Mean, Median and Mode: Which One to Trust and When

The arithmetic mean adds every value and divides by the count. The median is the middle value. The mode is the commonest. On the fifty month record the arithmetic mean is 0.50 per cent while the median and the mode both sit at 1.00 per cent, because a few bad months drag an average down and cannot move a middle. Compounding needs a fourth figure, the geometric mean, at 0.3788 per cent.

Three numbers, all of them called the centre, and on the very same fifty months they do not agree. The disagreement is not a defect in any of them. Each one answers a different question, and the whole skill is knowing which question is being asked before the choice is made. Each of the three is defined below in full, then set against the others on this record to show why they part company and in which direction, and a fourth centre follows that none of the three can stand in for.

One thing is already built. A generatorThe written down rule that produces the values, with every value it can take and the weight sitting on each. Because it is written down first, its properties are known rather than estimated. called the Nakshatra unit was defined earlier: an invented traded unitA thing carrying a price that moves. Since this one exists only in these notes, the phrase simply marks the figures beside it as price movements rather than temperatures or scores. whose monthly changeHow much the price moved over one month, written as a percentage of where it started that month. Fifty of them make the record used here. takes one of five values, minus 9.00, minus 4.00, 1.00, 6.00 and 11.00 per cent, with weights of 0.08, 0.18, 0.48, 0.18 and 0.08. Fifty months were then drawn from it, and that draw is the fifty month record. Its tallyFive numbers standing in for fifty, each one saying how often its value came up. Nothing is lost in the swap as long as the order the fifty arrived in does not matter to what is being computed. is 5, 9, 25, 8 and 3 months on those five values, adding to fifty. The tally is the only input below, and every centre that follows is worked out of those five counts.

And here is the advantage almost no real dataset has: the truth is known. The rule existed on paper first and the fifty months were pulled out of it afterwards, so the rule's own centre sits at 1.00 per cent and its own spread at 5.00 per cent as facts rather than as estimates. Every centre computed below can therefore be laid beside the answer it was reaching for, and precisely how far short it fell becomes visible. A known truth is what turns the disagreements between the four centres from interesting into checkable.

A word on the numbers before they start arriving. The five values and their weights were chosen so the working could be followed by hand, the counts of 5, 9, 25, 8 and 3 were chosen the same way, and everything below is those counts put through a calculator. Every percentage below is a method demonstrated, and every rupee amount is the same method demonstrated in money.

What is the Arithmetic Mean, and what question does it actually answer?

Add up everything, divide by how many things there were. The definition is the whole of it, and it has no hidden clauses. Fifty monthly changes, added together, divided by fifty.

Doing that fifty times over would be tedious, so the tally does the work. Five months came in at minus 9.00 per cent, which contributes minus 45.00 to the total. Nine months at minus 4.00 per cent contribute minus 36.00. Twenty five months at 1.00 per cent contribute 25.00. Eight months at 6.00 per cent contribute 48.00. Three months at 11.00 per cent contribute 33.00. The five products, minus 45, minus 36, 25, 48 and 33, come to 25.00. Divided by fifty months, that is 0.50 per cent.

Every one of the fifty months gets a vote, and the size of each vote is the size of the month. That sentence is the arithmetic mean in full. A month at 11.00 per cent pushes eleven times as hard as a month at 1.00 per cent, and a month at minus 9.00 per cent pulls nine units in the other direction. Nothing is capped, nothing is trimmed, and nothing is ignored. The arithmetic mean is the most complete of the three centres in the sense that it uses every last drop of information in the record, and that completeness is exactly what makes it vulnerable.

The everyday version helps. A household of five people pools its earnings into one pot at the end of each month and divides the pot by five to see what each person effectively brought in. Nobody in that household is asking who is typical. The household is asking what the pot came to, shared out. The arithmetic mean answers the pot question, and only the pot question.

What is the median, and what does it answer?

The fifty months line up in order, smallest on the left, largest on the right, and the count walks to the middle. Whatever value sits at that middle is the median. No multiplication, no adding, no division. Just counting to the middle of a queue.

Use the tally as a running count. The first 5 months are the ones at minus 9.00 per cent, so they take positions 1 to 5. The next 9 are at minus 4.00 per cent, taking positions 6 to 14. The next 25 are all at 1.00 per cent, and they take positions 15 to 39. Then 8 at 6.00 per cent take positions 40 to 47, and 3 at 11.00 per cent take positions 48 to 50. The running count is therefore 5, 14, 39, 47 and 50.

