Rounding Explained: Half Up, Half Even, and Why 2.5 Rounds Two Ways

Rounding replaces a number with a simpler one nearby: 3.14159 becomes 3.14 to two decimal places, 1,473 becomes 1,500 to the nearest hundred. The rule is simple except at the midpoint — the number that ends exactly in 5 at the digit being dropped — where two conventions disagree: half up rounds 2.5 to 3, and half even, or banker’s rounding, rounds 2.5 to 2. The difference is one unit in the last place, and it is the whole subject of this guide: the conventions, the places, the floating-point trap, and the error that rounding accumulates.

The conventions: half up, half even, and the rest

Half up is the schoolroom rule: look at the digit being dropped; if it is 5 or more, round up; if it is less than 5, round down. 2.5 → 3, 3.5 → 4, 2.4 → 2, 2.6 → 3. The name is slightly misleading: “up” means away from zero for the magnitude, and the rule is usually applied symmetrically, so −2.5 → −3. Half up is simple, memorable, and biased: it rounds every midpoint upward, so a large set of numbers rounded half up drifts upward on average, by half a unit per midpoint. For a single measurement the bias is invisible; for a million transactions it is real money.

Half even — banker’s rounding — rounds a midpoint to the even neighbour: 2.5 → 2, 3.5 → 4, 4.5 → 4, 5.5 → 6. Half the midpoints go up and half go down, so the bias cancels over a large set, and the average is preserved. It is the default in IEEE floating-point arithmetic, in Python’s round(), and in financial standards that care about unbiased totals. Its cost is surprise: a reader who expects 2.5 → 3 will call 2.5 → 2 an error, and the surprise is legitimate, because the convention was not stated. Half even is the better default for computation and the worse default for communication, unless the convention is named.

The other conventions have their niches. Truncation drops the digits without rounding — 2.9 → 2 — and is used where a value must never exceed the original, such as a remaining balance. Ceiling rounds always up, floor always down; they bound a quantity rather than approximating it. Half down rounds a midpoint toward zero (2.5 → 2, 3.5 → 3), the mirror of half up. Each is a choice, and the choice should be recorded wherever the rounded figure will be checked, audited, or added: a total of rounded figures that was rounded half up will not match a total of the same figures rounded half even, and the difference is the convention, not an error. The Rounding Calculator states its mode — half up — on the page, and offers the other modes for comparison.

Places, significant figures, and the magnitude

Decimal places count from the decimal point: 3.14159 to two places is 3.14, to three places is 3.142, to zero places is 3. The place is the precision, and it should match the measurement: a length measured to the nearest millimetre should not be reported to the nearest micrometre, because the extra digits are fiction. Significant figures count from the first non-zero digit: 0.00456 to two significant figures is 0.0046, and 1,473 to two significant figures is 1,500. The two systems answer different questions — places fix the absolute precision, significant figures fix the relative precision — and a figure should be rounded by the system that matches how it was measured.

Large numbers round by place value, not by decimal places: 1,473 to the nearest hundred is 1,500, to the nearest ten is 1,470, to the nearest thousand is 1,000. The midpoint rule applies at each place: 1,450 to the nearest hundred is 1,500 under half up and 1,400 under half even, because the hundred’s digit, 4, is even. Trailing zeros after rounding are significant when they mark the place: 1,500 to the nearest hundred has the hundreds place as its last significant figure, and writing it as 1.5 thousand loses the statement of precision. The notation should carry the precision it claims.

Money rounds to the cent, and the cent is the place where the conventions collide most visibly. A total of $10.005 rounds to $10.01 half up and $10.00 half even, and a ledger that mixes the two will not balance. Financial systems therefore specify the mode — half even is common in banking, half up is common in retail — and the specification belongs in the system’s documentation, not in the programmer’s memory. Where the mode is unspecified, half up is the reader’s expectation, and half even is the statistician’s, and the difference should be resolved before the first transaction, not after the first audit.

Floating point, double rounding, and accumulated error

Computers store most decimals in binary floating point, and most decimal fractions have no exact binary form. 2.675 in binary is slightly less than 2.675 — 2.67499999… — so rounding it to two places in binary gives 2.67, not the 2.68 that decimal arithmetic gives. The error is in the representation, not in the rounding: the computer rounded correctly the number it actually held, which was not the number typed. The defence is decimal arithmetic — storing money as integer cents, or using a decimal type — for any quantity that will be rounded and checked. A float is fine for a simulation; it is the wrong tool for a ledger.

