Simple, Weighted and Exponential Moving Averages Compared

A simple moving average gives every month in its window the same weight. A weighted moving average gives the newest month the most weight along a straight line. An exponential moving average never drops a month at all and shrinks its weight instead. On the last twelve months of one invented record the three read Rs 196.8859/-, Rs 182.2555/- and Rs 186.7689/-, a spread of Rs 14.6304/- on the same day.

Two things are needed before any moving average can be built, and both are already in hand. The first is the arithmetic mean: add the numbers up, divide by how many there were. Nothing below needs anything harder than that. The second is the six year record of the Nakshatra unit, an invented monthly series built earlier. The record holds 72 dated months. The earliest ends on 31 January 2019, the latest ends on 31 December 2024, and the prices are all measured away from an opening markA value a record is anchored to, dated one step earlier than its earliest observation, so every later figure reads as a distance travelled from it. An opening mark is picked rather than measured, and picking a different one shifts every price while leaving every monthly change exactly where it was. of Rs 100.00/- fixed a day ahead of the first of them. The last month stands at Rs 187.4539/-. The record is a level seriesA record of what something stood at on each date, rather than a record of how much it changed between dates. Prices are a level series. The difference between reading one and reading the other is covered separately., which is to say it records what one unit stood at on each date rather than how much it moved between dates. Every figure below comes off the Nakshatra record and off nothing else.

The genuinely new idea is small, and it is not the arithmetic. Three people can average the same twelve numbers, all of them correctly, and produce three answers that are Rs 14.6304/- apart. The gap between them is not an error in anybody's sum: it is a weighting decision made before the sum started, and the question worth learning to ask is what that decision was.

What question is a moving average actually answering?

Consider first what it is not answering. A moving average does not report the price; the record already carries that. On 31 December 2024 the unit stood at Rs 187.4539/-, and no averaging was needed to establish it.

A moving average answers a different and slightly awkward question: where has this been sitting lately? The method takes a stretch of recent months, combines them into one number, and hands that number back as a description of the stretch. Then it does the same thing again next month with the stretch shifted forward by one. Recomputing month after month is the moving part of the name, and it is the whole of the mechanical idea.

Here is the everyday version. A tea stall outside one office gate keeps a small notebook. Every Saturday the owner adds up the last twelve weeks of takings and divides by twelve, and writes the answer at the bottom of that week's entry. The figure written at the bottom of the entry is not this week's takings. The figure describes a stretch of weeks, of which eleven are already finished and only one is fresh. The date on it is this Saturday, because that is when it was computed, and the period it describes is mostly behind.

The dating is a choice somebody made, and it is the entire reason every moving average sits behind the series it is drawn over. How far behind it sits is measured by counting steps backwards through the record, an operation called taking a lagHow far back in a record a step has been taken. A figure lagged by one month is that figure as it read a month earlier, lined up against today's row. How it is measured is covered separately., and that measurement is covered separately. Dating the figure to the middle of its window is a perfectly reasonable thing to do instead. Separating direction from noise takes that arrangement up separately. Dating it to the last month of the window instead has one enormous practical advantage: at the moment it is written down, every number that went into it was already known. Nothing has been borrowed from a month that has not happened. The habit of checking that no computed column has quietly reached forwards in time is called guarding against lookaheadLetting a figure that could not have been known yet leak into a row dated earlier. Lookahead is a dating mistake inside a computed column rather than a matter of judgement, and it is covered separately., and it is covered separately, but the dating convention used here exists because of it.

So the price of an honest date is that the answer is always a little behind. Notice what that means for the tea stall. If takings collapsed this week, eleven of the twelve weeks in the average are still the old, comfortable weeks. The average will slide, but it will slide gently, and it will keep sliding for another eleven Saturdays as the good weeks fall out one by one. The average is not being slow on purpose. The average is being exactly as slow as the width of the window it was told to use.

How does a simple moving average weight the months in its window?

Equally, and that single word is the whole definition. Add the last twelve prices, divide by twelve. Every month inside the window carries a weight of one in twelve, or 8.3333 per cent. Every month outside the window carries nothing at all.

Do it on the record. The twelve months ending December 2024 run from January 2024 at Rs 238.4049/- through to December 2024 at Rs 187.4539/-. The twelve prices add to Rs 2,362.6311/-, and dividing by twelve gives Rs 196.8859/-. Rs 196.8859/- is the simple twelve month average of the unit read at December 2024, and there is nothing else inside it.

Now the part most readers never notice, and it is worth slowing down for. Move forward one month and the average changes for two reasons, not one. A new month arrives at the right hand end of the window, and an old month falls off the left hand end. Both of them move the answer, and they move it by exactly the same amount per rupee.