Fifty is an even number, so there is no single month in the middle. The convention is to take the two months either side of the middle, at positions 25 and 26, and average them. Look at the running count: positions 15 to 39 are the long run at 1.00 per cent, and 25 and 26 both fall inside it. Both are 1.00 per cent, so their average is 1.00 per cent, and that is the median.

The median counts months rather than weighing them, and that single difference is the whole of its character. A month at minus 9.00 per cent and a month at minus 40.00 per cent occupy exactly one place each in the queue. To the median they are identical: both below the middle, and how far below is information it deliberately throws away. Take one of the five months at minus 9.00 per cent and make it minus 40.00 per cent instead. The five products become minus 40, minus 36, minus 36, 25, 48 and 33 once that month is split out, totalling minus 6.00, so the arithmetic mean falls from 0.50 per cent to minus 0.12 per cent and changes sign. The median does not twitch. The median is still 1.00 per cent. That month is still exactly one place in the queue, and still on the same side of the middle.

What is the mode, and when is it the only honest answer?

The mode is the value that turned up most often. On the fifty month record, 25 months came in at 1.00 per cent and no other value came close, so the mode is 1.00 per cent. The mode is read straight off the tally by finding the biggest count. There is no arithmetic at all.

What is the mode for? The mode answers what a typical month looked like, not what the months averaged to, and where a record has a real cluster in it, no other centre can promise to name a reading that actually turned up again and again. That last clause matters. Only five outcomes are available to the rule, and 0.50 is not one of them, so the arithmetic mean of 0.50 per cent is a reading that never once turned up in the fifty months and never could have. The median of 1.00 per cent at least happened. The mode of 1.00 per cent happened twenty five times.

Think of a vegetable seller asked what a bunch of coriander costs at her stall. She could add up the day's takings and divide by the number of bunches, and get some figure with paise in it that she has never charged anybody. Or she could say Rs 10/-, the price she charged forty of the fifty customers. The second answer is the mode, and it answers the question the customer asked far better.

The mode also has a failure the other two do not. The mode can be missing, and it can be doubled. If every value in a record turned up exactly once, there is no mode at all. If two values tie for the biggest count, there are two modes, and neither of them is the centre in any useful sense. Both happen in the panel below. At one particular setting the record has two modes sitting eleven percentage points apart, and a doubled mode signals that the question the mode answers has stopped having a single answer.

Mean vs Median: why do the three disagree here, and in which direction?

Lay the three side by side. The arithmetic mean is 0.50 per cent. The median is 1.00 per cent. The mode is 1.00 per cent. So the median and the mode sit above the arithmetic mean, by half a percentage point, on identical data.

The direction is not random and it is not a quirk of this particular record. The direction follows from the shape of the tally. Look at where the months are. Twenty five of them, half the record, sit at 1.00 per cent. Eleven sit above that, at 6.00 and 11.00 per cent. Fourteen sit below it, at minus 4.00 and minus 9.00 per cent. So there are more months below the cluster than above it, and the ones below reach further away: minus 9.00 per cent is ten points below the cluster while 11.00 per cent is only ten points above, but there are five months at the bottom against three at the top, and nine at minus 4.00 against eight at 6.00.

The middle registers only that five months at minus 9.00 per cent sit below it, never how far below, so those five pull hard on a total and cannot shift a middle. That is the mechanism, stated without metaphor. The arithmetic mean is a balance point: every month applies a force equal to its distance from the balance, so a month sitting 9.50 points to the left of 0.50 per cent applies nine and a half units of pull. A month sitting 0.50 points to the right applies half a unit. The median is a headcount: it moves only when the number of months on one side of it changes, and a month down 9.00 per cent supplies one head, exactly as a month down a hundredth of one per cent supplies one head.

The everyday version is a small mall. Nine shops each take Rs 20,000/- on a given day. The tenth has a bad day and loses Rs 40,000/-. The average day across the ten shops has collapsed to Rs 14,000/-, and it fell because the loss was deep, not because it was common. The typical day, the middle of the ten, is still Rs 20,000/-, and it stayed there because the tenth shop is still just one shop. A landlord planning rent wants the middle. A landlord working out total rent receipts wants the average. Neither is wrong and they must not be swapped.

