Percent error or percentage error expresses as a percentage the difference between an approximate or measured value and an exact or known value. It is used in science to report the difference between a measured or experimental value and a true or exact value. Here is how to calculate percent error, with an example calculation.

### Key Points: Percent Error

- The purpose of a percent error calculation is to gauge how close a measured value is to a true value.
- Percent error (percentage error) is the difference between an experimental and theoretical value, divided by the theoretical value, multiplied by 100 to give a percent.
- In some fields, percent error is always expressed as a positive number. In others, it is correct to have either a positive or negative value. The sign may be kept to determine whether recorded values consistently fall above or below expected values.
- Percent error is one type of error calculation. Absolute and relative error are two other common calculations. Percent error is part of a comprehensive error analysis.
- The keys to reporting percent error correctly are to know whether or not to drop the sign (positive or negative) on the calculation and to report the value using the correct number of significant figures.

## Percent Error Formula

Percent error is the difference between a measured or experiment value and an accepted or known value, divided by the known value, multiplied by 100%.

For many applications, percent error is always expressed as a positive value. The absolute value of the error is divided by an accepted value and given as a percent.

|accepted value - experimental value| \ accepted value x 100%

For chemistry and other sciences, it is customary to keep a negative value, should one occur. Whether error is positive or negative is important. For example, you would not expect to have positive percent error comparing actual to theoretical yield in a chemical reaction. If a positive value was calculated, this would give clues as to potential problems with the procedure or unaccounted reactions.

When keeping the sign for error, the calculation is the experimental or measured value minus the known or theoretical value, divided by the theoretical value and multiplied by 100%.

**percent error = [experimental value - theoretical value] / theoretical value x 100%**

## Percent Error Calculation Steps

- Subtract one value from another. The order does not matter if you are dropping the sign (taking the absolute value. Subtract the theoretical value from the experimental value if you are keeping negative signs. This value is your "error."
- Divide the error by the exact or ideal value (not your experimental or measured value). This will yield a decimal number.
- Convert the decimal number into a percentage by multiplying it by 100.
- Add a percent or % symbol to report your percent error value.

## Percent Error Example Calculation

In a lab, you are given a block of aluminum. You measure the dimensions of the block and its displacement in a container of a known volume of water. You calculate the density of the block of aluminum to be 2.68 g/cm^{3}. You look up the density of a block of aluminum at room temperature and find it to be 2.70 g/cm^{3}. Calculate the percent error of your measurement.

- Subtract one value from the other:

2.68 - 2.70 = -0.02 - Depending on what you need, you may discard any negative sign (take the absolute value): 0.02

This is the error. - Divide the error by the true value:0.02/2.70 = 0.0074074
- Multiply this value by 100% to obtain the percent error:

0.0074074 x 100% = 0.74% (expressed using 2 significant figures).

Significant figures are important in science. If you report an answer using too many or too few, it may be considered incorrect, even if you set up the problem properly.

## Percent Error Versus Absolute and Relative Error

Percent error is related to absolute error and relative error. The difference between an experimental and known value is the absolute error. When you divide that number by the known value you get relative error. Percent error is relative error multiplied by 100%. In all cases, report values using the appropriate number of significant digits.

## Sources

- Bennett, Jeffrey; Briggs, William (2005),
*Using and Understanding Mathematics: A Quantitative Reasoning Approach*(3rd ed.), Boston: Pearson. - Törnqvist, Leo; Vartia, Pentti; Vartia, Yrjö (1985), "How Should Relative Changes Be Measured?",
*The American Statistician*,**39**(1): 43–46.