Percentage Difference Calculator

The denominator is the average, and that is what makes it symmetric

Every comparison of two numbers has to divide by something, and the choice of divisor is not a detail — it is the question. Percentage change divides by whichever value came first in time, because a change is measured from where you started. Percent error divides by whichever value is already known to be correct. Percentage difference is for the case where there is no such value: two thermometers, two quotes, two lab duplicates, two branches’ sales. Nothing distinguishes them, so nothing should be privileged in the arithmetic either, and the neutral choice is their midpoint.

The consequence is symmetry, and it is easy to check. Twenty and thirty differ by 40%, and so do thirty and twenty. The percentage change for the same pair is +50% one way and −33.3333% the other — two different numbers describing the same two values, because each is measured against a different base. Press Swap on the calculator above and you can watch the difference stay put while the changes trade places.

One property worth knowing: because the denominator sits between the two values, a percentage difference is always between the two percentage changes in size, and for two positive numbers it can never reach 200% — that ceiling belongs to the case where one value is zero. A percentage change has no such limit.

Four steps, worked through

The calculator shows this working for whatever numbers you enter; here it is once in full for the pair the page teaches from.

Step two is the only one that differs from every other percentage calculation, and it is where the mistakes happen. Dividing by 20 instead of 25 gives +50% and dividing by 30 gives 33.3333% — both real numbers, neither of them the percentage difference. If your answer changes when you swap the inputs, you have divided by one of the values rather than by their average.

One test decides which comparison you want

Ask whether swapping the two numbers ought to change the answer. If it ought to — because one number came first, or one of them is the correct one — you want a percentage change or a percent error, and the base is obvious once you say why. If it ought not to, because the two numbers are peers, you want the percentage difference.

Some worked cases:

• Last year’s revenue and this year’s. Time gives an order, so this is a percentage change measured from last year. The Percentage Change Calculator owns this question, along with CAGR and multi-period tracking.
• Your measurement and the textbook value. One of them is right, so this is a percent error measured against the textbook. That is the Percent Error Calculator.
• Two suppliers’ quotes for the same job. Neither came first and neither is correct — percentage difference, and the answer is the same whichever quote you type in first.
• Two readings from two instruments. Same again, unless one instrument is the calibrated reference — in which case it is a percent error and the reference is the base.

The distinction matters most when the answer is quoted rather than used. “These quotes are 40% apart” and “the second is +50% more than the first” describe the same two numbers and read very differently, and only the second one commits you to saying which is the base.

The same pairs, all three ways

Every row is computed by the same engine as the calculator above. The last two columns are what those numbers would be if you had reached for the other tool — worth scanning once, because the gap between them is bigger than most people expect:

Across a whole set the same idea gives the spread. Four quotes of 1,250, 1,399, 1,180 and 1,425 average 1,313.5, and the widest percentage difference in the set — between the cheapest and the dearest — is 18.81%. That figure is exactly the two-value answer for the extremes, which is the check the engine holds itself to.

Two percentages have two honest differences, and they are not the same size

When both numbers are already percentages the comparison splits, and the split has consequences. A rate moving from 4% to 5% has risen by +1 percentage point, which is a subtraction, and by +25% relatively, which is a division. Its percentage difference — symmetric, against the average of the two rates — is a third figure again, 22.2222%. All three are correct. They answer different questions and they differ by a factor of twenty-five.

This is why the word point exists. Statistical agencies and central banks report rate moves in percentage points precisely because “unemployment rose 25%” and “unemployment rose one point, from 4% to 5%” sound like different events and are the same one. When both figures are small the relative version is always the more dramatic, which is what makes it worth naming the units out loud whenever a rate moves.

The one asymmetry to watch: a rate of zero has no relative change, because nothing is a percentage of nothing. Going from 0% to 5% is five percentage points and no relative rise at all. The point difference survives every case; the relative one does not.

Negatives, zeros, and where the answer stops existing

Two negative values have a negative average, and dividing a positive gap by a negative number would return a negative percentage difference for a gap that plainly exists. The size of the average is used instead, so −10 and −12 differ by 18.1818% — the same figure as 10 and 12, which is the answer that keeps a magnitude a magnitude.

A pair that straddles zero is a different matter. Five and −5 average exactly zero, and there is no percentage of zero, so the percentage difference does not exist and the calculator says so rather than returning a very large number. Just either side of that point the answer is unstable rather than wrong — 5 and −4.9 average 0.05 and differ by 19,800% — which is a real property of the measure and the reason it is not used on data that crosses zero. Quote the absolute difference there instead.

A single zero is fine. Zero and ten average 5, so they differ by 200% — the ceiling for any pair of non-negative numbers. The percentage change from zero, by contrast, does not exist at all, which is one of the few cases where the difference is defined and the change is not.

In a spreadsheet, it is one ABS over an AVERAGE

With the two values in A1 and B1, each figure here is one short formula. The trap in all of them is multiplying by 100 in a cell already formatted as a percentage, which inflates the answer a hundredfold:

• Percentage difference: =ABS(A1-B1)/ABS(AVERAGE(A1,B1)), with the result cell formatted as Percentage. The outer ABS on the average is what keeps two negative values from returning a negative answer.
• Percentage change from A1 to B1, for comparison: =(B1-A1)/A1. Swapping the cell references changes this answer and not the one above it.
• Widest difference in a range A1:A20: =(MAX(A1:A20)-MIN(A1:A20))/AVERAGE(MAX(A1:A20),MIN(A1:A20)) — note that the denominator averages the two extremes, not the whole column.
• Percentage points between two rates: just =B1-A1, formatted as a number rather than a percentage, and labelled “pp” so nobody reads it as a relative change.

There is nothing to download for any of this and no account to create: the calculator above is a free web page that runs entirely in your browser, on desktop or mobile, and the numbers you type are never sent anywhere.

Sources and methodology

The arithmetic cites no authority because none exists to cite — it is written out in full above rather than asserted. What is cited below is the part that is somebody else’s convention rather than mathematics: how a spreadsheet behaves with these formulas, and the percentage-point usage the section above describes.

• Microsoft Support — calculate percentages in Excel
• Google Docs Editors Help — TO_PERCENT (Sheets)
• NIST/SEMATECH e-Handbook of Statistical Methods
• How we calculate — methodology

Method. Every figure here — the calculator, the step-by-step working, the comparison table, and the worked examples in the prose — is produced in your browser by one engine, src/lib/percentage-difference.ts, so nothing on the page can disagree with the tool above it. That engine is checked on every change against 76 hand-written assertions, and the central one is a property rather than a number: swapping the two inputs must leave the percentage difference untouched, on every pair tested, while the percentage change must not. The others cover the cases most likely to be got wrong — that two negative values return a positive magnitude, that a pair averaging zero has no answer at all rather than an enormous one, that the widest difference across a set equals the two-value answer for its extremes, and that percentage points, relative change, and percentage difference stay three separate numbers. The count and the per-case breakdown are published on the formula verification page.

Read the guide

The asymmetry this page keeps pointing at — that 20 to 30 is a rise of 50% while 30 back down to 20 is a fall of only 33.33%, because the base changes underneath you — has a guide of its own. Why a Percentage Loss and an Equal Percentage Gain Don’t Cancel Out follows that same mechanism into money, where it does real damage: a 25% loss needs a 33.33% gain to get back to level, and the gap widens fast as the loss deepens. It is the case for using a symmetric comparison when the two values are genuinely peers, made with numbers rather than argued.

Related calculators

The other ways to compare two numbers:

More in Math, or browse all calculators.