What a Timeframe Is

A bar is not something the market produced. It is a rule for grouping the tape, and the rule has two settings: how wide each group is, and where the first one starts. Only the first is ever quoted. Regroup the sixty bars of the last four lessons four at a time and the sample high and low do not move at all, while the swing count falls from fifteen to two and lesson 36’s ratio moves from 0.082 to 0.157. Then leave the width alone and move only the starting bar, and that same ratio runs from 0.104 to 0.249 on identical tape.

Prerequisites: Lesson 36, whose ratio is one of the two quantities re-measured here, and lesson 32, whose swing rule is the other: both were quoted in bars, and this lesson asks what a bar was.

The same sixty bars at six intervals

The grouping arithmetic is worth doing once by hand. Take bars 1 to 4. Their highs are 100.8, 101.6, 101.1 and 102.9, so the group high is 102.9. Their lows are 100.0, 100.7, 100.2 and 101.5, so the group low is 100.0. The group closes where the last of the four closed, at 102.8. One bar, spanning 2.9, in place of four bars spanning 0.8, 0.9, 0.9 and 1.4. Repeat fifteen times and the sixty-bar chart is a fifteen-bar chart.

Below, that done at every width from one to six. The swing column applies lesson 32’s rule — a high with two lower highs either side, a low with two higher lows either side — to the regrouped bars. The ratio column is lesson 36’s: net displacement over the whole sample divided by the total path walked between consecutive closes.

Bars grouped	Bars on the chart	Mean bar range	Swing points	Efficiency ratio
1	60	1.35	15	0.082
2	30	2.02	4	0.161
3	20	2.54	2	0.245
4	15	2.89	2	0.157
5	12	3.33	2	0.312
6	10	3.71	1	0.140

There is a column missing from that table, and it is missing because it would have been the same number six times. The sample high is 107.4 in every row and the sample low is 97.7 in every row. Not approximately: identically, to the tenth, at every width. The whole of the movement is still there. What changed is every description of it.

The swing count is the one that should be uncomfortable. Lesson 32 found fifteen swing points on this series and spent a lesson on the fact that the number depended on how many bars you required either side. It depends at least as much on something lesson 32 never mentioned: at a width of six, the same rule on the same tape finds one. Fourteen of the fifteen structural points a trader would have marked, argued about and placed stops against are not on the chart at all. They were never in the tape; they were in the grouping. Run the same rule on the closes alone, which are the part of this series that was published rather than generated, and the count goes seventeen, eight, four, two, one, one. The same collapse.

The ratio is worse, because it does not even fall in order. It runs 0.082, 0.161, 0.245, 0.157, 0.312, 0.140. Lesson 36 built a regime measurement out of that number and lesson 37 spent a lesson choosing a confirmation length for it, and both were right about what they measured. Neither said that the quantity nearly quadruples between the first width and the fifth, then falls to under half of that at the sixth, purely from how the same tape was grouped. It is not a property of the market. It is a property of the market and the grid, and the grid was a dropdown.

Why the range grows like a root and the count falls like one over

The mean range column is the one with an argument behind it. Grouping four bars into one does not make the new bar four times as tall. It makes it 2.14 times as tall on this series. Group six and you get 2.75 times, not six. The ratios across the table are 1.00, 1.50, 1.88, 2.14, 2.47 and 2.75, and the square roots of the widths are 1.00, 1.41, 1.73, 2.00, 2.24 and 2.45. Every ratio sits a little above its square root and none of them is remotely near the width itself.

That is the square-root-of-time scaling, and it is what you get when the steps are close to independent: the distances add, but the displacements partly cancel, so the total grows like the square root of the number of steps rather than the number of steps. It is the single most useful fact in this lesson, because of what it does to cost.

Suppose a fixed charge per round turn, and suppose you take one trade per bar. At a width of one there are sixty bars and 81.0 of summed range to work with, which is 1.35 of movement bought per charge paid. At a width of four there are fifteen bars and 43.3 of summed range, which is 2.89 per charge. The coarser chart offers 47 per cent less total movement and asks you to pay a quarter as many charges, so the cost of a unit of movement falls to 0.47 of what it was. Not to a quarter. To roughly a half, which is one over the square root of four.

This is the honest version of a claim that gets made badly everywhere. A higher timeframe does not improve your win rate, and nothing in this lesson or anywhere else measures that it does. What it does is arithmetic: it cuts the number of times you pay in proportion to the width, and it grows the size of what you are trading in proportion to the square root of the width, and the ratio of those two is the whole of the effect. A fourfold coarser interval halves your friction per unit of movement. It does not quarter it, and anybody who tells you the improvement is proportional to the timeframe has not multiplied it out.

