Arithmetic Mean vs Geometric Mean: Which One Compounds

Two means, two questions. The arithmetic mean reports what a single month taken at random is worth, and on the fifty month record it reads 0.50 per cent. The geometric mean reports the steady rate that would have carried money through all fifty months to the same ending place, and it reads 0.3788 per cent. Neither is more accurate. The smaller one is not a cautious version of the larger.

Both readings come off one invented tally. The Nakshatra unit is a made up traded unitSomething with a quoted price that people buy and sell. Here it is nothing more than a label to hang a run of numbers on. whose monthly changeThe move across one calendar month, expressed against whatever the value happened to be when that month opened. can land on five values, and the fifty month record is fifty of those months tallied as 5, 9, 25, 8 and 3. Adding those fifty months and sharing the total out gives one number. Multiplying them and sharing the growth out gives a different one. Everything below is the distance between adding and multiplying, and what that distance costs when the wrong one is quoted.

What question does the arithmetic mean actually answer?

Start with the question rather than the formula. The formula is the easy half. The arithmetic mean answers this: reach into the fifty month record, pull out one month without looking, and ask what that month is worth. Not the run of fifty. One month, taken on its own, with nothing carried into it and nothing carried out of it.

The arithmetic is the one everybody already knows. Total the fifty monthly changes and divide by fifty. Multiply each value by how many months landed on it, and the two falling values contribute minus 45 and minus 36 while the three rising ones contribute 25, 48 and 33. The five products total 25.00 per cent, and 25.00 shared across fifty months gives every month 0.50 per cent. The arithmetic mean is the answer to a question about addition, and addition is the correct operation whenever the quantities sit beside each other rather than on top of each other.

Consider a vegetable seller counting a month of takings. Thirty daily totals go into a tin, and at the end of the month the tin holds the sum. Tuesday's takings did not make Wednesday's takings any larger or any smaller; they simply landed in the same tin. The total in the tin divided by thirty is a fair figure for a day. The tin figure is an arithmetic mean doing exactly the job it was built for, and no other average would answer the question better.

What is compounding, and what question does the geometric mean answer?

Nothing is assumed here, so the word comes first. Rs 100/- goes into something whose value moves month by month. One month of 10.00 per cent leaves Rs 110/-. Whatever the next month does then works on Rs 110/-, not on the Rs 100/- it started from. Compounding is exactly that: every period acts on whatever the periods before it left behind. A savings balance nobody withdraws from does it. So does the cost of a wedding that rises with everything else each year. The eight per cent added this year is eight per cent of a bill that already grew last year.

Once quantities behave that way, months stop adding and start multiplying, and a new question becomes askable. Which one unchanging monthly rate, applied fifty times over, would take a starting amount through this record and set it down precisely where the fifty real months set it down? The steady rate that does this is the geometric mean. On this record it comes to 0.3787920792 per cent a month, printed everywhere below as 0.3788. The geometric mean is the answer to a question about multiplication, and multiplication is the correct operation whenever each period acts on what the period before it left behind.

One property of multiplication removes an objection before the objection forms, and the property is worth pausing on. The fifty months carry no order. The record is a tally rather than a diary, and no month in it is the first. Multiplying the same fifty factors in any order lands on the same product, so the missing order does not matter in the slightest. The geometric mean therefore needs no ordering to be well defined, and shuffling a record cannot change it.

Where exactly do the two means differ?

The two means differ in the question, and nowhere else. Both are computed off the same fifty months, both use every one of those months, and neither throws anything away. There is no extra care in one and no carelessness in the other. The two means differ in the operation their question requires, not in how honest or how careful the arithmetic behind them is.

The common framing gets this backwards, and the point is worth stating flatly. The geometric mean is often called the conservative figure, the prudent one, the one to quote when a record should not be flattered. The conservative framing produces the right habit for the wrong reason, and the wrong reason breaks the moment the question changes. The geometric mean is not a haircut applied to an optimistic number. The geometric mean answers a different question altogether, and put to an adding question it is simply wrong.

Here is a case where reaching for it would be costly. Suppose an arrangement pays whatever a single month of the record did, on a fixed Rs 100/-, and then resets, fifty times over. Nothing compounds; each payment lands in the tin and stays there. The fifty payments come to fifty times the arithmetic mean, and quoting the geometric mean there would understate the total. Being smaller does not make a number safer. Smallness only makes a figure the answer to a question nobody asked.

Why can the geometric mean never be the larger of the two?

Take the smallest record that shows it. Two months: one up 10.00 per cent, one down 10.00 per cent. Add them and share the total out and the arithmetic mean is exactly 0.00 per cent. Now carry Rs 100/- through them. The rise takes Rs 100/- to Rs 110/-. The fall of 10.00 per cent then works on Rs 110/-, so it takes away Rs 11/- and leaves Rs 99/-. The record lost a rupee on every hundred while its arithmetic mean read zero.

A fall works on the larger amount that the rise created, so a rise and a fall of the same size never cancel, and the gap between the two means opens the moment a record has any bounce in it at all. The geometric mean of those two months is minus 0.5013 per cent, and it is the figure that matches the Rs 99/-: two months at minus 0.5013 per cent, compounded, land on Rs 99/- exactly.

Now take the case at the other end, the case that makes this a proof rather than an example. Suppose every month in a record is identical, say 1.00 per cent, fifty times over. Nothing ever pulls ahead of anything else, so there is no larger amount for anything to work on. The arithmetic mean is 1.00 per cent, the geometric mean is 1.00 per cent, and the two are equal. Identical months are the only way the two can be equal. Every record with any variation in it at all has a geometric mean strictly below its arithmetic mean, and the more variation, the further below.