Now the part that must not be skipped. The rule that produced these months has a centre of 1.00 per cent, and both the median and the mode landed dead on it while the arithmetic mean landed at half of it. Reading the coincidence as proof that the median is the better estimator is very tempting. The coincidence proves nothing, and it belongs to this particular fifty months. The arithmetic mean is the estimator that targets the generator's true centre; over many records it lands on 1.00 per cent on average, and this record simply happened to draw an unlucky handful of bad months. The median targets the generator's middle value. For this generator that middle value also happens to be 1.00 per cent, so the two targets coincided here, and for a lopsided generator they would not. Draw a different fifty months and the ranking can flip. How far a centre computed from fifty months can sit from the truth, and how a range would be put around it, is covered separately.

What is the Geometric Mean, and why does compounding need it?

First, the word. Compounding means each month's change applies to whatever the previous months left behind, not to the amount put in at the start. Put Rs 100/- in. A month of 10.00 per cent leaves Rs 110/-. The next month's change is then a percentage of Rs 110/-, not of Rs 100/-. Months multiply rather than add.

Now ask a question the arithmetic mean is not built to answer. What single steady monthly rate, repeated fifty times, would have carried Rs 100/- through the fifty month record to exactly where it actually ended? The question has one right answer, and the geometric mean is that answer.

Work it. Turn each monthly change into a factor: minus 9.00 per cent becomes 0.91, minus 4.00 becomes 0.96, 1.00 becomes 1.01, 6.00 becomes 1.06 and 11.00 becomes 1.11. Multiply all fifty factors together. On the tally that means 0.91 to the power 5, times 0.96 to the power 9, times 1.01 to the power 25, times 1.06 to the power 8, times 1.11 to the power 3. The product is 1.2080871. Take the fiftieth root of the product, giving 1.0037879, and subtract one. The geometric mean is 0.3788 per cent a month.

A build that cannot be reversed is a build not worth trusting, so the check runs the other way round. Raising 1.0037879 to the fiftieth power gives 1.2080871 again, exactly the product it came from. Fifty months at a steady 0.3788 per cent lands in precisely the same place as the fifty actual months did.

The geometric mean is not a different opinion about the same quantity, it is the answer to a different question, and the question is the one anybody compounding money is asking. The arithmetic mean answers what the fifty monthly changes add up to, shared out. The geometric mean answers what they multiply out to, shared out. Where nothing is compounded, the second question does not arise. The moment a rupee is carried forward through more than one period, it is the only question there is.

What does choosing the wrong mean cost, in rupees?

A gap of about a tenth of a percentage point a month sounds like nothing, so put it in money.

Rs 100/- carried through the actual fifty months, month by month, at the changes the record actually contains, ends at Rs 120.81/-. The same Rs 100/- compounded fifty times at the arithmetic mean of 0.50 per cent ends at Rs 128.32/-. The overstatement is Rs 7.51/- on a starting Rs 100/-, a little over six per cent of the starting amount, produced by nothing more than quoting one correct number in place of another correct number.

The Rs 7.51/- gap is not an approximation error and it is not rounding. Nobody made an arithmetic mistake anywhere. The arithmetic mean of 0.50 per cent is exactly right as an answer to the adding question, and the reader who compounded it was asking the multiplying question. Two correct answers to two different questions were swapped, and the swap costs Rs 7.51/-. The swap costs more the longer the run gets, and more still the more the record bounces around, for the reason set out under the gap between the two means.

One caution on the picture below. Multiplication does not care about order, so the ending value of Rs 120.81/- is the same whatever order the fifty months are put in. The jagged line is drawn in one stated order, chosen by dealing the five values round in a fixed rotation until the counts run out, purely so there is a line to look at. Change the order and the line changes shape while both endpoints stay exactly where they are.

Where does the gap between the two means come from?

The gap is not arbitrary and it is not a property of this record's personality. The gap has a size that can be predicted before computing it, and the ingredient that predicts it is how much the record bounces around.

