Frequency: Daily, Weekly, Monthly and What Changes Between Them

Frequency is how often a series is observed, and changing it changes almost everything about the figures. Aggregating the Nakshatra unit's six year record from 72 monthly observations to 24 quarterly ones and then 6 yearly ones takes the mean from 1.00 to 3.2199 to 12.3657 per cent and the spread from 5.00 to 10.6652 to 16.3527 per cent. The record did not change.

Two things are taken as settled here and neither is rebuilt. The first is the mean and the spread, worked out earlier and used without alteration. The second is compoundingApplying one change on top of another, so that the second one acts on a base the first has already moved. Two rises of ten per cent do not make twenty., in the plain sense that two changes applied one after the other combine by multiplying rather than by adding. Everything else below is arithmetic worked in the open.

What does the frequency of a series mean, and what fixes it?

Frequency answers exactly one question: how often is this thing written down? The Nakshatra unit's six year record answers it once a month. There are 72 observations in it, running from the month that ended on 31 January 2019 to the month that ended on 31 December 2024, measured out from an opening markA starting value placed just ahead of a record's first observation. It gives the first movement something to be measured against, and is not itself counted as an observation. of Rs 100.00/- dated 31 December 2018. Monthly is its frequency, and nothing done to the record afterwards alters that.

Picture a tea stall outside an office gate. The stall writes a slip for every cup it sells, drops the slips into a tin, and empties the tin at closing time. Its record is daily. Want a weekly figure? Add seven days. Want a monthly one? Add thirty. The daily slips are sitting underneath all of those totals, so every one of them is available for the asking. Now suppose the stall keeps only the weekly totals and burns the slips at the end of each week. Tuesday has gone. The detail was destroyed and not merely folded away. No arithmetic anybody has ever written will bring Tuesday back.

A record can always be aggregated to a lower frequency and can never be split into a higher one, so the frequency a record arrives at is a ceiling rather than a starting point. That single sentence settles a surprising number of arguments before they begin. If someone asks for a daily figure and the record arrives monthly, the honest answer is that the daily figure was never recorded and cannot be manufactured, however reasonable the request sounds. If someone asks for a yearly figure and the record arrives monthly, the yearly figure is available at any time, and everything below is an account of what making it does to the numbers.

What happens to the count when the frequency falls?

Aggregating a record comes down to two decisions: how long a block is, and how the observations inside a block are combined. Blocks of three months turn the six year record's 72 observations into 24 quarterly ones. Blocks of twelve turn them into 6 yearly ones. Nothing is discarded and nothing is counted twice. Every one of the 72 months sits inside exactly one block. Only block lengths that divide 72 exactly are used anywhere in this guide.

The arithmetic is trivial and what follows from it is not. Say those three counts out loud. Seventy two. Twenty four. Six. The count falls by the same factor that the block lengthens by. The fall is far faster than most people picture when they agree to move a record to a lower frequency. Going from monthly to yearly sounds like a change of presentation. The change throws away nine tenths of the rows and then most of what is left.

The yearly record leaves little, and the little it leaves is quickly stated. Every figure computed on the yearly record rests on six numbers, and six numbers cannot support very much at all. Here they are in full: the six year record's yearly changes are 3.8112, 18.8496, 14.1452, 28.2317 and 28.1689 per cent, and then minus 19.0128 per cent in the final year. One year in six fell. The entire yearly record fits in the head at once, pleasantly enough. Any statement of the form "how often does a year fall" is then being answered by a count of one.

The everyday version is a caterer who works six weddings a year. Ask how variable the takings are and there are six numbers to look at, one bad monsoon can be half the story, and a seventh wedding would move every figure. Ask the same question about the 72 individual meals served across those six years and the answer rests on something far sturdier. The caterer did not become more or less variable. Only the counting changed.

Why is the yearly mean not simply twelve times the monthly one?

Here are the three means, computed on the same record at the three frequencies. Monthly, 1.00 per cent. Quarterly, 3.2199 per cent. Yearly, 12.3657 per cent. Three times 1.00 is 3.00 and the quarterly figure is higher than that. Twelve times 1.00 is 12.00 and the yearly figure is higher than that too. The quarterly mean is more than three times the monthly one and the yearly mean is more than twelve times it, and the reason is that changes applied one after another combine by multiplying rather than by adding.

Feel it on a stall before doing it on the record. Takings rise a tenth in April, then a tenth again in May. May's tenth is charged on a base April already lifted, so the stall has not risen a fifth over the two months. The takings have risen by twenty one hundredths of where they started. Small over two months. Not small over seventy two.

