Can percent error be negative?

Only if you leave out the absolute value on purpose. The standard formula uses |measured − accepted|, so the result is never negative. Dropping the bars gives a signed percent error, where a negative number means you measured below the accepted value and a positive one means you measured above it. A 98.4 °C boiling point against 100 °C is 1.6% error, or −1.6% signed. Both describe the same measurement.

What is the difference between percent error and percent difference?

Percent error compares a measurement against a known accepted value and divides by that accepted value. Percent difference compares two measurements when neither is the reference, and divides by their mean. For 8.78 and 8.96, percent error against 8.96 is 2.01%, while percent difference divides 0.18 by the mean 8.87 and gives 2.03%. Use percent error when one value is authoritative.

What counts as a good percent error?

There is no universal threshold. What matters is how the percent error compares with the precision of your instrument. A 2% error from a ruler marked in millimetres is unremarkable; the same 2% from an analytical balance means something went wrong. Your lab protocol or instructor sets the acceptable range for a given experiment, so check it rather than guessing at a number.

What if the accepted value is zero?

Percent error is undefined, because the formula divides by the accepted value. Report the absolute error in its own units instead, or divide by a meaningful reference such as the full scale of the instrument and say so explicitly. This also applies when the accepted value is so close to zero that the percentage explodes into a meaningless number.

What is the percent error formula?

4 August 2026

Short answer. Percent error = |measured value − accepted value| ÷ accepted value × 100. Subtract the accepted value from your measurement, drop the sign, divide by the accepted value, then multiply by 100. A boiling point measured at 98.4 °C against an accepted 100 °C gives 1.6 ÷ 100 × 100 = 1.6% error.

Percent error answers one question: how far off was I, as a share of the right answer? The formula is three operations long. Almost every mistake people make with it happens in the division, because they divide by the wrong number.

The formula

                |measured − accepted|
percent error = ─────────────────────── × 100
                       |accepted|

The accepted value is the one you trust: a published constant, a certified reference, the true count. The measured value is what your experiment produced. The denominator is always the accepted value, never your measurement, because the point is to size your error against reality rather than against your own possibly wrong number.

The absolute value bars do only one job, which is to make the result positive. More on when to leave them off further down.

A worked example

You measure the density of a copper sample and get 8.78 g/cm³. The accepted density of pure copper at 20 °C is 8.96 g/cm³:

Measured        8.78 g/cm³
Accepted        8.96 g/cm³

Subtract        8.78 − 8.96     =  −0.18
Drop the sign   |−0.18|         =   0.18      the absolute error
Divide          0.18 ÷ 8.96     =   0.0200892857...
× 100                           =   2.01%

Notice the middle line. Before it becomes a percentage, 0.0200892857 is the relative error, and it is the number that actually carries the meaning. Multiplying by 100 is cosmetic. It just moves the decimal point two places so the answer reads as 2.01 instead of 0.02.

Rounding is your call, but be consistent: 2.0089% rounded to two decimals is 2.01%, and to one decimal it is 2.0%. Do not quote more decimal places than your instrument earns.

Does percent error go negative?

Not with the bars in place. Drop them and you get a signed percent error, which keeps one extra piece of information: which way you missed.

(98.4 − 100)  ÷ 100  × 100  =  −1.6%    measured low
(9.55 − 9.81) ÷ 9.81 × 100  =  −2.65%   measured low
(262 − 250)   ÷ 250  × 100  =  +4.80%   measured high

Chemistry courses usually want the unsigned magnitude. Physics and engineering work often wants the sign, because a thermometer that always reads low is a calibration problem, while one that scatters either side of true is just noise. Read the question before you throw the sign away.

Percent error is not percent difference

These get swapped constantly. Percent error needs a value you trust. Percent difference is for when you have two measurements and no authority between them, so it divides by their mean instead:

percent error       |8.78 − 8.96| ÷ 8.96   × 100  =  2.01%
percent difference  |8.78 − 8.96| ÷ 8.87   × 100  =  2.03%
                                       ↑
                             the mean of 8.78 and 8.96

Close, here, because the two values are close. The gap widens as the values move apart. If one of your numbers came out of a reference table, you want percent error.

Quick reference

Accepted	Measured	Absolute error	Percent error
100boiling point, °C	98.4	1.6	1.60%
8.96copper density, g/cm³	8.78	0.18	2.01%
9.81gravity, m/s²	9.55	0.26	2.65%
250counted items	262	12	4.80%
1.50concentration, mol/L	1.53	0.03	2.00%

The last two rows are worth comparing. An absolute error of 12 sounds much worse than one of 0.03, and yet 0.03 out of 1.50 is 2.00% while 12 out of 250 is 4.80%. Absolute error on its own tells you almost nothing. That is the entire reason percent error exists.

It works outside the lab too

Any estimate you can later check against the real figure has a percent error. Budget 4,800 for a job that lands at 5,150:

|4800 − 5150| ÷ 5150 × 100  =  350 ÷ 5150 × 100  =  6.80%

The actual cost is the accepted value here, because it is the one that turned out to be true. Divide by 4,800 instead and you get a flattering number that measures your estimate against itself.

The subtraction that calculators fumble

Percent error starts by subtracting two numbers that are, if the experiment went well, nearly equal. That is precisely the operation binary floating point handles worst. Try the boiling point example on most calculators:

Exactly     100 − 98.4  =  1.6
                        →  1.6 ÷ 100 × 100  =  1.6%

In binary   100 − 98.4  =  1.5999999999999943
                        →  1.5999999999999945%

98.4 has no exact representation in binary, so the machine subtracts the nearest thing it can store and the leftover garbage floats up into the visible digits. It is the same fault as the famous 0.1 + 0.2 = 0.30000000000000004. Most calculators hide it by rounding the display, which is fine right up until you paste the result somewhere. Prism computes with exact rational arithmetic, so 100 − 98.4 is 1.6 and 0.1 + 0.2 is 0.3, and the percentage that comes out the other side has nothing to hide. Fractions get the same treatment: see how to add fractions.

Or let the percentages tool do the dividing

Prism has a percentages calculator built in, along with six other tools. Percentages are free forever, and there are no ads and no subscription. See the app.