Dispersion Measures: Variance, Deviation and Range

The calculator takes a run of values, or that record written as values with counts, and returns the range, the mean absolute deviation, the variance and the standard deviation, every squared distance printed. On the invented fifty month record it returns a variance of 24.74, a deviation of 4.97 per cent, a mean absolute deviation of 3.52 per cent and a width of 20.00 per cent, alongside 24.25 and 4.92 on the count denominator.

Underneath every one of those readings is the same short run of arithmetic: find the centre, measure how far each value sits from it, square those distances, weight each square by how often it occurred, add them, divide, and take a root of the result. Nothing is fitted, nothing is assumed about the shape of the record, and nothing is looked up. The Nakshatra unit is an invented traded unitA single thing whose price is quoted and which changes hands. Used here only as a label for the invented object the record is attached to. written for teaching, and its fifty month record is one invented list of what its monthly changeThe change in price across one month, written as a percentage of where the price started that month. did over fifty months.

What does this calculator compute?

Five readings come out of one run of values, and two of them are then printed a second time. The mean, the centre every distance is measured from. The range. The mean absolute deviation. The variance. The standard deviation. Then the variance and the standard deviation again on the other denominator. Those two are the only readings a denominator can move.

The range does not care which denominator is used, and that single fact is a hint about how little of the record the range actually reads. A denominator is a division by a count, and the range never divides by anything. The range takes the largest value, takes the smallest value away from it, and stops. Fifty months or five thousand, the arithmetic is identical and the answer is identical. Forty eight of the fifty months took no part in producing it.

Here is the everyday version. The tallest and shortest people in a wedding hall stand up, the gap between them is measured, and that gap is the range of heights in that hall. Four hundred more guests of middling height arrive and the figure does not move at all. Asked instead how far the typical guest stands from the average height, every one of those four hundred changes the answer. The distance between those two answers is the difference between a reading built from two cases and a reading built from all of them, and the calculator prints both so the disagreement is visible.

Where does each typed number come from?

The panel at the top takes the run itself, one entry for every period, and the panel further down takes the same record written the short way: the distinct readings in one column, how many times each occurred in the other. Both forms are typed in, and the note beside each field says which part of the record to read it off and then stops there.

The arithmetic estimates nothing and assumes nothing about the shape of the record: it adds, squares, divides and takes a root on exactly what was typed. It does not smooth anything, it does not throw out a value for being far from the others, it does not fill a gap, and it does not check whether the record leans one way or carries heavy extremes. A wrong count gives flawless arithmetic and a wrong answer.

Writing a record as values with counts rather than as fifty separate rows is called a tallyA compressed way of writing a record: every distinct reading listed once, with a number beside it saying how many cases carried that reading., and it is only a shorthand. Fifty rows reading minus 9.00 five times and minus 4.00 nine times, and so on, produce every figure here identically. The tally is quicker to type and much quicker to check. The one thing to confirm is a single addition.

How is the variance built, step by visible step?

Six steps, and the panel at the top prints all six. A tool that shows only its answer cannot be checked. First the mean, 0.50 per cent here. Then each value minus that mean, giving the distances. Then each distance squared. Then each square weighted by how often its value occurred, so a value that happened twenty five times weighs twenty five times as much as one that happened once. Then the running total. Then the division.

On the fifty month record the distances are minus 9.50, minus 4.50, 0.50, 5.50 and 10.50. Squared, they are 90.25, 20.25, 0.25, 30.25 and 110.25. Multiplied by the counts 5, 9, 25, 8 and 3, they give 451.25, 182.25, 6.25, 242.00 and 330.75, and those five products add to 1,212.50.

A reader with a pen can reproduce every line of that, and this calculator is built for exactly that. The last two steps are the only place a choice appears. Dividing 1,212.50 by 49 gives 24.744898, printed as 24.74. Dividing the same 1,212.50 by 50 gives 24.250000, printed as 24.25. The square roots of those two are 4.974424 and 4.924429 per cent, printed as 4.97 and 4.92.

The third row of the ladder is worth a pause. Twenty five of the fifty months sat at 1.00 per cent, half the record, and they contribute 6.25 to a total of 1,212.50. The whole of that middle group contributes half of one per cent of the answer, from half of the months. Squaring a distance of 0.50 leaves 0.25, and a distance that small all but disappears once it has been squared. Whether that is a good property or a bad one is argued separately. The ladder prints the number so the effect is visible.

