Shift Where the Bars Start and the Widest Bar of Four Months Nearly Doubles

A price series is a list of observations. A chart is not. Between them sits an aggregation rule — how many observations go into one bar, where the first bar begins, and what closes it — and that rule is chosen by the person reading the chart, not supplied by the market. Change the rule and the same underlying numbers produce different bars, different bar counts, and different answers to questions as basic as whether the last stretch closed higher than it opened.

The claim is easy to nod along to. Running the arithmetic on a public series, with the grouping rule held constant and only the starting point moved, makes it concrete.

A bar is a four-number summary of a window

An open, high, low and close are not four measurements. They are four order statistics computed over a set of observations that someone decided belonged together. The open is the first value in the window. The close is the last. The high and low are the extremes. Everything else inside — the sequence, the intermediate prints, the number of observations — is discarded at the moment the bar is drawn.

That discarding is the whole point of a bar, and it is also the source of the problem. Many different internal paths reduce to identical bars.

Because the summary is computed over a window, it inherits the window. Two windows overlapping by four observations out of five still produce different opens, different closes, and often different extremes. Nothing in the market changed between them.

One series, one rule, five starting points

The test series is the daily nominal 10-year Treasury constant maturity yield published by the Board of Governors of the Federal Reserve System in statistical release H.15, series identifier RIFLGFCY10_N.B, quoted in percent per year. The window runs from 2026-04-17 through 2026-08-20, the last observation in the H.15 release dated 2026-08-21.

The file carries 90 business-day rows. Three of them are marked ND with no published value: 2026-05-25, 2026-06-19 and 2026-07-03. Those three dates are the observed federal holidays for Memorial Day, Juneteenth National Independence Day, and Independence Day — 4 July 2026 falls on a Saturday, so the Friday before it is the observed date. That leaves 87 published observations.

The rule applied to them is deliberately plain. Take consecutive runs of five observations. Each run becomes one bar: open is the first value, close is the fifth, high and low are the extremes of the five. Any leftover observations at the start or end that cannot fill a complete bar are dropped. The only thing that varies across the five trials is the offset — whether grouping begins at the first observation, the second, the third, the fourth or the fifth.

Nothing is resampled or interpolated. The same 87 numbers go in every time.

The two panels are the same series. The bars are not the same bars.

Grouping starts at	Complete bars	Close above open	Close below open	Widest bar	Narrowest bar
observation 1	17	11	6	12 bp	8 bp
observation 2	17	10	7	17 bp	5 bp
observation 3	17	9	8	17 bp	3 bp
observation 4	16	11	5	21 bp	4 bp
observation 5	16	11	5	20 bp	4 bp

Three results are worth separating out.

The balance of up and down bars moves. Starting at the first observation gives 11 bars closing above their open against 6 closing below — close to two to one. Starting at the third gives 9 against 8, which is close to a coin. A description of the period as “mostly higher closes” or “evenly split” is settled by the offset, not by the yield series.

The widest bar of the period nearly doubles. Under the first grouping the largest single bar range in four months is 12 basis points. Under the fourth it is 21 basis points, on the bar covering 2026-05-13 to 2026-05-19. The 21 bp move exists in the data either way. Whether it lands inside one bar or is split across two is decided by the offset.

The narrowest bar collapses. The smallest range under the first grouping is 8 bp. Under the third it is 3 bp, less than half. Any rule that treats a narrow bar as a distinct condition is reading a property of the grid as much as a property of the market.

The bar holding the sample high

The highest published value in the window is 4.75 on 2026-07-31. Every grouping contains it, because it is in the data. What the groupings disagree about is the bar it sits in.