The setting nobody writes down

Now hold the width fixed at four and move only the start. Below are the four possible phases of a four-bar grid on this series, each truncated to fourteen chart bars so that every row covers fifty-six bars of the same tape and the only difference between them is where the boundary fell.

Four-bar grid starting at	Bars of tape covered	Efficiency ratio	Tallest bar	Swing points
bar 1	1 to 56	0.188	4.2	2
bar 2	2 to 57	0.249	5.3	2
bar 3	3 to 58	0.104	4.5	2
bar 4	4 to 59	0.200	5.9	1

Same instrument, same width, same number of bars on the screen, fifty-six bars of the same tape in every row. The regime reading runs from 0.104 to 0.249, a factor of nearly two and a half. Set a threshold anywhere between those two and these four charts of one instrument do not agree about which side of it the tape is on; set it at exactly 0.200 and the fourth row is a tie of precisely the kind lesson 37 spent a section on. The tallest bar on the chart runs from 4.2 to 5.9, so a stop sized as a multiple of recent bar height is being sized in units the phase invented. One of the four grids loses a swing point that the other three keep.

The high and the low, again, are 107.4 and 97.7 in all four rows. They survive because the bars that made them — bar 24 for the low, bar 53 for the high — fall inside all four spans. That is not luck so much as the reason levels are worth marking: an extreme is a trade that happened, and it is still there whatever you group it into.

Notice one more thing in passing. The first row of this table reads 0.188, and the fourth row of the previous table, which is the same grid, reads 0.157. Nothing changed but the truncation from fifteen bars to fourteen: four bars came off the end of the sample. Dropping four bars out of sixty moved the reading by nearly a fifth. That is the same lesson in miniature, and it is why a reading quoted without its span is not a number.

What this does to the last four lessons

It does not retract them. Every figure in lessons 32 through 37 is correct for the interval it was computed on, and each of those lessons said which interval that was. What this lesson adds is that the interval belongs in the quote. Lesson 32’s fifteen swings, lesson 35’s sweep counts, lesson 36’s ratio and lesson 37’s confirmation lengths are all measured in bars, and a bar was a choice.

The practical form of that is short. When you write down a rule, write the width and the phase beside it, the way you would write a currency beside a price. When you compare two results, check they were computed on the same grid before you conclude that one method beat the other. And when a reading and a level disagree, remember which of the two you would still have if somebody changed the dropdown.

Problems

1. Regroup your own bars and recompute one number you rely on. Take two hundred bars of your own instrument at whatever width you normally use, and group them into twos, threes and fours by hand or in a spreadsheet: high is the highest high, low is the lowest low, close is the last close. Then recompute one number you actually act on — a swing count, an average range, a moving-average crossover date, an indicator reading. Write the four answers in a row. The spread between them is what the dropdown was worth, and until you have seen it on your own instrument you have been quoting a price without a currency.
2. Move the boundary and measure the phase risk. Keep the width fixed and compute the same reading at every possible starting offset — four of them for a four-bar grid, six for a six. Every one is a legitimate chart and somebody is looking at each. The range of answers is your phase risk, and it is the part of your uncertainty that no amount of data collection reduces, because it is not noise: each answer is exactly right about a different grid.
3. Test the root on your own tape. Compute the mean bar range at your width, then at two, four and nine times that width. Divide each by the first. If your instrument behaves like the series above you will get numbers near 1.4, 2.0 and 3.0 — the square roots. If instead you get numbers near 2, 4 and 9, the moves are trending across bars rather than cancelling, and that is a genuine and rare finding worth far more than the exercise cost. Either way, How to Collect a Base Rate is how the count is kept honest.

Sources. Holbrook Working, “Note on the Correlation of First Differences of Averages in a Random Chain” (Econometrica, 1960), for the original demonstration that aggregating a series into wider intervals changes its statistical properties in ways that have nothing to do with the underlying process — three pages, and it settles the question. Robert C. Merton, “On Estimating the Expected Return on the Market: An Exploratory Investigation” (Journal of Financial Economics, 1980), for the asymmetry underneath the root scaling: sampling more finely sharpens what you know about variance and does essentially nothing for what you know about drift. Torben G. Andersen and Tim Bollerslev, “Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts” (International Economic Review, 1998), for the working version of the same point — the measured quantity depends on the interval you measured it over, so the interval has to be part of the answer.

Knowing that the width and the phase are settings does not tell you which to use, and the usual advice is to use several: a wide one for direction, a middling one for structure, a narrow one for entry. That advice is not wrong, and it is also not free. Lesson 39 takes it apart: what reading three charts actually buys, what it costs in decisions, and the arithmetic for how many trades a filter has to remove before it has paid for the ones it removed by mistake.

Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.