Percent Error Calculator

1. What is percent error?
2. Using the percent error calculator
3. Percent error formula
4. How to calculate percent error
• Example #1: Percentage error from observed and true value
• Example #2: Absolute error to percent error
• Example #3: Absolute to percentage error
• Error of a scale or measurement tape

    What is percent error?

Percent error, also percentage error, is a measure of the accuracy of a measurement relative to a true or estimated value, sometimes referred to as "theoretical" value. It is percentages making it much easier to communicate and understand how big of a concern the error is, unlike the absolute error, which is expressed in whatever the measurement units are and requires more domain expertise. Additionally, saying 1% error is more convenient than using the corresponding relative error 1/100 and our percentage error calculator will quickly compute it.

For example, if I tell you that the absolute measurement error is 500 grams, you are not sure if that is a big or small difference, relative to what you are measuring. If you were measuring weights in a market, say, you might be selling potatoes, a 500 gram (0.5 kg or about 1 pound) error would be quite significant as the percent error from a 2 kg (~4 pounds) bag of potatoes would be 25% and you will not be allowed to sell scales with such a low level of accuracy. However, 500 grams is a standard error in many home scales made for measuring your body weight, since 0.5 kg is just 1% for a 50 kg man (110 lb), and only 0.5% error from a 100 kg (220 lb) man - good enough for the purpose. The same absolute error results in vastly different relative errors, demonstrating their usefulness.

Percent error is not to be mistaken with standard error (of the mean), standard deviation (of the sample) or margin of error, as these are more advanced statistical concepts.

    Using the percent error calculator

Simply enter the observed value from any given measurement or observation, and then provide the true value, if known. If it is not known, enter your best available estimate. The relative error calculator will then produce the error in percentages, positive or negative, depending on the direction of the observed discrepancy from the true value.

    Percent error formula

There are several ways to write the percent error formula. The most straightforward is, assuming V stands for Value:

Percentage error = (V_observerd - V_true) / V_true x 100

When calculated as we show above, the sign of the error will be positive when the observed value is larger than the true. It will be negative otherwise, so the sign of the error is useful in understanding the direction of the error. Note that for chemistry and most sciences it is customary to preserve the sign of the error as its direction matters. The percent error calculator will output negative errors with the minus sign in front of the percentage.

In some places the formula is written with (V_true - V_observerd), which is incorrect, or at least inconvenient, as this reverses the sign. If you do not care about the directionality of the error, then just take the absolute: |V_observerd - V_true| in the numerator.

If you have calculated the absolute error using the formula below:

E_absolute = V_observerd - V_true

then you can write the percent error formula like so:

Percentage error = E_absolute / V_true x 100

If you know the relative error:

E_relative = (V_observerd - V_true) / V_true

Then the percent error is simply:

E_percent = E_relative x 100

As you can see there are many ways to calculate it, but the math is simple enough to be done by hand in many cases.

    How to calculate percent error

Below are several examples using different starting information and measurement units. All of them can be checked using our calculator.

Example #1: Percentage error from observed and true value

Let's take an example from everyday life. Say you are purchasing wooden laths to make a treehouse. You order laths with a length of 10 feet (304.8 cm), but upon receival you get laths that are 10 ft 1 inch (307.34 cm) long. How much is that as a percent error?

You need to apply the formula like so:

(10 ft 1 in - 10 ft ) / 10 ft x 100 = (121 in - 120 in) / 120 in x 100 = 1 / 120 x 100 = 0.0083 x 100 = 0.83% error

Example #2: Absolute error to percent error

Let's get back to our scales example from paragraph one. If we know the allowed absolute error of our home scale is 0.5 kg or 1.1 pound, how do we calculate the percent error if we expect our true weight to be 100 kg (220 lb)?

In kg, we need to divide 0.5 by 100 to get the relative error and then multiply by 100 to convert it to percent. In pounds we need to divide 1.1 by 220 to get the relative error, and then multiply by 100 to convert to percent.

In kg: (0.5 / 100) * 100 = 0.005 * 100 = 0.5% error

In lb: (1.1 / 220) * 100 = 0.005 * 100 = 0.5% error

Example #3: Absolute to percentage error

Let us go back to the treehouse example, but with a spin. Now, when purchasing, the producer tells you that due to the production process, a difference of up to 0.5 inches (12.7 mm) versus your specification is expected. How much is that as a percent error?

To calculate that, you need to know the required path length, for example 10 feet (304.8 cm). Then, you need to apply the formula based on known absolute error like so:

0.5in / 10 ft x 100 = 0.5 in / 120 in x 100 = 0.00416 x 100 = 0.416%.

Error of a scale or measurement tape

Say a student’s scale recorded his weight as 160 lb. but his weight is known to actually be 164 lb. What is the percent error of the scale? - the answer would be (164 - 160) / 164 = -4 / 164 = 0.0244 which equals -2.44% as a percentage

A craftsman measured a line to be 11.1 inches long. If the actual length of the line is 11.6 inches, then what was the percent error of the measurement? - to answer this calculate (11.6 - 11.1) / 11.6 = 0.5 / 11.6 = -0.0431 so the error is -4.31%

Want to work out your own examples? Make sure to check the results using our percentage error calculator.