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Relative Error

Relative Errors Explained With a Definition, Equation, And Examples

Relative error is estimated by comparison, unlike an absolute error. This Buzzle article lists the equation and method to compute relative error.
ScienceStruck Staff
Last Updated: Jun 3, 2018
A lot of experiments in different branches of science, including chemistry, physics, and biology involve determination of various types of data. Many of these experiments have standard data values. While performing these experiments, one needs to compare the observed values with the standard ones, to determine how accurate the experiment was! This brings us to the term 'error'. What does 'error' mean? Error is nothing but the deviation of the observed value from the true value of the measured quantity. There are two types of errors: absolute error and relative error.
Definition
Absolute error is the difference between the magnitude of the true value and the observed one. It gives us the exact number with the units of the quantity that are deviated from the true one. Unlike absolute error, relative error is expressed in terms of percentage, and it helps us to compare how incorrect a quantity is from the value considered to be true. Relative error is defined as the absolute error divided by the true value. It is generally expressed as percentage and helps us to calculate the ratio between absolute error and true value.
Equation of Relative Error
Relative error is determined by using the following formula:

Relative error = (x - x0) ÷ x

Where, x = true value of a quantity,
x0 = observed value of the quantity,
x - x0 = absolute error.

But as described earlier, this calculation is often expressed as percentage. To find the percentage, just multiply the answer that you get by hundred.

Percent Relative Error Formula

Percent relative error = Relative error × 100

Calculating Relative Error

Simply substituting the values in the above formula and calculating accordingly will give you the exact value in no time. Let us take an example for a better understanding. Consider the following values of a given quantity:

True value = 50
Observed value = 47.5

Relative error = Absolute error ÷ True value
Relative error = (true value - observed value) ÷ True value
= (50 - 47.5) ÷ 50
= 2.5 ÷ 50
Relative error = 0.05
Percent relative error = 0.05 × 100 = 5%

Let us take another example. Consider a quantity with 44 as its true value. While performing the experiment, observed value was reported to be 43.9.

True value = 44
Observed value = 43.9

Relative error = Absolute error ÷ True value
Relative error = (true value - observed value) ÷ true value
= (44 - 43.9) ÷ 44
= 0.1 ÷ 44
Relative error = 0.002273
Percent relative error = 0.002273 × 100 = 0.2273%

Note that if the observed value of a quantity is greater than the true value, the absolute error will be negative. In this case, you will have to obtain the modulus of the absolute error.

│Absolute error│= Absolute error, when the value is greater than or equal to zero,
│Absolute error│= -1 × absolute error, when the value is less than zero.

So, if your true value is 50 and the observed value is 51, absolute error will be 50 - 52 = -2 and│absolute error│= -1 × -2 = 2
Now, relative error = 2 ÷ 50 = 0.04 and relative error percentage = 4%

It was pretty simple, wasn't it? Just keep the formula in mind and follow the simple steps explained above.