Percent Error Calculator

The percent error formula

Percent error compares a value you measured with a value that is treated as correct. The formula is the absolute difference divided by the accepted value, multiplied by 100.

Percent error = |measured - accepted| / |accepted| x 100

The accepted value goes in the denominator because it is the reference. Dividing by the measured value instead gives a different number, and that number answers a different question: how far the accepted value sits from your result, expressed as a share of your result. For small errors the two are close, and for large ones they diverge quickly.

The vertical bars mean absolute value, so the usual percent error is never negative. The BIPM guide to the expression of uncertainty in measurement is the international reference behind most national metrology rules. The units cancel in the division, which is the point: 0.23 g/cm3 over 7.87 g/cm3 is the same 2.92 percent as 0.23 kg/m3 over 7.87 kg/m3, as long as both values use the same unit.

Worked example: the density of iron

A student measures the density of an iron sample at 8.10 g/cm3. The accepted density of iron at room temperature is about 7.87 g/cm3. The arithmetic runs in four short steps.

So the measurement is 2.92 percent above the reference. That is a reasonable result for a school lab with a graduated cylinder, where reading the meniscus alone can cost a percent or two. It would be a poor result for a certified analytical method, where the tolerance is often tighter than 1 percent.

Notice what the number does not say. It does not say the accepted value is perfect. Handbook densities vary with temperature and purity, and the 7.87 figure is itself a rounded average. Percent error measures the gap between two numbers, not the quality of the reference.

Two more cases: boiling point and a pipette

Two more cases, both common in lab reports and in quality checks.

Boiling point. A thermometer reads 99.1 degrees Celsius for water at sea level, against an accepted 100 degrees. The difference is 0.9, and 0.9 / 100 = 0.009, so the percent error is 0.90 percent. A thermometer that reads 101.4 degrees gives 1.40 percent in the other direction. Both are within the drift of a cheap glass thermometer, which is why the sign matters when you are deciding whether to recalibrate.

Volume delivered by a pipette. A 250 mL flask is filled to a mark that should hold 250.0 mL, and a check weighs the contents and finds 253.75 mL. The error is 3.75 mL, and 3.75 / 250 = 0.015, so the percent error is 1.50 percent. Turn it around: if a method is allowed 1.5 percent error on a 250 mL target, the acceptable band is 250 x 0.015 = 3.75 mL, so anything from 246.25 to 253.75 mL passes.

Gravity. A dropped-ball experiment returns 9.65 m/s2 against the standard 9.81 m/s2. The gap is 0.16, and 0.16 / 9.81 = 0.0163, so the percent error is 1.63 percent. Timing with a stopwatch by hand usually costs more than that, which is why the result is judged against the method, not against zero.

Signed error and what direction tells you

The absolute value in the formula throws away direction, and sometimes direction is the whole finding. The signed relative error keeps it: (measured - accepted) / accepted x 100.

Two instruments with the same 2.92 percent error can need opposite fixes. One reads high and needs a negative correction; the other reads low and needs a positive one. Report the absolute percent error when the question is how big the gap is, and the signed version when the question is which way it points.

Bias shows up in the signed column. If ten trials all land positive, the instrument has a systematic offset, not random scatter. If the signs alternate, the error is random and averaging more trials will shrink it. Percent error on a single trial cannot tell you which one you have.

Percent error, percent difference and percent change

Three similar-looking percentages answer three different questions. Mixing them up is the most common error in lab reports and in business writing.

Percent difference exists for the case where neither value is a reference. Compare two lab groups that each measured the same sample and you have no basis for calling either one true, so the average is the neutral denominator. For 8.10 and 7.87 the average is 7.985, and 0.23 / 7.985 = 2.88 percent, a little smaller than the 2.92 percent error against the handbook figure.

Percent change is a movement in time, and it has a direction built in. A price that goes from 40 to 46 rose 15 percent; a price that goes from 46 to 40 fell 13.04 percent. Same 6 units, different percentages, because the base changed. The percent increase calculator covers that case, where there is no accepted value and the old number is the base by definition. This page is for the other case, where a reference value exists and the question is how far a measurement sits from it.

What counts as a small error

There is no universal threshold. What counts as small depends on the field, the instrument and the cost of being wrong.

Percent error is also not the same thing as uncertainty. The National Institute of Standards and Technology draws a firm line between the two: error is the difference between a measured value and the true value, while uncertainty describes the range within which the true value is believed to lie. A result can have a small error and a large uncertainty, or the reverse, and a measurement result is complete only when the uncertainty is stated alongside it. NIST's glossary entry for measurement uncertainty and its guidelines for expressing uncertainty spell out the difference.

That distinction matters when a number is used for a decision. A 0.5 percent error on a single reading looks excellent, but if the instrument's uncertainty is 3 percent, the agreement was luck. Report the gap and the uncertainty, and let the reader see both.

Five mistakes that change the answer

Five mistakes account for most wrong answers, and each one is easy to spot once you know the shape.

Dividing by the measured value. For 8.10 against 7.87 that gives 0.23 / 8.10 = 2.84 percent instead of 2.92 percent. The gap is small here and grows with the size of the error: at a 50 percent error the two bases differ by 33 percent of the answer.

Dropping the absolute value and then reporting a negative percent error as if it were smaller. A signed error of -5 percent is a 5 percent gap. The sign is information about direction, not about size.

Rounding before dividing. Round 0.23 to 0.2 and the answer becomes 2.54 percent instead of 2.92 percent. Keep full precision through the division and round only the final percentage.

Comparing percentages with percentage points. An error that moves from 2 percent to 4 percent rose by 2 percentage points, which is a 100 percent increase in the error itself. Both statements are true and they mean different things.

Using percent error when there is no accepted value. If two methods are being compared and neither is a standard, percent difference is the honest measure. Calling one of them "accepted" to make the formula work turns an assumption into a result.

How to report a percent error

Report the number with the same discipline as the measurement it describes.

Give the percent error to two significant figures, or to one if the measurement itself is coarse. A 2.92 percent result from a graduated cylinder implies more precision than the cylinder can deliver; 3 percent is the honest version. State which value you used as the reference and where it came from, because a percent error without a named reference cannot be checked.

Include the sign when direction matters, and say plainly that the measured value was high or low. If the result is part of a set, report the percent error for the mean and the spread across trials, not just the single best run. A mean percent error of 1.2 percent with trials ranging from -3 to +4 percent describes a different instrument from one that lands between 1.0 and 1.4 every time.

Finally, put the number in context. A 2.92 percent error is good for a school lab and poor for a calibration certificate, and the reader needs to know which situation applies. The calculator above returns the percent error, the signed error, the absolute difference and the percent difference, so the three measures can be compared side by side for the same pair of values.