The same test runs the other way through the append field at the top. One further month of 46.00 per cent on the end of the fifty makes the record fifty one months with a centre of 1.39 per cent, and that single month carries 61.37 per cent of the squared total on its own. One month in fifty one, and just over three fifths of the answer. Squaring is what lets it do that, and it is the same property working in both directions: the months near the centre all but vanish, and the months far from it come to dominate.

Variance vs Standard Deviation: how are the two outputs read together?

The variance and the standard deviation are one quantity printed twice, at two sizes, in two units. The variance on the fifty month record is 24.74, carried in per cent squared. Every distance was squared before anything was added. The standard deviation is 4.97, carried in per cent, the same unit as the months themselves. The square root undid the squaring at the end. Reading them together means reading one number and one conversion, not two facts.

The check worth running every single time is that the standard deviation squared must give the variance back. 4.974424 multiplied by itself lands on 24.744898. A tool reporting 24.74 and 5.42 has taken one of the two from somewhere else, and that is established in four seconds without seeing any of the underlying months. The check costs nothing and it catches a whole class of copy and paste faults.

The second check is a sanity check on size. The standard deviation should sit comfortably below the range and should almost never come close to it. Here it is 4.97 against 20.00, roughly a quarter. The relationship holds for a plain reason. The range is a gap between the two furthest apart months, the standard deviation is a typical distance from the middle, and a typical distance cannot be the whole width. A standard deviation that is larger than the range is not a surprising record, it is an input error, every time.

The third thing to read is the size of the variance relative to the standard deviation, and it changes direction depending on the numbers. Here 24.74 is much larger than 4.97, and people take the ordering for a rule. The ordering is not a rule. Squaring a number below one makes it smaller, so a record whose standard deviation is 0.99 per cent has a variance of 0.98. The flip appears in the panel further down: at the lowest setting of the factor the same record returns a standard deviation of 0.50 per cent against a variance of 0.25.

Why does the mean absolute deviation not match the standard deviation?

Both readings start from the same fifty distances, and both have to get rid of the direction before they can go anywhere. The signed distances add to exactly nil, and an average of nil measures nothing at all. The two readings get rid of the direction in two different ways, and the difference lands in the answer.

The mean absolute deviation drops the direction and keeps the size untouched. The five distances of 9.50, 4.50, 0.50, 5.50 and 10.50, weighted by their 5, 9, 25, 8 and 3 months, add to 176.00, and 176.00 over fifty months is 3.52 per cent. The standard deviation squares instead. Squaring destroys the direction just as thoroughly, but it is not a neutral way of destroying it: a distance of 10.50 becomes 110.25 while a distance of 0.50 becomes 0.25, so the far months come out weighing very much more than their count of months would give them. The same fifty months return 3.52 per cent one way and 4.92 per cent the other on the count denominator, and neither figure is a correction of the other.

The gap between the two is a reading in its own right. The gap says how uneven the distances are. Record B in the panel at the top has every one of its months sitting exactly 4.92 per cent from its centre, so dropping the direction and squaring it come to precisely the same thing there, and its mean absolute deviation and its standard deviation are both 4.92 per cent. The fifty month record has a heavy middle and a handful of far months, and its two readings sit 1.40 per cent apart. Which of the two answers which question is argued separately, and this calculator prints both rather than choosing.

What changes when the denominator is switched?

Two of the four readings, and by a specific amount. On the fifty month record the variance is 24.74 on the 49 denominator and 24.25 on the 50, a gap of 0.4949. The standard deviation is 4.97 against 4.92, a gap of 0.0500. The mean does not move and the range does not move. Those two gaps are the whole of the difference, and both figures came out of a total of 1,212.50 that never changed at all.

Both rows are printed here. A tool that silently picks one is the reason two people holding the same record get different answers and cannot find the difference. Neither row is a correction of the other. Why the two denominators exist and which one answers which question is argued in full separately, and it is worth reading before either figure is quoted to anybody.

Spotting which one has been handed over is straightforward. The 49 row is always the larger of the two, on every record. A smaller denominator divided into the same total gives a bigger answer. The ratio between the two variances is fixed at fifty divided by forty nine, or 1.020408, whatever the months happen to be. So two figures differing by almost exactly two per cent in the variance or one per cent in the standard deviation mark a denominator difference and not a data difference.

