fitting the line, so the residual standard error divides the same 218 by eight rather than ten. Its square root is 5.2202 per cent. The residual standard error is the figure most often reported as the typical miss, and on these ten months the typical miss is 4.6690, not 5.2202. The two are 0.5511 percentage pointsThe unit a change in a per cent figure gets counted in. Going from four per cent to six per cent gains two percentage points, and that is not the same claim as gaining two per cent. apart, on the identical set of misses, purely because of the denominator.

What does the last row show that most fit reports leave out?

The lag one reading of the misses comes back as 0.4862 on the default months. The reading is computed by pairing each miss with the one before it, multiplying the two, adding the nine products to 106, and dividing by the 218 already in hand. Nothing exotic. But it is the only row of the seventeen that would give a different answer if the ten pairs were shuffled, and that is precisely why it earns its place.

Sixteen of the seventeen rows throw the running order away, and this one keeps it. Reordered any way at all, the ten months leave the slope at 1.5000, the correlation at 0.8694 and R squared at 0.7559, and every error measure holds to the last decimal. All of them are built from sums, and a sum does not care what order things are added in. Sorted by the outcome column, though, the same ten pairs take the last row down to 0.1422. Sorted by the input column instead, it reads minus 0.2982. Sorted by the misses themselves, smallest to largest, it climbs to 0.6193. Three different orderings, three different readings, one unchanged set of numbers.

On the record as it actually happened, the 0.4862 says that the misses left over from a fit reporting an R squared of 0.7559 are not independent of one another: a month landing above the line is unusually likely to have a month above the line sitting next to it. Whether that finding is fatal, mild or a symptom of something else is a proper argument with its own working, and it is covered separately. Most fit reports never print the number at all.

What does the whole table read on the ten default months?

Here it is in full, printed as ordinary text so that it exists whether or not the tool is able to run. The ten pairs are the ones in the first figure, in the order they were recorded, and they are never sorted. Every row below was computed from the two columns.

And the ten misses themselves, month by month, in the order they happened: 1.00, 9.00, 8.00, 0.00, minus 5.00, 0.00, minus 3.00, minus 3.00, minus 2.00 and minus 5.00 per cent. Add them and nothing is left. The zero reads like a tick in a box and is not one. A fitted line always leaves misses that cancel, so a total of nothing confirms that the sums were done and says not one thing about whether the line is worth having.

What is the one thing this calculator cannot settle?

Whether the two columns have anything to do with each other. The calculator will fit a line to whatever is typed in, and it will fit that line just as willingly to two columns with no connection of any kind. The willingness is not a flaw to be patched but the arithmetic itself, and a tool that pretended otherwise would be lying. The misreading this calculator exists to prevent is treating a good fit as a claim that the input causes the outcome, or that the eleventh occasion can be read off the line. Both halves of that are drivable in the instrument at the top, and it is worth pressing the two buttons rather than taking the next four paragraphs on trust.

Press the chair count button and the input column becomes the Kadamba count: how many chairs were set out in a made up community hall that month, running 58, 79, 66, 72, 53, 45, 60, 59, 53 and 55. Nothing links chairs in a hall to a traded price. Fitted against the same Vasant column, the calculator returns a correlation of 0.9502, an R squared of 0.9029 and only 86.7349 left over, against the traded unit's 0.8694, 0.7559 and 218.0000.

The chairs win on every figure this calculator can produce, and there is no arithmetic anywhere in the seventeen rows that can say they should not have. Splitting the record in half is no rescue either. The pattern holds in both halves. The separation being sought is a question about mechanism, and a mechanism is not a number, so it will never appear in a returned table.

The swap button makes the same point from the other side, and it is the harder of the two to argue with. Press it and the Vasant column becomes the input while the Nakshatra column becomes the outcome. A slope is measured in units and the units have just been exchanged, so the slope moves: 450 over 300 becomes 450 over 893, and 1.5000 becomes 0.5039. The correlation stays at exactly 0.8694 and R squared stays at exactly 0.7559. Fit strength reads identically in both directions. Reading the same both ways is the plainest proof available that fit strength carries no direction at all. If the arithmetic cannot tell which column is acting on which, nothing it returns can be evidence that either one is.

The claim about next month fails on related ground. The line was built to sit as close as it can to occasions that have already been recorded, and none of the six steps looked forward. An eleventh month is in none of the sums, so the line has no hold on it, and the R squared of 0.7559 reports how the line did on the very record it was fitted to. The argument about prediction is worked through under patterns with no mechanism, covered separately.

What is checked before trusting anything it prints?

Four things, and each one has a specific way of going wrong. First, confirm the pairs are in the order they actually happened. Get that wrong and sixteen of the seventeen rows are still perfectly correct while the last one is quietly meaningless, and nothing anywhere on the screen will flag it.

Second, both columns are confirmed to carry the units they are believed to carry. The covariance and the slope both move when units move: a column recorded as 0.065 rather than 6.50 changes the slope by a factor of a hundred and the covariance by the same. The correlation and R squared do not budge. Two people can therefore agree completely about the strength of a relationship and disagree wildly about its size, and neither is making an error of arithmetic.

Third, confirm nothing was sorted, dropped or patched after the first look at the data. A month deleted because it seemed odd is a decision about the answer disguised as a decision about the data, and the returned table cannot see the difference.

Fourth, look at the count. Ten pairs is a thin sampleThe readings that happen to be in hand, as against all the readings that could have been taken. Ten months is a sample, not a complete record of anything., and this calculator prints every figure to four decimal places regardless. The tool has no way of knowing how many pairs deserve that many. Four decimals on ten pairs is a statement about the arithmetic, never about how firmly the answer is known. The printing is exact; the knowing is not, and the tool cannot mark the difference.

How is the whole table rebuilt by hand?

In six lines, and it is worth doing once by hand even though the instrument above will do it unaided. An analyst who has never done it reads a fit report as a black box that emits figures; an analyst who has done it once can see immediately three things: the figures that move when the units move, the figures that move when a month is added, and the figures that cannot move at all.

Line one: add each column and divide by ten, giving averages of 1.00 and 2.00 per cent. Line two: take each input away from 1.00, square each deviationThe distance from one reading to the centre of its own column, carrying a direction as well as a size., add the ten squares, giving 300. Line three: multiply each month's two distances together and add the ten products, giving 450. Line four: take each outcome away from 2.00, square, add, giving 893. Line five: 450 over 300 is the slope of 1.5000, and 2.00 less 1.50 times 1.00 is the intercept of 0.5000. Line six: take each outcome away from what the line said, giving the ten misses.

Everything else in this calculator is those six lines rearranged. At line three, what happens to the 450 if the outcome column is recorded in decimals rather than percentages? The 450 shrinks by a factor of a hundred, so the covariance shrinks and the slope shrinks with it. Then the correlation divides that shrunken 450 by a spread that shrank by exactly the same factor. Nothing happens to it at all. Written out by hand once, that stops being a fact to memorise.

The Nakshatra unit, the Vasant unit and the Kadamba count are invented.
Educational material. Not advice on any investment, tax, budget or market position.