Grouping starts at	Bar span	Open	High	Low	Close	Direction
observation 1	07-29 to 08-04	4.67	4.75	4.63	4.63	down
observation 2	07-30 to 08-05	4.68	4.75	4.63	4.63	down
observation 3	07-31 to 08-06	4.75	4.75	4.63	4.69	down
observation 4	07-27 to 07-31	4.65	4.75	4.61	4.75	up
observation 5	07-28 to 08-03	4.61	4.75	4.61	4.70	up

Under the third grouping the bar opens at 4.75 and gives ground afterwards. Under the fourth it closes at 4.75 as the top of a run. Two opposite-looking events, built from overlapping slices of one undisputed series.

The vocabulary attached to bars — a close at the extreme, a long upper shadow, an inside range — describes the grid. It is not wrong to use, but it does not survive the boundary moving.

What no regrouping can change

Aggregation is not arbitrary in every respect. The fixed part is worth stating precisely.

• The endpoints. The first published value in the window is 4.26 on 2026-04-17 and the last is 4.69 on 2026-08-20. Net change over the window is +43 bp under every grouping tested.
• The extremes of the whole window. The maximum is 4.75 and the minimum is 4.26. No grouping can create a higher high or a lower low than the observations contain.
• The observation count. 87 published values, whatever the bars do with them.

Everything else in the table above is a function of the rule. The distinction is practical: a statement about the window as a whole is robust to regrouping, and a statement about any individual bar is not. Conclusions drawn from “the last three bars” are conclusions about a grid that another reader may not be using.

Unit size changes it again

Moving the starting point is the smaller of the two levers. Changing how many observations go into a bar moves the same series further. Holding the offset at the first observation and varying only the bar size gives this.

Observations per bar	Bars	Close above open	Close below open	Mean range	Widest bar
3	29	15	14	6.4 bp	13 bp
5	17	11	6	9.7 bp	12 bp
10	8	6	2	14.9 bp	22 bp
21	4	3	1	24.0 bp	33 bp

At three observations per bar the period reads as almost balanced, 15 up against 14 down. At twenty-one it reads as three up bars against one. Both are correct summaries of the identical 87 numbers.

One detail in that table is worth not glossing over. The widest bar at three observations per bar is 13 bp, wider than the 12 bp widest at five observations per bar, even though the larger unit covers more ground. Larger windows usually contain larger ranges, but not necessarily, because a boundary can fall in the middle of a move and hand half of it to each neighbour. Aggregation does not merely coarsen a series; it redistributes what the series looks like it did.

The clock rule and the counter rule

Everything so far used a counting rule: close the bar after a fixed number of observations. The alternative is a clock rule: close the bar at a fixed boundary in time and accept whatever arrived inside it. On the same 87 observations the two rules do not line up.

Grouped by calendar week, the window contains 19 week bars. Fourteen of them hold five published observations. Three hold four, because a federal holiday sat inside them. The first and last are clipped by the edges of the window. Grouped in fixed runs of five observations, the same data gives 17 bars.

The two grids drift apart immediately and only occasionally coincide. Of the 17 five-observation bars, 3 close on a date that is also the close of a calendar week. The rest close mid-week, because once a holiday removes an observation the counter no longer lands on Friday.

The drift shows up in duration too. Each of the 17 bars holds exactly five observations, and their calendar spans run from 4 to 7 days: twelve span six, three span seven because a holiday sits inside, two span four because the run starts on a Monday. Same contents, nearly double the clock length.

The last week of July shows the divergence at close range. The calendar-week bar for 2026-07-27 to 2026-07-31 opens at 4.65 and closes at 4.75, its highest value, and reads as an up bar. The five-observation bar running from 2026-07-29 to 2026-08-04 opens at 4.67 and closes at 4.63, and reads as a down bar. There is no contradiction to resolve. Two rules were applied and two answers came back.

Where a 144-tick bar gets its shape

A tick bar is the counting rule with the transaction as the unit. A 144-tick bar closes when 144 trades have printed, regardless of how long that takes. A time bar is the clock rule: a 5-minute bar closes five minutes after it opened, regardless of how many trades printed inside it.