There is one more figure worth putting beside those two, and this record allows something almost no real record does. The five values and their weights were settled before the record existed, so the true spread of the generatorThe written down rule saying which values can occur and how often each is expected to. Here it was fixed first and the record was drawn from it afterwards, which is why a true figure exists at all. behind this record is known, and it is 5.00 per cent exactly. Both of the calculator answers sit below it: 4.97 and 4.92. Neither of them is the truth, and the gap between them is smaller than the gap between either of them and the truth.

What can a spread figure not report, however carefully it was computed?

A spread figure reports how far the values stand from the middle of their own record, and reports nothing beyond that. A spread figure does not say the record is symmetric. Nor does it say the extremes are rare or common. Nor does it say the months arrived in any particular order. And it does not say anything whatsoever about a month that has not happened yet.

Two records can both report a standard deviation of 4.97 per cent and look nothing alike. The fifty month record has a heavy middle, with twenty five of its months sitting at 1.00 per cent and a thin scatter either side. Now take a second invented record of fifty months where twenty five months read minus 4.42 per cent and the other twenty five read 5.42 per cent, and nothing sits anywhere near the middle. Both have a centre of 0.50 per cent. Both report a standard deviation of 4.97 per cent to two decimals. One has a middle and the other has a hole where its middle should be.

Their ranges are not even close: 20.00 per cent against 9.84 per cent. So the second record is tighter at the extremes and more spread in the body, and one figure was never going to carry both of those facts. Whether a record leans to one side, and whether its far months are heavier than they look, are measured by separate figures that are covered separately.

How is a spread figure handed over by somebody else checked?

Spread figures arrive already computed far more often than they get computed from scratch. A lender reads a spread on a borrower to see how much the monthly receipts jump about before deciding what a repayment schedule can survive. An analyst reads one on a set of monthly changes before quoting any summary of it. A household does the same arithmetic without naming it when it looks at twelve months of grocery bills and asks how far a bad month runs from a normal one. In each case the figure arrives finished, and four checks cost about a minute.

Confirm the counts, square the deviation back, confirm the answer sits below the range, and ask which denominator produced it. Three of those four are free arithmetic on figures already in hand. The fourth has to be put to a person, and it is the one that gets skipped.

There is a fifth question worth asking when the range is the figure being quoted: how many cases are carrying it. A range of 20.00 per cent built from two months out of fifty says that two months were far apart, and the other forty eight took no part in producing it. The standard deviation of 4.97 per cent used all fifty. The count behind a figure is not an argument for one reading over the other. The count is a fact about what each figure was built from, and it changes what a single unusual month can do to the answer.

And note what none of these checks can do. The four checks confirm that a figure is internally consistent with the record it claims to come from. Not one of them can confirm that the record is the right record, that the months were transcribed correctly, or that a month was not quietly left out. An estimatorAny recipe that turns a record into a figure meant to stand for the truth behind that record. Which recipes are trustworthy, and in what sense, is covered separately. is only as good as what was fed to it, and the arithmetic here is entirely happy to be fed the wrong thing.

What does the whole worked record look like in one place?

Here is the default set of inputs and every reading it produces, written out as static text so the arithmetic exists in the guide and not only inside the panel. Its values are minus 9.00, minus 4.00, 1.00, 6.00 and 11.00 per cent, occurring 5, 9, 25, 8 and 3 times, and those counts add to fifty months.

Beside those five readings sits one figure the calculator did not produce and could not produce. The true standard deviation behind this record is 5.00 per cent, and it is known because those five values, and the weights sitting on them, were settled before the first month was ever drawn. The 5.00 per cent is a stated property of the generator, not a measurement taken off anything. The calculator returned 4.97 and the truth is 5.00, so a perfectly executed run of arithmetic on fifty real months landed 0.0256 short, and no check available here would have revealed that. Almost no real record permits that sentence, and this one does only because the answer was written down first.

What has to be looked up before this calculator can be trusted?

Nothing at all. There is no document to fetch, no series to download and no institution whose word would make a variance of 24.7449 any more correct than the ladder above already makes it. A tool that adds, squares, divides and takes a root can be checked by anyone holding a pen, and every step is printed above.

The Nakshatra unit, its fifty month record and the second record set beside it are invented.
Educational material. Not advice on any investment, tax, budget or market position.