How to Calculate Percent Error

To calculate the percent error, you take the difference between a value collected in an experiment and a known or exact value. Percent error is most frequently used in chemistry and the other physical sciences. The purpose of determining the percent error is to ascertain the difference between exact/standard values and experimental values. The calculation for percent error is as follows:

The absolute value of the accepted value minus the experimental value divided by the accepted value times 100.

Or to put that another way, you calculate percent error by doing this:

|accepted value – experimental value| \ accepted value x 100

Understanding The Formula For Percent Error

To calculate percent error you will first want to find the difference between the value that has been measured and an accepted/standard value. The value found measured in the experiment referred to as the experimental value. The standard value is the value that is typically regarded as true by scientists and is also called the accepted value or theoretical value.

“An error does not become a mistake until you refuse to correct it.” — John F. Kennedy

Quite often you’ll see the percent error given as a positive value. You will then take the absolute value of the difference between the accepted value and the experimental value and divide it by the accepted value. Multiplying by 100 just gives the number as a percentage.

In the sciences, particularly chemistry, it is often standard to keep the value negative. It does matter whether or not the error is negative or positive. As an example, if you were comparing the theoretical yield of a chemical reaction one would expect a negative percent error. If a positive percent error showed up as the result of the calculation, it’s a clue that something has gone wrong with the experiment and that there were either unaccounted for reactions or an error in procedure.

If the sign for error is kept negative, the calculation for error is expressed as the measured value/experimental value minus the theoretical/known value, all divided by the theoretical value. Finally, the result is multiplied by 100.

Steps For Calculating Percent Error

A way of calculating percent error. Photo: Public Domain

To calculate the percent error, follow the steps listed below:

Start off by subtracting one of the values from the other value in the equation. It’s critical to remember that it matters whether or not the sign is kept in the calculation. If the negative sign is being dropped than the order you subtract in doesn’t matter. However, if the negative sign is being kept, then you must be sure to subtract the theoretical value from the experimental value, and not the other way around. After you’ve subtracted one value from another value, you have your value error.

“From error to error, one discovers the entire truth.” — Sigmund Freud

Now you divide the value error by the other value, and this will give you a decimal number that to multiply by 100 to convert it into a percentage. Now you have your percent error.

Examples of Percent Error Calculation

If you were dealt a cube of copper and asked to find the percent error between the determined mass of the cube and the cube’s expected/accepted density, how would you do it? Let’s suppose that the density of the copper returns as 8.78 grams/cm3 when placed on a scale. The expected density of a copper cube is 8.96 g/mc3.

So if accepted/expected value is 8.96 g/cm3 and the experimental value is 8.78 g/cm3, you should begin by subtracting the expected value from the experimental value. Subtracting 8.96 g/cm3 from 8.78 g/cm3 gives us -0.18 g/cm3. Since the formula for percent error specifies that this should be an absolute value, we’ll just take the absolute value of -0.18 g/cm3, which is 0.18 g/cm3.

Let’s take a look at one more example. Let’s suppose you are provided with a cube of aluminum and are instructed to determine the percent error. Placing the cube in a container of water and measuring its displacement, then comparing this with its dimensions would give you its volume. The volume of the aluminum cube is calculated at 2.68 g/cm3. This is in contrast to the accepted value of a room temperature chunk of aluminum’s density of 2.70 g/cm3.

So if your experimental value is 2.68 g/cm3 and your accepted value is 2.70 g/cm3, you just need to subtract the accepted value from the experimental value, and doing so gives the value -0.02. Let’s choose to discard the negative sign here and take the absolute value, so it’s just 0.02. Now we’ll have to divide the absolute value of 0.02 by 2.70, which gives us 0.0074074. Multiplying this 0.0074074 by 100% will get us 0.74%.

How Is Percent Error Different From Relative Error/Absolute Error?

Percent error is a similar concept to other forms of error calculation like absolute error and relative error, but it does differ in important ways. Unlike percent error, absolute error is found by determining the difference between the experimental value and known value, and the relative error is determined by dividing the absolute error by the standard value. So percent error is what you get when you multiply the relative error by 100%, meaning that relative error and absolute error are necessary to determine the percent error.

“Error is not a fault of our knowledge, but a mistake of our judgment giving assent to that which is not true.” — John Locke

Summing Up

Percent error is calculated in order to determine the amount of difference between a measured value and a true/standard value. This equation for percent error can be given as:

|accepted value – experimental value| \ accepted value x 100

Or to put that another way it is the difference in experimental value and accepted value, divided by the accepted value and multiplied by 100.

Some fields always express percent error as a positive value, dropping the negative sign. Meanwhile, other fields of study will work with either a negative or positive value. If you are keeping the negative sign it is done to find out if the experimental values are consistently coming in either above or below the expected values.

Percent error is calculated alongside relative error and absolute error for a comprehensive analysis of errors.

Finally, you should always know whether or not you are dropping the negative sign during the calculations. You’ll also want to know how many significant figures of the value you need to report (how precise the measurement is).