Double rounding is rounding twice: first to an intermediate precision, then to the final one. 2.4495 rounded to two places is 2.45; rounded to one place from 2.45 it is 2.5; but rounded directly to one place from 2.4495 it is 2.4, because 2.4495 is below the 2.45 midpoint. The intermediate rounding created a midpoint that was not in the original number, and the second rounding acted on the fiction. The rule is to carry full precision through every intermediate step and round once, at the end, to the precision the answer needs. The same rule governs the calculator: it computes at full precision and rounds only the display.

Accumulated rounding error is the sum of many small roundings. A thousand figures each rounded half up drift upward by up to 500 units in the last place; a thousand figures rounded half even drift by roughly the square root of that, because the errors cancel. The drift is why totals should be computed from unrounded figures and rounded once, rather than computed from rounded figures: the sum of the rounded parts is not the rounded sum of the parts, and the difference, though small per figure, is systematic. A total that disagrees with the sum of its displayed parts by a unit in the last place is usually this effect, and the footnote “totals may not add due to rounding” exists for exactly this case.

Rounding in spreadsheets: ROUND, ROUNDUP, and ROUNDDOWN

Spreadsheets offer four rounding functions, and they implement four different conventions. ROUND(number, places) rounds half away from zero — 2.5 to 3, −2.5 to −3 — the half-up rule of the schoolroom. ROUNDUP always rounds away from zero: 2.4 becomes 3, and 2.1 becomes 3, because “up” means away from zero, not toward the next integer above in the usual sense for negatives. ROUNDDOWN always rounds toward zero: 2.9 becomes 2, and −2.9 becomes −2. MROUND rounds to the nearest multiple of a stated significance: MROUND(7, 5) is 5, and MROUND(8, 5) is 10. Each is a choice, and the formula that uses the wrong one is wrong by the choice, not by the arithmetic.

The display is not the value. A cell formatted to two decimal places shows 2.68 for a value of 2.675, while the stored value remains 2.675, and a sum of displayed values will differ from the displayed sum of the stored values by the rounding — the accumulated-error effect, in spreadsheet dress. The cure is the same: compute from stored values, format only the display, and footnote the difference where the parts are shown. The ROUND function, by contrast, changes the stored value, and a chain of ROUND calls is double rounding made easy: round once, at the end, and let the intermediate cells carry full precision.

Floating-point representation bites in spreadsheets too. A cell containing 2.675, typed as a decimal, is stored in binary as slightly less than 2.675, and ROUND(2.675, 2) returns 2.67 in Excel, not 2.68 — the same representation error as in any binary float, wearing a familiar interface. The defence is the same: money as integer cents, or a decimal-aware tool, for quantities that will be rounded and checked. A spreadsheet is fine for a model; it is the wrong ledger for a business, and the rounding surprises are the symptom that proves it.

Rounding and measurement: the last digit earns its keep

A measurement’s last digit is a claim about its precision, and rounding should preserve the claim, not inflate it. A length measured with a tape to the nearest millimetre is reported to the nearest millimetre; reporting it to the nearest tenth of a millimetre invents a precision the tape never had, and the extra digit is fiction however accurately it is computed. The rounded figure should carry the precision of the measurement, and the measurement’s precision should be stated with the figure: “12.4 cm ± 0.1 cm” is a measurement; “12.43 cm” from the same tape is a pretence.

Calculated quantities inherit the weakest input’s precision. A rectangle measured as 12.4 cm by 8.2 cm has an area of 101.68 cm² by multiplication, but the inputs have three and two significant figures, so the area is honestly 1.0 × 10² cm² — two significant figures — and the extra digits are arithmetic, not information. The rule is not pedantry: it prevents a result from looking more certain than the evidence, and it is the reason scientific results carry their uncertainties explicitly. A calculation that reports six digits from two-digit inputs has not computed more; it has rounded less, and the difference matters to anyone who must rely on the result.

Uncertainty rounds too, and it rounds up. An uncertainty of ±0.14 is reported as ±0.2, not ±0.1, because understating an uncertainty is worse than overstating it, and the value is then rounded to the same place as the uncertainty: 12.43 ± 0.14 becomes 12.4 ± 0.2. The rounding of the uncertainty is conservative by design, and it is the one place where rounding up is not a convention but a duty: the error bar must contain the error, and a bar that is too small fails at its only job.

Frequently asked questions