The same arithmetic on three real months of the record makes the point. July, August and September 2023 changed by 6.00, 8.00 and 6.00 per cent. Adding them gives 20.00 per cent. The three months written as growth factors are 1.06, 1.08 and 1.06, and multiplying them together gives 1.213488, a quarterly change of 21.3488 per cent. The addition is short by 1.3488 per cent on one quarter alone, and there are twenty four quarters.

One honest qualification. A reader who takes away a multiplier will misuse it. If every month were exactly 1.00 per cent, twelve of them would compound to 12.6825 per cent. The record's yearly mean is 12.3657 per cent. The yearly mean sits above twelve times the monthly figure and below the steady compounding figure. Compounding sets the direction with certainty but supplies no exact conversion. The months are not all equal to each other, and the arithmetic is sensitive to how they are spread. The direction and the reason carry forward, never a factor to be reused on the next record. There is also a way of writing changes so that they do add up cleanly across periods, called a log differenceA way of recording a movement: take the natural logarithm of where a period ended over where it began. Useful because consecutive ones can simply be added up., and it is covered separately.

What happens to the spread when the frequency falls?

The spread grows as well, and this is where aggregation earns its keep. Monthly, the spread of the six year record's changes is 5.00 per cent. Quarterly, it is 10.6652 per cent. Yearly, it is 16.3527 per cent. All three divide by how many observations there are, not by one less than that. Six whole years are being described in full rather than sampled to guess at something wider.

Set the two growths beside each other and the point falls out. Going from monthly to yearly multiplies the mean by 12.3657 and multiplies the spread by only 3.2705. The spread grows, and grows more slowly than the mean. The gap between the two growths is the entire reason anybody aggregates a record. A monthly figure of a 1.00 per cent centre against a 5.00 per cent spread looks like noise with a rumour of direction inside it. A yearly figure of 12.3657 against 16.3527 is still wide, but the direction is now large enough to be worth discussing.

The household version: an electricity bill jumps around from month to month for reasons that feel random, one hot week here, guests staying there. Compared across two whole years, the month to month wobble has partly cancelled itself out while the underlying difference in how much the household uses has not. Nothing about the household changed between the two comparisons. Only the length of the block did.

Where does the square root of time rule miss on this record?

The shortcut in that question has a name, the square root of time rule, and it is everywhere. The rule says that to move a spread from a short period to a long one, multiply by the square root of how many short periods fit inside the long one. Applied to the six year record's monthly spread of 5.00 per cent, it predicts 5.00 times the square root of three, or 8.6603 per cent, for a quarter, and 5.00 times the square root of twelve, or 17.3205 per cent, for a year.

The record itself gives 10.6652 per cent for a quarter and 16.3527 per cent for a year. On one record, at one moment, the rule reads 2.0050 per cent too low at a quarter and 0.9678 per cent too high at a year. Both misses are computed results rather than cautions. Both predictions and both actual figures come out of the same 72 numbers.

The rule rests on two assumptions and this record breaks both, in opposite directions. The first assumption is that one period says nothing whatever about the next. Consecutive months in the six year record do lean on each other. The leaning is a property built and measured separately under the name autocorrelationA reading of how much a series resembles itself at an earlier position. High means an observation carries information about the one that follows it.. When neighbouring months tend to move the same way, three of them stacked together come out wider than three unrelated months would. The leaning pushes the quarterly spread above the prediction.

The second assumption is that there is no pattern repeating on a fixed period. The six year record has such a pattern and it is large. Once a block is a full twelve months long, that pattern is present in its entirety inside every single block, so it stops making the blocks differ from one another and starts cancelling inside each of them instead. The cancelling pulls the yearly spread back below the prediction. Two effects, pulling opposite ways, biting at different block lengths. A rule that reads low in one place and high in another on the same record cannot be repaired with a correction factor. There is no single direction to correct in.

What does aggregation look like when the block length moves?

Reading that the spread grows more slowly than the mean is one thing. Watching the bars merge, the count collapse and the gap between the rule and the record change sides is another. The panel below holds the identical 72 monthly changes at every setting. The single control cuts them into blocks of the chosen length, combines each block by multiplying growth factors, and recomputes all three readings from scratch. The faint bars behind the solid ones are the original months, left on screen to show exactly what got folded into what.

Two things stand out in particular. The reading marked as held does not move at any setting. Whatever length the record is cut into, multiplying all the blocks back together returns the same six years. And the gap between the rule and the record starts at nothing, opens on one side, then crosses over and opens on the other.

What happens to a repeating calendar pattern as the frequency falls?

