What percent error measures
Percent error compares a measured value with a reference value and expresses the gap as a percentage of the reference. The reference is the number you trust: a textbook value, a manufacturer's specification, a standard weight or a theoretical result. The measured value is the number you obtained from an experiment, a sensor or an estimate. The smaller the percent error, the closer the measurement is to the reference. Percent error is also called percentage error, and teachers often call the two values "experimental" and "accepted" (or "observed" and "true").
This calculator takes the two values, shows the raw difference, the absolute difference and the percent error with every step written out, tells you whether the measurement is above or below the reference and draws both values on a number line. It uses exact decimal arithmetic, so there is no floating-point noise in the digits.
The formula, and why the reference is the denominator
The formula is percent error = 100 × |measured − reference| ÷ |reference|. The vertical bars mean the absolute value, which removes any minus sign. The reference goes on the bottom because it is the fixed standard you are judging the measurement against: an error of 2 units means something very different against a reference of 9 than against a reference of 900. The same absolute difference gives 22.222222% for a measured 7 against a reference of 9, but 28.571429% if you swap the two values (a measured 9 against a reference of 7). Swapping the roles is the most common mistake, so check which number is the standard before you type.
A worked example
A student measures a quantity as 56.891 and the accepted value is 62.327. The difference is -5.436, the absolute difference is 5.436 and the percent error is 100 × 5.436 ÷ 62.327 = 8.721742%. The calculator keeps the exact value of the fraction and rounds only the displayed percentage, to six decimal places, and it marks the result with "about" when rounding was needed. In a lab report you would normally round further to the number of significant figures your instructions ask for.
Above or below: the signed percent error
The absolute percent error hides the direction of the mistake. The signed percent error keeps it: 100 × (measured − reference) ÷ |reference|. A measured 7 against a reference of 9 is -22.222222%, so the measurement is below the reference, while the absolute percent error is 22.222222%. A measured 9.72 against a reference of 9.8 is -0.816327%, a small shortfall. The calculator shows both, together with the words "above" or "below" and the size of the difference.
Negative reference values
Some quantities can be negative, such as temperatures in degrees Celsius, charges and altitudes below a datum. If you divide by the reference itself rather than by its size, the sign of the answer flips and can disagree with the direction of the error. With a measured value of −8 and a reference of −10 the measurement is above the reference by 2, and the calculator reports 20% (20% in absolute terms). Dividing by the absolute value of the reference keeps the signed result consistent with "above" and "below", so the page states that convention rather than leaving it hidden.
When the reference is zero
Percent error divides by the reference, so a reference of 0 makes it undefined. The calculator does not show infinity or 0%. For a measured 5 against a reference of 0 it reports the raw difference (5), the absolute difference (5) and a clear message that the percentage is undefined. When the true value really is zero, compare the absolute difference with the resolution of your instrument or with the scale of the quantity instead. Equal values give a percent error of 0%, and the page says the measurement equals the reference.
Percent error, percent difference and percentage change
Three similar-sounding measures answer different questions. Percent error judges a measurement against a trusted reference. Percent difference compares two measurements when neither is the standard: it divides the absolute difference by the average of the two values, so 7 and 9 differ by 25%. Percentage change describes how far a value moved from a starting point to a new one, and the starting point plays the role of the reference. For percentage change, increases and decreases, use the percentage calculator. If your values are very large or very small, write them in powers of ten with the scientific notation calculator first, and enter them here as plain decimals.
Typing numbers correctly
Use a dot for the decimal point and no thousands separators. A comma is rejected on purpose: some tools read "1,5" as 15, which would quietly give a wrong answer, so this calculator asks you to write 1.5 instead. Scientific notation such as 4.5e-4 is not accepted either; write 0.00045. Each value can have up to 20 digits and up to 12 decimal places, and a blank field is an error, never a zero. If you have a list of repeated measurements, find their mean with the average calculator first and use the mean as the measured value.
Percent error questions
What is percent error?
It is the size of the gap between a measured value and a reference value, written as a percentage of the reference. For 7 measured against 9 it is about 22.222222%.
How do I calculate percent error?
Subtract the reference from the measured value, take the absolute value, divide by the absolute value of the reference and multiply by 100. The steps in the results panel show each stage with your numbers.
Can percent error be negative?
The usual definition uses the absolute value, so it is never negative. The signed version keeps the direction: a negative signed percent error means the measurement is below the reference, a positive one means it is above.
What is a good percent error?
It depends on the experiment and on the precision of the instrument. A percent error close to 0 means the measurement agrees closely with the reference, but the limit you should aim for is set by your course, lab manual or measurement standard, not by this page.
What if the true value is zero?
Percent error is undefined because it would divide by zero. The calculator still shows the raw and absolute differences so you can report the size of the error in the original units.
What is the difference between percent error and percent difference?
Percent error uses a trusted reference as the denominator. Percent difference compares two measurements by dividing by their average, because neither one is the standard.