Statistics Definitions > Relative Error

**Contents**:

## Relative Error as a Measurement of Precision

Relative error (RE)—when used as a measure of precision—is the ratio of the absolute error of a measurement to the measurement being taken. In other words, this type of error is relative to the size of the item being measured. RE is expressed as a percentage and has no units.

As a formula, that’s:

**RE = absolute error / measurement being taken.**

For example, let’s say two people measure a rug with a meter stick. One person measures the length and the other person measures the width. The meter stick is accurate to within 1 mm, which means the absolute error is ±0.001 m.

- The length of the rug is measured at 3.215 meters. RE = 0.001 m / 3.215 m = 0.0003%.
- The width of the rug is measured at 4.075 meters. RE = 0.001 m / 4.075 m = 0.0002%.

Although the absolute error of 0.001 m is the same for each, the relative error for the width is smaller.

## Different Units

The relative error is very useful when you want to be able to compare things that are measured in different units. For example, let’s say you’re measuring height and weight of a dog. The height of the dog is measured as 84 cm with an absolute error of ±3 cm. The weight of the dog is 35 lbs with an absolute error of ± 1 lbs. Which is more precise?

You can find the answer by figuring out the relative errors for both:

- RE
_{height}= 3 cm / 84 cm = 0.04% - RE
_{weight}= 1 lb / 35 lb = 0.03%

The weight measurement is more precise.

## Relative Error as a Measure of Accuracy

Nine times out of ten, RE is a measure of precision, as in the examples above. However, the same term can (confusingly) also be used to describe accuracy; Specifically, how accurate a measurement is compared to the true value. You can only find RE_{accuracy} if you know the actual “true” measurement—something that’s difficult to do unless you’re measuring against the atomic clock. The formula is:

**RE**

_{accuracy}= (Absolute error / “True” value) * 100%.For example, if your bathroom scale weighs you at 165 lb but you known your “true” measurement (from the doctor’s office) is 172 lb, then:

RE

_{accuracy}= (165 lb / 172 lb) * 100% = 0.96

When expressed as a percentage (i.e. 96%), this is also called percent error.

If you don’t know the “true” measurement, you can use the first definition —precision —as a substitute.

## References

Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York.

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.

Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.

Wheelan, C. (2014). Naked Statistics. W. W. Norton & Company