Relative Error Calculator

Suppose you’ve ever wondered what the difference between relative and absolute error is. In that case, our relative error calculator is for you. When you need to compare items measured in different units or math problems it is also useful. The absolute and relative error formulas and easy-to-follow examples are provided in the following text. There is also included a brief overview of the distinctions between the two sorts of errors and why the relative error is regarded as more helpful. You can also use our calculator and these errors in statistics and math.

What is Relative Error?

As with the case of standard error, the relative error is a measurement of how similar a given parameter is to the true theoretical value and we can express it as an approximation error. We commonly use it to put errors into perspective. For example, let’s say an error of 5cm would be significant when the total height of an object is 20cm. Still, it would be of minor importance if the said object is 20 kilometres. Therefore, with that in mind, the smaller the result is, the larger the relative error becomes. 

Absolute error formula

Absolute error (\Delta x) is called absolute accuracy error, and even though their formulas may differ, they are the same, just with different names.        (\Delta X) = x_1 – x

• x_1 is the measurement,
• x is the true value.

Moreover, if we are dealing with multiple measurements, we may see the formula above with absolute symbol ( | | ),     (\Delta X) =| x_1 – x |

We may use this symbol sometimes because the result can be negative numbers. At the same time, we express them as positive numbers, therefore, the need for an absolute symbol.

Relative error formula

In contrast to the error mentioned above, we can calculate the relative error in percentages. Following that, we have the formula given:

\newline relative \, error = \left| \frac {actual \,value - measured \,value }{ actual \,value } \right| \cdot 100\%

Whether you calculate size, temperature, length, the unit isn’t of importance, as we express them in percentages, but we can also use this formula without percent. Moreover, the units are dropped off, so the result is unitless. 

How to calculate the absolute error and relative error

As mentioned before, we can calculate both errors with provided formulas. Furthermore, our calculator can calculate both errors, and below, we will explain how to use it. 

You can use the same measurements for both errors to make them easier to remember. For example, let’s take the measured value of 44cm, and the true or actual value is 49cm. We have two ways to calculate them, and one is easier than the other. Of course, the easier one would be our calculator, where you only need to type in the given values; after that, it will do the rest.

For absolute error, we can calculate it by applying the formula, 

  (\Delta X) = x_1 – x=> (\Delta X)=|44 - 49| = 5.00 cm

as we can see, even though the result should be negative, due to the absolute symbol used, we have a positive value; furthermore, because the unit used is centimetres, the result will also be in the same unit.

On the other hand, relative error uses a formula that will end as a percentage as a result of the mentioned formula above:

\begin{aligned} relative \, error = \left| \frac {actual \,value - measured \,value }{ actual \,value } \right| \cdot 100\% \end{aligned}  

\begin{aligned}\newline \Delta X= \left| \frac {49 - 44}{49} \right| \cdot 100\% \end{aligned}

\newline \Delta X =10.20%

One note that we should be aware of is that when the true value is zero, the relative error is undefinable. Furthermore, relative error makes sense only when a measuring scale begins at a true zero. This makes logic for the Kelvin temperature scale, but not for Fahrenheit or Celsius!

Relative vs. Absolute Error

We have mentioned what both terms are and what they represent. Still, this may not be enough for some to understand the meaning and their actual use in real life, so here we will explain how they differ and their key points. First, the absolute error indicates how close the measured value is to the actual real value. It’s the difference between what we’ve measured and what we should have measured, in other words. 

On the other hand, the relative error tells us if the measured parameter is large or small, compared to the true value. So, for example, when we measure in larger units, like kilometres or miles, a few meters or millimetres error isn’t significant. So that was a case of small relative error. 

Is my absolute error too high?

In contrast to relative error, which is easy to determine whether given results are large or small, it is more difficult to determine if the absolute error is accurate or too high. So, to find out if it is too high, we will take an example with measures of 1kg of error in the case of weighing ourselves and an object that weighs 5 tons. Our case makes a significant difference whether we are one kg less or more, while we don’t care if the object is 1kg heavier than it is. With this as a reference, we can see that the accepted value of absolute error is higher when the real value is bigger. 

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