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Relative Error Calculator

Table of contents

What is the absolute error?What is the relative error?How to calculate the absolute error and relative errorIs my absolute error too high?FAQs

If you ever wondered what's the difference between relative and absolute error, our relative error calculator is right up your alley. In the following text, you'll discover the absolute and relative error formulas, together with easy-to-follow examples. We also prepared a short section on the differences between the two types of error, as well as one on the reason why the relative error is considered to be more useful.

What is the absolute error?

The absolute error also called the approximation error, is the absolute value of the difference between the actual value and the measured value. The absolute error formula is

absolute error = |actual value - measured value|

The actual value is otherwise known as the real or true value. On the other hand, the measured valueis an approximation.

What is the relative error?

Relative error (or percent error), on the other hand, expresses the error in terms of a percentage. You can use the following relative error formula:

relative error = |absolute error / actual value| = |(actual value - measured value) / actual value|

We typically express both the absolute and relative errors as positive values, hence the use of absolute values.

How to calculate the absolute error and relative error

You can use our relative error calculator to estimate both the absolute and relative error for any measurement or calculation. Let's analyse the difference between these two types of error with an example.

Let's say you want to determine the value of the square root of two. The value you find online is 1.41421356237, but you wonder how accurate it would be to simply write it rounded to two significant figures. Note, that this is different to decimal points - see the significant figures calculator for more information.

  1. To find out the absolute error, subtract the approximated value from the real one:

    |1.41421356237 - 1.41| = 0.00421356237

  2. Divide this value by the real value to obtain the relative error:

    |0.00421356237 / 1.41421356237| = 0.298%

As you can see, the relative error is lower than 1%. In many cases, this is considered a good approximation.

Is my absolute error too high?

The main advantage of the relative error is that, since it can only take values between 0-100%, it's easy to evaluate whether an error is big or small. It's much more challenging to determine whether a specific absolute error is of sufficient accuracy. For example, let's imagine you take measurements of weight with an absolute error of 1 kg:

  • If you are weighing apples in a grocery store, and you are planning on buying 2 kg of apples, an absolute error of 1 kg can result in buying up to 50% more or less than you need. You wouldn't want to use such scales in a store, would you?
  • When you weigh yourself at home, a 1 kg error makes a substantial difference - after all, you'd like to know whether you weigh 75 or 76 kg. Nevertheless, this error feels more acceptable than in the case of apples.
  • However, if you want to weigh a 20-meter-long steel beam that weighs approximately 2 tonnes, you're not interested in a difference of one kilogram - it's a relative error of about 0.05%, which can easily be neglected.

As you can see, the bigger the real value, the higher the accepted absolute error.

FAQs

Is the relative error the same as absolute error?

The absolute error is the discrepancy between your measurement and the true value, while the relative error is the ratio between the absolute error and the absolute value of the true value.

The relative error helps us asses how accurate the measured value is when compared to the true value.

Is there another name for relative error?

Relative error is known under several different names:

  • Relative uncertainty;
  • Approximation error;
  • Fractional error; and
  • Percentage error.

What is the relative error if I measured 42 and the true value is 40?

The answer is 0.05 or 5%. To arrive at this result, we apply the relative error formula:
relative error = |(actual - measured) / actual|.
Plugging in actual = 40 and measured = 42, we obtain
relative error = |(42-40) / 40| = 1/20 = 0.05.

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