Percent Error Calculator

Enter the measured value and the accepted value

The percent error calculator has two boxes. Experimental value is the number you measured or calculated in the lab. Theoretical (accepted) value is the reference you are checking against, such as a density from a data table or the accepted acceleration due to gravity. There is no calculate button. As soon as both boxes hold numbers, the line underneath writes the formula with those numbers substituted, and the percent error appears under that.

Decimals are allowed, and so are negative quantities, which come up with temperature changes, voltages, and other signed measurements. The formula takes the absolute value of the difference and the absolute value of the accepted value, so a result that lands below the accepted figure still produces a positive percent error.

The formula, with your numbers filled in

Percent error is the size of the mistake divided by the value that was supposed to be true, then turned into a percentage: the absolute value of experimental minus theoretical, divided by the absolute value of theoretical, times 100.

A density lab makes the substitution easy to follow. Aluminum measured at 2.63 g/cm³, with an accepted density of 2.70 g/cm³, is written |2.63 − 2.70| / |2.70| × 100. The absolute difference is 0.07. Divide 0.07 by 2.70 and multiply by 100 and the percent error is 2.5926%. The page shows that filled-in formula first, then the result, so you can check the arithmetic before you copy it into a report.

The order of the two numbers inside the absolute value does not change the answer. The denominator does. Divide by the theoretical value. Dividing by the experimental value, or by the difference itself, is a different calculation.

Percent error and percent difference

Percent error compares a measurement with a value treated as correct, and divides by that accepted value. Percent difference compares two measurements when neither one is the accepted answer, and divides the absolute gap by the average of the two values. Two lab partners who each find a density for the same metal, and who do not have a table value, are in the percent-difference case. Using their average in the denominator will not match the percent error a worksheet asks for.

Percent change is a third comparison. It measures a new value against an old one and keeps the sign, so a drop comes out negative. Percent error drops the sign on purpose, because the question is how far off the measurement was, not whether it landed high or low.

Density, boiling point, and gravity

Copper's accepted density is 8.96 g/cm³. A measurement of 8.40 g/cm³ is |8.40 − 8.96| / |8.96| × 100, which is 6.25%. Water measured at 101.3 °C against an accepted boiling point of 100 °C is |101.3 − 100| / |100| × 100, or 1.3%. A motion lab that records g as 9.75 m/s² against the accepted 9.81 m/s² gets |9.75 − 9.81| / |9.81| × 100, which is 0.6116%.

The two values have to be in the same unit before you type them. Percent error has no unit of its own, because the grams, degrees, or seconds cancel. Mixing g/cm³ with kg/m³, or a Celsius reading with a temperature you have not converted, changes the percentage even when the underlying measurement was fine. A small percent error means the measurement sits close to the accepted value. It does not by itself prove the method was careful, and a large one does not by itself name the mistake, but it is the figure most lab reports ask you to show next to the data.

When the accepted value is zero

If the theoretical value is 0, percent error is undefined. The formula divides by that value, and division by zero has no number at the end of it. The calculator says so in words. It does not print an infinity symbol. A theoretical mass of 0 g, or any other accepted baseline of zero, cannot be turned into a percent error. You would need a different comparison, agreed with your teacher, because a percentage of zero is not a scale you can read.

A negative accepted value is a normal case, not an error. A temperature change of −3.6 °C measured against an accepted change of −4.0 °C is |−3.6 − (−4.0)| / |−4.0| × 100, which is 10%. The absolute value in the denominator is what makes a negative baseline usable. The displayed percentage is rounded to four decimal places, or two when the result is 100% or more, and trailing zeros are dropped. A very small or very large result is shown in scientific notation so it does not collapse to 0%.