Watch it happen on this record. At November 2024 the simple average read Rs 200.5532/-. At December 2024 it reads Rs 196.8859/-, so it fell by Rs 3.6673/-. December 2024's price of Rs 187.4539/- came in. December 2023's price of Rs 231.4611/- went out. The difference between them is minus Rs 44.0072/-. One twelfth of minus Rs 44.0072/- is minus Rs 3.6673/-, the entire move to the last paisa. A simple moving average can turn downwards because of something that happened a year ago, and nothing on the chart shows that this is what happened.

Equal weighting is not a flaw to be fixed. The effect follows unavoidably from giving every month in the window the same weight, and a month that carries 8.3333 per cent on the way in carries the same 8.3333 per cent on the way out. But it does mean that an explanation of a movement in a simple average which points at recent news should be checked against the other end of the window first. The paisaOne hundredth of a rupee, the smallest unit in the amounts written here. Four decimal places are shown so that sums can be checked against each other rather than because anything is measured that finely. that moved may have been leaving rather than arriving.

How does a weighted moving average differ from a simple one?

A weighted moving average uses the same twelve months and refuses to treat them as equals. The oldest month in the window takes a multiplier of 1. The one sitting after it takes a 2. The multipliers climb the window like that up to the newest month, and the newest month takes a multiplier of 12. The twelve products are then added up and divided by the total of the multipliers. The total is 1 plus 2 plus 3 and so on up to 12, or 78.

Run it on the same twelve months ending December 2024 and the answer is Rs 182.2555/-. The oldest month in the window, January 2024 at Rs 238.4049/-, now carries only 1 in 78, or 1.2821 per cent. The newest month, December 2024 at Rs 187.4539/-, carries 12 in 78, or 15.3846 per cent. The newest month has twelve times the pull of the month at the other end.

Look at what that does here. January 2024 was the highest price inside the window. Holding it back to 1.2821 per cent while pushing the recent, lower months forward is exactly why the weighted answer of Rs 182.2555/- comes out Rs 14.6304/- below the simple answer of Rs 196.8859/-. Neither method made a mistake. The two methods were asked to describe the same twelve months and were given different instructions about which ones matter.

The ramp of weights from 1 to 12 is a decision somebody made, not a property of the record. There is nothing in the data that says the newest month deserves twelve times the pull of the oldest. Somebody chose a straight line because a straight line is simple to state and simple to check, and any other shape would have been just as arithmetically legal and would have produced a different number. The choice of shape is the first hint that a number handed over depends on a decision nobody mentioned.

What does an exponential moving average do that neither of the others does?

An exponential moving average stops using a window at all. Both methods so far draw a boundary: twelve months in, everything else out. The exponential moving average draws no boundary. Instead it holds one running figure and, every month, nudges that figure a fixed fraction of the way towards the new price.

In words, and this is the whole rule: keep 0.846154 of the figure already held, and add 0.153846 of the new price. The two fractions add to exactly one, so nothing is created and nothing is lost. The fraction taken from the new price is called the smoothing factor. Here that factor is 0.153846, or two divided by thirteen.

Why two divided by thirteen? Because the setting here is a twelve month one, and the conventional way to turn a window length into a smoothing factor is to take two over that length with one added to it. Two over the window length plus one is a convention rather than a law, and it is a well chosen one for a reason worth seeing. The ages of the months inside a simple twelve month average run from 0 to 11 and every one carries the same weight, so the average age is 5.5 months. The average age of the exponential weights at a smoothing factor of two over thirteen is also 5.5 months, exactly. The conventional smoothing factor is the one that puts the exponential average's centre of gravity in the same place as the simple average's. The two methods lean on the same middle. The two methods just distribute themselves around that middle very differently.

Here is the everyday version of the rule. A household is trying to hold a picture of what it spends in a month. At the end of every month it does not throw out last year's picture and start again. The household takes the picture it already had and nudges it a little way towards what actually happened this month. A big month nudges it up. A quiet month nudges it down. Nothing is ever fully forgotten and nothing is ever fully believed. The nudging is an exponential moving average, and most people run one in their head without a smoothing factor anywhere in sight.

Now the structural difference, and it is the reason this method behaves unlike the other two. No month ever leaves. The price from four years ago is still inside the December 2024 figure. Forty eight steps behind the newest month, its weight is 0.0051 per cent. Small is not the same as zero. Compare that with the other two, where a month at 8.3333 per cent or at 1.2821 per cent one day is at exactly nothing the next day, the moment it falls past the edge of the window. The exponential average has no edge to fall past. Its weights fade; they never stop.

