How to Calculate Relative Error with Examples

Calculating relative error is an important skill for anyone working in science engineering, statistics, or other quantitative fields. While absolute error tells you how far off your measurement is from the true value relative error tells you the error relative to the magnitude of the true value. This allows you to compare the precision of measurements that may be in different units or on vastly different scales. In this comprehensive guide, I’ll explain what relative error is, when to use it, and step-by-step how to calculate it.

What is Relative Error?

Relative error is a measure that allows you to compare the error of measurements that are in different units or scales, It expresses the absolute error as a percentage of the true value

For example, let’s say you are measuring the length and width of a rug. The length measures 5 feet with an absolute error of ±0.1 feet. The width measures 3 feet with the same absolute error of ±0.1 feet. While the absolute errors are the same, the relative error will be different since the measurements are different sizes.

To find the relative error, you divide the absolute error by the true value and multiply by 100%. This gives you the error as a percentage of the true value, allowing an apples-to-apples comparison.

Relative Error Formula

The formula for relative error is:

Relative Error = (Absolute Error / True Value) x 100%

Where:

• Absolute Error is how far off your measurement is from the true value
• True Value is the accepted or known true value

Let’s break this formula down using an example:

You measure the width of a table as 4.5 feet. Later, you look at the specs and see the table width is actually 4.7 feet.

• True Value = 4.7 feet
• Measured Value = 4.5 feet
• Absolute Error = |4.7 – 4.5| = 0.2 feet

Plug this into the relative error formula:

Relative Error = (0.2 / 4.7) x 100% = 4.3%

So your measurement was off from the true value by 4.3%.

When to Use Relative Error

The main times you’ll want to use relative error rather than absolute error are:

• Comparing measurements in different units: As in the rug example above, you can compare the precision of length versus width measurements.

• Measurements on different scales: You can compare the error of small versus large measurements, like the width of a pencil versus the length of a football field.

• Checking precision of an instrument: You can see if an instrument is accurate to within a certain percentage of the true value.

• Determining statistical significance: In inferential statistics, relative error helps determine if results are statistically significant or due to chance.

Steps for Calculating Relative Error

Here is a step-by-step process for calculating relative error:

1. Measure the item and record this as your measured value.

2. Determine the true, accepted, or known value from a reliable source.

3. Subtract the measured value from the true value.

4. Take the absolute value of this difference. This is your absolute error.

5. Divide the absolute error by the true value.

6. Multiply this result by 100% to convert to a percentage.

7. Report your relative error! Be sure to include the % symbol.

Let’s walk through an example:

1. You measure the length of a board as 3.7 meters.

2. The actual length according to the specs is 3.9 meters.

3. 3.9 m – 3.7 m = 0.2 m

4. |0.2 m| = 0.2 m

5. 0.2 / 3.9 = 0.051

6. 0.051 x 100% = 5.1%

7. The relative error is 5.1%.

Tips for Using Relative Error

Here are some key tips to keep in mind when using relative error:

• Be sure your true value is accurate. Using an incorrect true value will invalidate your calculation.

• Use significant figures appropriately in your calculation based on the precision of your instruments.

• For very small true values, the relative error can be large and possibly meaningless. Use judgment whether relative error is appropriate.

• Report your relative error result with relevant context, like the measurement it pertains to and instruments used.

• If a relative error is over 10%, it may indicate systemic errors in your measurement methods. Troubleshoot your process.

• Relative error is a useful tool, but it does not give you information about accuracy. An inaccurate measurement can still have small relative error if close to the true value.

Common Mistakes

Some common mistakes to avoid when calculating relative error:

• Mixing up absolute error and relative error in the calculation. Double check your formula!

• Using the wrong value for the true or accepted value. This will make your calculation meaningless.

• Forgetting to take the absolute value of the difference between measured and true.

• Multiplying by 100% at the end to convert to a percentage.

• Reporting relative error without proper context or units. Make sure to communicate your full process.

Real World Examples

Let’s look at some examples of relative error calculations in real world contexts:

• A bathroom scale shows your weight as 165 pounds. Your weight at the doctor’s office yesterday was 172 pounds. The relative error of the scale is:

  • Absolute Error: |172 – 165| = 7 lb
  • Relative Error: (7/172) x 100% = 4.1%

• You measure the length of a bolt as 3.25 inches. The packaging states the bolt length is 3.30 inches. The relative error is:

  • Absolute Error: |3.30 – 3.25| = 0.05 in
  • Relative Error: (0.05/3.30) x 100% = 1.5%

• A GPS unit measures the distance between two points as 9.41 miles. A road atlas says the distance is 9.34 miles. The relative error is:

  • Absolute Error: |9.34 – 9.41| = 0.07 mi
  • Relative Error: (0.07/9.34) x 100% = 0.75%

When Relative Error Is Not Appropriate

While relative error is very useful in many cases, it does have some limitations. Times when it may not be appropriate include:

• The true value is zero or near zero. Dividing by zero or a tiny number will skew your calculation.

-Measurements are on an interval or ratio scale like temperature in °F or °C. The zero point is arbitrary so relative error is meaningless.

-You want to compare total error between measurements, not error percentages. Use absolute error.

-You are calculating error margins for a statistical analysis like a confidence interval. Use standard error instead.

-The measurement errors are asymmetric around a true value. Relative error assumes symmetric error.

Being able to properly calculate and interpret relative error is a valuable skill for engineers, lab technicians, data analysts, and scientists of all kinds. By expressing error relative to the true value, you can better understand the quality, precision, and meaning of your measurements. With the simple formula, step-by-step process, and real world examples provided in this guide, you should have a solid foundation for applying relative error calculations in your own work. Whether you are taking experimental measurements, assessing instruments and processes, or analyzing large datasets, relative error provides an indispensable metric for quantifying accuracy.