The consequence is structural rather than empirical. On a US equity venue the core session runs 9:30 a.m. to 4:00 p.m. ET, which is 390 minutes, so a 5-minute grid produces exactly 78 bars every full session. On a scheduled early close at 1:00 p.m. the session is 210 minutes and the same grid produces 42 bars. The count is known before the session opens, because it is a property of the clock and not of the tape.

A 144-tick rule offers no such guarantee. The number of bars in a session is whatever the trade count divides by, and the duration of any individual bar is set by how fast trades arrive while it is open. When arrivals slow, one bar spans a longer stretch of clock. When they accelerate, several bars close inside a minute. That is the same arithmetic demonstrated above with holidays: a fixed count of events stretches across a variable amount of calendar.

Two things follow, and both are commonly overstated.

• A tick bar's boundaries are set by transaction count, so its duration varies. This is definitional, not an empirical finding.
• How much duration varies depends on the intraday arrival profile of the specific instrument and session. That profile was not measured here. The Treasury demonstration establishes the mechanism at daily resolution; it does not quantify intraday behaviour on any equity or futures contract, and no figure in this piece should be read as doing so.

The narrower point does hold. A tick bar and a time bar on the same tape are different aggregations, so every open, high, low and close differs between them. A read taken from one grid has to be re-derived on the other, not carried across.

What Would Invalidate This

The sensitivity shown above is real for this series and this window. Several conditions shrink it or remove it.

• A constant arrival rate with no boundary splitting a move. If every directional move happens to sit fully inside one bar, offsetting the grid changes little. The 12 bp versus 21 bp gap exists because boundaries cut through moves.
• Quantisation. H.15 constant maturity yields are published to two decimals, so one basis point is the smallest distinguishable step and small regrouping differences do not round away. On a series quoted more coarsely relative to its typical move, some of these differences would disappear into the rounding.
• Sample specificity. This is 87 observations of one series over four months. That per-bar statistics depend on the grouping rule is arithmetic and general. The magnitude is specific to this window and is not a typical figure.
• Rules anchored to a fixed calendar boundary. A method that only ever reads month-end or quarter-end values has no free offset to vary. The issue does not arise, and none of the above applies to it.
• Bar-independent methods. Anything computed directly from the observation series without bars — a cumulative change, a count of days above a level — is unaffected by all of this.

This piece does not establish that any bar size, unit or anchor produces better decisions than another, and does not attempt to. The measurement is sensitivity, not usefulness. A grid that moves a lot under offsetting is not thereby worse, and one that moves little is not thereby better. Those are separate questions and this data does not answer them.

Concrete Framework

1. Write the bar rule down as three parts. Unit (transactions, minutes, observations), size (144, 5, 5), anchor (session open, wall-clock grid, first row of the file). Most disagreements about what a chart shows are disagreements about part three, which is the part nobody states.
2. Re-derive the same read at anchor shifts of −2, −1, +1 and +2 units. Five versions of the same conclusion. This costs one loop in whatever tooling is already producing the bars.
3. Record the spread, not just the base case. In the test above the spread on “bars closing above their open” was 9 to 11 out of roughly 17, and the spread on the widest bar was 12 bp to 21 bp. A conclusion is worth what its weakest offset says.
4. Flag any result that changes sign under a one-unit shift. The 2026-07-31 bar flips from down to up between the third and fourth anchors. A sign flip at one unit of shift means the result is a statement about the grid.
5. Check holiday and half-day handling explicitly. Three missing observations in four months were enough to knock a five-observation counter permanently out of phase with the calendar week, leaving only 3 of 17 bars closing on a week boundary. Confirm whether the data source omits the day, forward-fills it, or emits an empty bar, because all three are in use.
6. Confirm where the first bar of the session is anchored. A grid anchored to the session open and one anchored to a wall-clock boundary differ whenever the session does not start on a round interval. That is a configuration question with a definite answer.
7. Keep window-level and bar-level statements apart. Net change, window high, window low and observation count survive regrouping. Bar direction, bar range, bar sequence and anything counted in bars do not. Label which of the two any carried-over backtest figure came from.