How big is the gap here, and what decides its size?

Measure it directly on the fifty month record. The arithmetic mean stands at 0.50 per cent while the geometric mean stands at 0.3787920792 per cent, so the gap between them is 0.1212079 percentage pointsThe unit for a difference between two percentages. A move from 3 per cent to 5 per cent is a rise of two percentage points. A rise of two per cent is a different thing.. Set that beside the record's own varianceThe average of the squared distances between each reading and the middle of the readings. How it is computed, and why it has two possible divisors, is covered separately.. The variance is 24.2500 and carries units of per cent squared. Halve it, giving 12.125, and put that back onto the per cent scale, giving 0.1212500 percentage points.

The gap and half the variance agree to four decimal places. The two are not equal, and the difference of minus 0.0000421 makes the relationship an approximation rather than an identity. Half the variance is the first and largest term of the gap, not the gap itself.

The reason is worth having in plain words. Turning a percentage change into a form that can be added rather than multiplied costs a small correction, and the leading piece of that correction is half the square of the change. Averaged across the fifty months, that leading piece is half of the mean squared distance, and half the mean squared distance is precisely half the variance. But the correction does not stop there. Smaller pieces follow, built on the cube of each change and on higher powers still, and those leftovers are exactly what stops the two numbers from matching. On a calm record the leftovers are minute. On a lively one they are not.

The approximation can be watched failing. With the same fifty months stretched away from the record's middle by a factor, the tally never changes and the arithmetic mean stays pinned at 0.50 per cent, but the record gets calmer or livelier. The true gap and half the variance then print side by side.

Read the last column downward. The difference is small and negative on a calm record, crosses zero at a stretch of about 1.045 times, and then grows steadily positive. The fifty month record happens to sit almost on top of that crossing, and the crossing is precisely why its gap and its half variance agree so beautifully to four decimals. Move to a different record and they would not. A quantity that drifts as a record gets livelier, and that changes sign along the way, is an approximation and cannot be an identity.

The consequence is the part to carry away, and it is simple. Since the gap tracks the variance, a calm record has almost no gap and the choice between the two means barely changes the answer. A lively record has a wide gap and the choice decides the answer. The penalty for picking the wrong mean grows with how much a record bounces, and has nothing to do with how high or low its readings sit.

What does the wrong mean cost, once it is put in money?

Twelve hundredths of a percentage point a month reads as nothing at all until it is converted into rupees. Rs 100/- carried through the fifty months of the record finishes at Rs 120.81/-. Put another way, the fifty monthly factors multiply to 1.2080871. A reader moving quickly does something else naturally. Take that arithmetic mean, the 0.50 per cent, and compound it fifty times instead. Rs 100/- finishes at Rs 128.32/-, overstating by Rs 7.51/- against the same starting hundred, and both amounts are arithmetic on an invented record rather than anything anybody received.

Two things drive that overstatement and neither is the level of the record: how many periods the figure gets applied across, and how much the months move about. The overstatement is therefore smallest in exactly the situations where nobody would have worried about it. Ten months in, the two paths are Rs 1.26/- apart. At twenty five months, Rs 3.37/-. At forty, Rs 5.75/-. At fifty, Rs 7.51/-. The overstatement of each month is itself compounded along with everything else, so the gap does not merely accumulate. The gap accelerates.

What do the fifty months give when both means are worked?

Every figure used so far is set out below, with the operation that produced it. Nothing has to be taken on trust. The last three rows read as one thought: the geometric mean is the rate that lands on Rs 120.81/-, and the arithmetic mean is the rate that lands somewhere the record never went.

One thing about this record is unusual and should not be passed over quietly. The rule behind these fifty months existed first and the months came out of it afterwards: five outcomes with five fixed weights, a generatorA rule for producing readings, set out first and then used. Its properties are facts about the rule itself rather than anything measured afterwards. in the plainest sense. The rule has a true average of 1.00 per cent a month and a true standard deviationThe usual measure of how far readings sit from their middle, on the same scale as the readings themselves. How it is computed is covered separately. of 5.00 per cent, not as measurements but as arithmetic on the rule itself. So something can be said here that no ordinary collection of data ever permits: the fifty month record's arithmetic mean of 0.50 per cent, used here as an estimatorA recipe for turning readings into a guess at some quantity that cannot be seen directly. How well one behaves, and how far its answer can sit from the truth, is covered separately. of that true average, is known to be exactly half the truth. The record is not lying, it is just short. The gap between a sample and the whole populationEvery reading the rule could ever produce, as against the handful actually drawn. Which one a figure describes, and how far apart the two can be, is covered separately. is a different subject, covered separately, but it is worth knowing that both means here are computed on a record that already sits below its own source.

How is it decided which mean a question needs?

Two questions settle it, and the second one is only a rewording of the first for people who find it easier to think in money.

One. Does each period act on what the period before it left behind? If yes, the question is about multiplying and the geometric mean answers it. Two. Is the quantity a rate of change applied to an amount that carries forward? If yes, same answer, for the same reason. Anything else is an adding question. Rent collected across twelve months, claims paid across a year, units sold across a quarter, or a single month pulled out of a record at random: all of these sit side by side and none of them builds on the last.

The reverse case matters just as much: for the expected size of one period taken on its own, or for any quantity being added rather than carried forward, the arithmetic mean is correct and the geometric mean is the wrong answer. The skill is not learning to prefer one mean over the other. The skill is learning to notice which of the two questions is actually being asked, and to notice it first.

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