Percent Error Calculator

Enter the value you measured and the value it should have been, and this calculator returns the percent error along with the absolute error, the relative error and the working behind them. An observed value of 10 against a true value of 11 gives a percent error of -9.0909%, or 9.0909090909091% error once the sign is dropped. The number line under the inputs redraws as you type, so the gap between the two values is something you can see rather than only read.

What percent error is

Percent error measures the gap between an observed value and a true one, written as a percentage of the true value. The observed value is whatever you measured or worked out. The true value is the one that is accepted, expected or already known, which is why it is also called the accepted value or the theoretical value. The comparison answers a plain question: how far off was I, given the size of the thing being measured?

Measurements drift from the truth for ordinary reasons. A ruler is read at an angle, a balance was never zeroed, a stopwatch is started a fraction late, an instrument has a limit to how finely it can resolve anything. None of that is unusual, and none of it is avoidable in full. What percent error does is put a size on the discrepancy so you can judge whether the result is usable.

A small percent error means the measurement sits close to the accepted value. A large one means it does not, and that either the measurement or the method behind it needs another look. If a reading is out by 90 percent, something has gone wrong that no amount of careful rounding will fix, and the sensible response is to check the technique rather than the arithmetic.

Computing percent error

Three related quantities do the work, and they build on each other. The absolute error is the plain difference between the observed and the true value. Dividing that by the true value gives the relative error. Multiplying the relative error by 100 gives the percent error:

$$\mathrm{Absolute\; error}=|V_{\mathrm{observed}}-V_{\mathrm{true}}|$$ $$\mathrm{Relative\; error}=\frac{|V_{\mathrm{observed}}-V_{\mathrm{true}}|}{V_{\mathrm{true}}}$$ $$\mathrm{Percent\; error}=\frac{|V_{\mathrm{observed}}-V_{\mathrm{true}}|}{V_{\mathrm{true}}}\times 100\%$$

Take an observed value of 56.891 against a true value of 62.327. The absolute error is 5.436, and dividing by 62.327 then multiplying by 100 gives a percent error of 8.722 percent:

$$\frac{|56.891-62.327|}{62.327}\times 100\%=8.722\%$$

The one part people get wrong is which value goes on the bottom. It is always the true value, never the observed one, because the true value is the reference the comparison is being made against. Dividing by the observed value instead gives a different answer and a quantity nobody asked for.

All of this assumes the true value is actually known. Often it is not, and the honest way to describe the error is then to repeat the measurement and quote the spread of the results, which the standard deviation calculator works out.

Negative percent error

With the absolute value in place and a positive true value, percent error always comes out positive, because the sign is thrown away before the division. Most of the time the size of the error is all anyone wants and the direction is beside the point.

Dropping the absolute value keeps the direction. When the observed value is smaller than a positive true value, the percent error is negative. An observed value of 7 against a true value of 9 works out like this:

$$\frac{V_{\mathrm{observed}}-V_{\mathrm{true}}}{V_{\mathrm{true}}}\times 100\%=\frac{7-9}{9}\times 100\%=-22.222\%$$

A negative percent error means only that the measurement came in under the expected figure, and a positive one that it came in over. Neither is better than the other. The best possible result is zero, where the observed and true values agree, so a negative error is not a sign that the measurement beat expectations. This calculator reports the signed figure as the headline and the absolute figure beside it, so you can quote whichever your report asks for.

Why relative error matters more than absolute error

An error of one centimetre sounds small until you know what was being measured. On a length of one metre it is a percent error of 1 percent. On a kilometre it is 0.001 percent, a thousand times better, for exactly the same absolute mistake. The raw difference on its own says nothing about quality.

This is the whole reason percent error exists. Dividing by the true value strips out the scale, which makes errors comparable across measurements that have nothing else in common. A chemist can compare a mass measurement against a volume measurement and an engineer can compare a resistance against a frequency, because both have been reduced to the same unit of nothing at all.