Count the fade on this setting. The newest month carries 15.3846 per cent. One month back carries 13.0178 per cent, two back 11.0150 per cent, eleven back 2.4492 per cent, twenty three back 0.3299 per cent, fifty nine back 0.0008 per cent. Add up everything inside the most recent twelve months and it comes to 86.5292 per cent. The remaining 13.4708 per cent sits further back than the window the other two methods use at all.

There is one small coincidence in that picture that is not a coincidence at all, and spotting it saves confusion later. The newest month carries 15.3846 per cent under the weighted average and 15.3846 per cent under the exponential average, the same figure to four decimal places. The match is arithmetic, not luck: 12 divided by 78 reduces to 2 divided by 13, the smoothing factor itself. Everything the two methods do behind the newest month is different, so they agree exactly on that month and still disagree by Rs 4.5134/- on the answer. The weighted average ramps steeply down and then stops dead at twelve months. The exponential average fades gently and carries on past the edge for ever.

Moving Average vs Exponential Moving Average: what actually separates them on one day?

Put the simple average and the exponential average side by side at December 2024 and the whole comparison sits in two numbers. The simple reads Rs 196.8859/-. The exponential reads Rs 186.7689/-. The gap between them is Rs 10.1170/-, on the same record, on the same day, at the same nominal twelve month setting.

The reason is one sentence long: the simple average is still carrying the first half of 2024 at full weight, and the exponential average has already shrunk it. Prices in the record ran from Rs 238.4049/- in January 2024 down to Rs 158.1089/- in October 2024. Under the simple average, January's Rs 238.4049/- carries the same 8.3333 per cent in December as December's own Rs 187.4539/- does. Under the exponential average, January 2024 is eleven months back and carries 2.4492 per cent, roughly a sixth of what the newest month carries. The high months are still in there. The high months have just stopped shouting.

Watch the exponential average actually make that journey through 2024. The exponential average starts the year at Rs 208.5704/-, its December 2023 value. January's Rs 238.4049/- is well above it, so the figure is pulled up by Rs 4.5899/-, or 0.153846 of the distance between them. Then the prices turn, and each month the figure is dragged down by a fraction of a widening gap: Rs 6.6944/- in July, Rs 7.8023/- in August. December's price of Rs 187.4539/- was almost exactly where the average already sat, so by December the figure has come to rest at Rs 186.7689/-, only Rs 0.1245/- above where November left it.

Neither figure is an error and neither is a correction of the other. The two averages answer slightly different questions. The simple average answers what the last twelve months averaged, treating them as a set. The exponential average answers where a running figure ends up if it is nudged towards every price the record has ever held, with the older nudges faded out. Two different questions have been asked, and there is no reason at all why two different questions should have the same answer.

The starting value, and the thing people get wrong about it

An exponential average is a rule for getting from last month's figure to this month's figure. Applied to the very first month, it has a problem: there is no last month's figure to start from. Something has to be put there by hand, and that something is the starting value. The exponential average here starts at Rs 101.2600/-, the simple average of the first twelve prices in the record, and every exponential figure below follows from that choice. Stating it is not a formality. The starting value is one of the three things needed before any exponential figure can be reproduced.

Now the part that is worth getting right. The obvious worry turns out to be the wrong worry. Two people who start their exponential average from different values might be expected to end up with different lines for ever. The two lines converge instead. Every step keeps 0.846154 of whatever came before it, and that includes whatever remains of the starting value, so any gap between two starting values shrinks by 0.846154 at every single step.

Put numbers on it. Suppose one person starts at Rs 101.2600/- and another starts 74 paise higher at Rs 102.0000/-, and both then roll the same sixty months of the record forward. After sixty steps, the 74 paise between them has become Rs 0.0000328/-. Both people read Rs 186.7689/- at December 2024. An exponential average forgets what it was started from, and the only question is how many steps it has had in which to forget.

Which points straight at the failure that does bite. The failure is not two different starting values. The failure is two columns of different lengths. Suppose one column carries sixty months of the record behind it and another column of the same method, with the same smoothing factor of 0.153846, was started inside 2024 at January's price of Rs 238.4049/- and has only had eleven steps since. The long column reads Rs 186.7689/- at December 2024. The short one reads Rs 190.7878/-, a gap of Rs 4.0189/-. After eleven steps, 15.9200 per cent of the short column is still nothing but its own starting value. After sixty steps the long column has 0.0044 per cent of its starting value left in it, and 0.0044 per cent is none.