The squared distance of each value from the arithmetic mean of 0.50 per cent, weighted by its count, summed and divided by fifty, gives the figure that follows. Five months are 9.50 away, squaring to 90.25 each. Nine months are 4.50 away, squaring to 20.25. Twenty five months are 0.50 away, squaring to 0.25. Eight months are 5.50 away, squaring to 30.25. Three months are 10.50 away, squaring to 110.25. Weighted, those come to 451.25, 182.25, 6.25, 242.00 and 330.75, adding to 1212.50. Divided by fifty, that is 24.25. The 24.25 is the record's varianceThe average of the squared distances from the centre. Its units are the square of the original units, so it cannot be read on the same scale as the values themselves. Variance is covered in full separately., and its units are per cent squared rather than per cent, which is why it reads as a large number sitting beside a small one.

Halve it. Half of 24.25 is 12.125, and once that is put back on the per cent scale it is 0.12125 per cent. Now measure the gap directly: the arithmetic mean of 0.50 per cent less the geometric mean of 0.3788 per cent is 0.1212 per cent. The gap and half the variance agree to three decimal places, and that is not a coincidence of this record.

Why do they match? Because turning a percentage change into something that can be added requires a small correction, and the correction is a squared term. For a small change, the compounding-friendly version of a change is very slightly less than the change itself, short by about half the change squared. Averaged over the record, the shortfall is about half the average squared distance, and half the average squared distance is half the variance. So the geometric mean sits below the arithmetic mean by roughly half the variance, always, and the approximation gets better the smaller the monthly moves are. Here the rule predicts 0.50 less 0.12125, or 0.37875 per cent, against a true geometric mean of 0.3788 per cent.

The consequence is worth stating in plain words. A calm record, one whose months cluster tightly, has almost no gap between its two means, and quoting the wrong one costs almost nothing. A wild record has a wide gap, and quoting the wrong one is expensive. The penalty for using the arithmetic mean where the geometric one was the right choice grows with the bounce in the data, not with the level of it. Spread itself, how it is measured, why there are two denominators for it and what the standard deviationThe square root of the variance, taken so the answer comes back onto the same scale as the values themselves. The standard deviation is covered in full separately. adds, is covered separately and nothing more is taught about it here.

What do all four centres say about the same fifty months?

Every centre above comes off one tally, so the whole build sits in one place below, with the true centre of the generator beside it to show which landed where.

Read together, the two tables make one thing stand out. Three of the four centres are correct answers to three different questions, and the fourth, the mode, is a correct answer to a fourth. Not one of them is wrong. The only wrong move available is to take an answer computed for one question and hand it to somebody asking another. The Rs 7.51/- is exactly what that move costs. The bottom row of the first table can exist for one reason only: somebody wrote the five values and their weights on paper, and only afterwards drew fifty months out of them, so the 1.00 per cent in that row is a fact about the rule rather than a guess about it. Given fifty real months instead, that row is simply missing. The habit of naming which centre was quoted therefore matters more than any individual reading.

How does anybody choose between them without guessing?

Four questions, asked in order, and none of them requires a rule of thumb. The four questions are the part a lender, an analyst or a household actually performs, and they take under a minute.

One. Am I adding these or compounding them? Adding covers total rent collected, total claims paid, total units sold in a year. Compounding covers anything carried forward through periods: an amount growing, a price rolling on, a loan balance. If the answer is compounding, the geometric mean is the only correct centre and everything else on this list is secondary.

Two. Is the record lopsided, and which way? The mean and the median, computed together, show which of the two sits higher. Mean below median means a tail of bad cases is doing the pulling, and that is the fifty month record's situation. Mean above median means a few very large cases are doing it. The comparison costs one extra calculation and settles whether the choice is going to matter at all.

Three. Is one case carrying the answer? Drop the largest observation and recompute. A household budgeting on an average monthly income that is being held up by one bonus month is budgeting on a number that will not arrive again. A lender averaging ten borrowers where one is enormous is not describing the other nine. If the mean moves a lot when one case leaves, quote the median beside it.

Four, and it is the one that decides the sentence written. Would the conclusion survive if the other centre were quoted instead? Swapped in, the sentence is read again. If the conclusion holds either way, the choice of centre was never doing any work and whichever is conventional will serve. If the conclusion flips, the choice of centre was carrying the argument, and it belongs in the sentence itself rather than in a footnote nobody reaches. On this record, saying the unit averaged 0.50 per cent a month and saying it typically returned 1.00 per cent a month are both true, and they lead a reader to different places.