The six year record carries a pattern that repeats every twelve months, and there is a direct way to see it. Average all six Januaries together, then all six Februaries, and so on down the calendar. The twelve figures come out as 5.00, 3.00, 1.00, minus 1.00, minus 2.00, minus 3.00, minus 2.00, minus 1.00, 1.00, 3.00, 4.00 and 4.00 per cent. The twelve figures make a shape rather than a scatter: strong at the turn of the year, sinking through the middle of it, climbing back through the autumn.

Now group the same 72 months by calendar quarter and average the monthly changes inside each group. January to March averages 3.00 per cent, April to June averages minus 2.00 per cent, July to September averages minus 0.6667 per cent and October to December averages 3.6667 per cent. The shape is still there and it is flatter. The monthly figures swing across 8.00 per cent from the highest to the lowest. Averaging three neighbouring months together partly cancels them against each other, and the quarterly figures swing across only 5.6667 per cent.

Go to yearly and it is gone completely. Not faint, not weak, gone. The twelve calendar values cancel to exactly zero across any twelve consecutive months, so each yearly block already contains one whole turn of the cycle and the pattern can no longer make one year differ from another on that account. Aggregation does not remove a calendar pattern, it hides it, and the frequency at which it disappears is exactly the length of the cycle.

Two warnings before leaving this. The first is that hiding is not removing. Taking a recurrence out of a record on purpose, so that whatever is left can be read free of it, is a seasonal adjustmentReworking a record on purpose so that whatever recurs on the same point of the calendar each time is taken out, and what is left can be read on its own.. Seasonal adjustment is a different operation with different consequences, covered separately. Aggregating to yearly hid this pattern by accident, and hiding a pattern by accident is not the same as having dealt with it.

The second warning matters more. Do not test whether a record has a repeating pattern by measuring how much a month resembles the month a lagA position counted backwards in a record. A lag of one is the observation immediately before; a lag of twelve is the observation twelve places before. of twelve earlier. On this record that reading comes out close to nothing while the calendar pattern printed above is large, real and recoverable exactly. The calendar averages ask the question directly instead of asking a related one and hoping. The calendar averages are the test used here, and the test worth trusting.

What changes in the other direction, towards weekly and daily?

One point of fact governs what follows. The six year record is monthly, no daily or weekly version of it exists, and one is not going to be invented here to fill the gap. A record cannot be split upwards, so what follows describes what changes at a higher frequency in general terms, without figures.

More observations cover the same span, the obvious half. The less obvious half is that each one carries a smaller spread, and turning that small spread back into a figure for a longer period is exactly the operation the square root of time rule was invented for and exactly the operation it has just been seen to miss, twice, in opposite directions.

Inside a short window there is often nothing much to record, so more readings sit at or very near nothing. A record full of near zeros behaves differently from one where every observation carries something.

A much larger share of the record is made of how it was collected rather than what happened. At a monthly frequency, small mechanical details wash out. At a daily one they dominate: which reading of the day was taken, what happens on a closure or a holiday, whether a week is five days or seven, and above all what the marker attached to each observation actually claims. The marker on each observation, a timestampThe date or date and time attached to an observation. What it claims can vary: the start of a period, the end of it, or the moment a value was recorded, and the three are not interchangeable., can mean the start of a period, the end of a period, or the moment somebody wrote it down, and mixing the three quietly ruins a record. Timestamps are covered separately, and timestamps are where most daily data actually goes wrong, not in the arithmetic but in the dating.

And there is more work. Recomputing a figure again and again as time moves forward, across a rolling windowA stretch of set length that steps forward through a record one observation at a time, with the figure worked out afresh wherever it stops., is a modest job on 72 monthly observations and a heavy one on thousands of daily ones. Rolling windows are covered separately too.

How does the whole record read at all three frequencies at once?

Everything above, in one place. Read the three rows as one record described three ways, not as three records.

The monthly mean and spread are exactly 1.00 and 5.00 per cent, printed to four places above only so the rows line up with the panel. The last row settles the whole table: cut the record into blocks of any permitted length, multiply all the blocks back together, and the six years still come to 87.4539 per cent, carrying Rs 100.00/- to Rs 187.4539/-. Every figure in the three rows above moved and the record underneath them did not.

One line to carry away, if only one: aggregating the six year record from 72 monthly observations to 6 yearly ones multiplies the mean by 12.3657 and the spread by only 3.2705, empties nine tenths of the rows, hides a calendar pattern that is still fully present, and leaves the six years themselves at exactly 87.4539 per cent. Every one of those changes came from the block length and none of them came from the record.