So the honest way to say it is this. The starting value is not what bites. The number of steps rolled since the starting value is what bites, and a short column is still mostly made of the number somebody typed into the top of it. When two people cannot reconcile an exponential average, the first question is not what the column started from, it is how far back the column goes.

Where do all three sit on the same day?

Here they are, on 31 December 2024, on the same record, at the same twelve month setting, with the price beside them for comparison.

Same day, same record, same window length, same arithmetic, and the only thing that differs anywhere is a weighting decision. Rs 14.6304/- is not a small disagreement either. Against a price of Rs 187.4539/- it is 7.80 per cent of the thing being described. Anybody who hands over one of these three numbers and calls it the moving average has handed over one of three answers and dropped the label that would distinguish them.

The price is the interesting part of that picture. At Rs 187.4539/- it sits below the simple average, above the exponential average by Rs 0.6850/-, and above the weighted average by Rs 5.1984/-. Three true sentences about where the price sits, pointing in two directions, on one afternoon.

Does one method always sit highest?

No. A reader who has seen only December 2024 walks away with a rule that is false, and no ranking of the three methods survives a change in the months underneath it.

Go back exactly twelve months. At 31 December 2023 the same three averages on the same record read: simple Rs 207.4234/-, weighted Rs 214.3590/-, exponential Rs 208.5704/-. The weighted average is the highest of the three. Twelve months later it is the lowest of the three. Nothing about the method changed. Nobody adjusted a weight or a smoothing factor. The stretch of months underneath it changed, and that was enough.

Why did it move? Because the weighted average leans hardest on recent months. The centre of gravity of each method gives the answer: the simple average's average month is 5.5 months old, the exponential average's is also 5.5 months old, and the weighted average's is 3.6667 months old. The weighted average is the most recent of the three by a clear margin. Through 2023 prices were rising hard into a peak of Rs 238.6196/- in November, so leaning on recent months pushed the weighted figure to the top. Through 2024 prices fell from Rs 238.4049/- in January to Rs 158.1089/- in October, so leaning on the same recent months dropped it to the bottom.

Reacting fastest means leading in both directions, and it never means being right. A method that moves first onto whatever is happening now will be the highest of the three while prices climb and the lowest while they fall, and both of those are the same property showing itself twice. The ranking of the levels is a fact about the stretch of months. The ranking of the responsiveness is a fact about the methods. Keep those two apart and the picture stops being confusing.

Across the whole record the three lines cross each other repeatedly rather than holding station. On this six year record, at a twelve month setting, the weighted average and the simple average change places four times, the simple and the exponential nine times, and the weighted and the exponential twice. The three lines are not parallel with a constant gap between them. The three lines are different descriptions of the same walk, pulling apart and closing up as the walk changes shape.

How should a price sitting above or below its average be read?

Carefully, and with three labels attached. Without them the sentence is not yet a measurement. Here is the practitioner version of that, and it has nothing to do with anybody buying anything.

A lender is looking at a small printing shop that wants a working capital facility. The shop's monthly takings jump around: a wedding season month is enormous, a monsoon month is thin. Nobody can assess that shop from one month, so the officer computes a moving average of takings and asks whether the latest month sits above it or below it. Comparing the latest month against a moving average of takings is a perfectly sensible thing to do, and it is exactly the operation set out above.

Now suppose two officers in the same lending team assess the shop in the same week. One runs a simple twelve month average. The other runs an exponential one at the conventional smoothing factor. On the Nakshatra unit's numbers, one of them would report a latest reading Rs 9.4320/- below the average and the other a latest reading Rs 0.6850/- above it. One writes that the shop is running below its recent run rate. The other writes that it is running slightly above. Both officers did correct arithmetic, so the disagreement between their opposite sentences cannot be settled by rechecking anybody's sums. That is the worst kind of disagreement to have inside a credit file, because there is nothing to correct.

The habit that prevents the disagreement runs to three fields. Whenever a price is quoted against an average, record the window length, the weighting scheme, and, if the scheme is exponential, what the column was started from and how many periods ago. A moving average figure without those three attached is an opinion wearing the clothes of a measurement. With them attached, anybody can rebuild it, and two people who disagree can find out which of the three fields they differ on in about a minute.

One more thing the officer should hold on to, and it is the honest limit of the whole exercise. A moving average here smooths a record and does nothing else. A moving average does not say what the shop will take next month, it does not say whether the shop is well run, and it is not a verdict. The method compresses a stretch of months into one number so that a stretch can be talked about, and the compression throws away information on purpose. The trendWhere a record is heading over its whole length once the month to month jumping about has been set aside. Prising the two apart needs machinery of its own and is covered separately. underneath a record, if it has one, is a separate question with separate machinery, and is covered separately.