It also explains why percent error grows unmanageable near zero. A true value of 0.5 measured as 1 is only 0.5 out in absolute terms, yet the percent error is 100 percent. Divide by something small enough and any error looks catastrophic.

Percent error against percent difference

Percent error needs one value to be the accepted one. When neither is, such as two labs measuring the same sample or two instruments disagreeing, percent difference is the right tool. It divides the gap by the average of the two values rather than by either one:

$$\mathrm{Percent\; difference}=\frac{|V_1-V_2|}{(V_1+V_2)/2}\times 100\%$$

Readings of 48 and 52 have a percent difference of 8 percent, and it stays 8 percent whichever order you feed them in. Treat one as the true value and the answer depends on which: 48 against 52 gives -7.692 percent, while 52 against 48 gives 8.333 percent. That asymmetry is a feature of percent error rather than a flaw, and it is precisely why the method only applies when one value genuinely is the reference.

What counts as an acceptable percent error

There is no universal threshold, and any source quoting one without naming a field is guessing. A school physics experiment measuring gravity with a metre rule and a stopwatch might return 9.79 m/s squared against 9.81, a percent error of -0.204 percent, and that is a good result for the equipment. An analytical chemistry assay would treat a fraction of a percent as unremarkable and a few percent as a failure. A rough field estimate might be fine at 10 percent.

Engineering usually sets the threshold in advance and calls it tolerance. A resistor marked 470 ohms at 5 percent is guaranteed to fall between 446.5 and 493.5 ohms, so a measured 462 ohms, a percent error of -1.702 percent, is a part working exactly as specified rather than a part that missed. The question worth asking is never whether an error is small, but whether it is small enough for the decision resting on it.

Where the error comes from

Errors split into two kinds, and percent error alone cannot tell them apart. Systematic error pushes every reading the same way: a balance reading 0.2 g heavy adds 0.2 g to everything you weigh, and repeating the measurement will not help. Random error scatters readings either side of the truth through reaction time, vibration, or the last digit an instrument cannot quite settle on, and averaging several readings does reduce it.

The practical tell is what happens when you repeat the work. Readings that cluster tightly around a wrong answer point to systematic error, and the fix is to calibrate. Readings that scatter widely around roughly the right answer point to random error, and the fix is more readings. This is also the difference between accuracy, which is closeness to the true value, and precision, which is agreement between repeats. A measurement can be precise and wrong.

When percent error stops working

Two situations break it. If the true value is zero, the division is undefined and there is no percent error to report, so quote the absolute error instead. This calculator says so rather than printing infinity.

The second is a scale whose zero point is arbitrary. A thermometer reading 1 degree Celsius when the true value is 0.5 gives a percent error of 100 percent, but the same half-degree gap expressed in kelvin, 274.15 against 273.65, is 0.1827 percent. Nothing about the measurement changed, only where the scale happens to start. Percent error is meaningful on ratio scales, where zero means none of the quantity, and misleading on interval scales such as Celsius, Fahrenheit and calendar years.

Reporting the number properly

Round the answer to a precision the measurement can support. If the observed value carried three significant figures, quoting a percent error to nine decimal places claims a certainty the data never had. Two or three significant figures is normal, so 8.72 percent rather than 8.7218035...; the rounding calculator handles the trimming. State whether the figure is signed or absolute, since the two differ by more than a minus sign in how they read, and say what you took as the true value and where it came from.

One more habit worth keeping: percent error compares a measurement against a standard, and percent change compares one value against an earlier one. The arithmetic is nearly identical and the meanings are not, so the label matters. The percent calculator covers the change side.

Reading the results on this page

The panel leads with the signed percent error, which keeps the direction of the miss, and lists the absolute version underneath for reports that want the size alone. Absolute error is the plain difference in the original units, and relative error is the same thing as a fraction rather than a percentage. The steps below reproduce the calculation line by line, from the formula to the substituted numbers to both final forms, so the answer can be checked by hand.