Absolute and Relative Error

Introduction

Committing Error is inevitable. There are no perfect measurements of values, especially when comparing these to actual sizes. Knowing this, we should not fear mistakes because coming up with different values doesn’t necessarily mean you are wrong. It is just a matter of accuracy.

By knowing errors, we can find ways to correct them. In this article, you will learn how to deal with errors in a mathematical sense. 

What is Absolute Error?

Definition

Absolute Error is the variance between a quantity’s actual value and the measured value. It measures how far-off a measurement is from the true value given.

How to Calculate Absolute Error?

To calculate the absolute error value, you have to subtract the measured value from the actual value.

Here is the formula to help you calculate absolute Error:

Absolute Error Formula

AE = AV – MV

where; 

AE = Absolute Error
AV = Actual Value
MV = Measured Value 

What are the Classifications of Absolute Error?

Absolute Error can be classified into the Absolute Accuracy Error and the Mean Absolute Error. Absolute Accuracy Error is the other term for Absolute Error. It has the same formula as the Absolute Error, which is the Actual Value subtracted by the Measured Value. On the other hand, the Mean Absolute Error is the mean or average value of all the measured Absolute Errors. To calculate the Mean Absolute Error, you must first calculate all the Absolute Errors. After that, you will add the computed values and divide by how many absolute errors were. In formula,

MAE=$\frac{sum\: of\: all\: absolute\: errors}{number\: of\: absolute\: errors}$

where,

MAE = Mean Absolute Error

What is Relative Error?

Definition

Relative Error compares how significant the difference between the absolute Error and the actual value is. It is expressed in percentages, giving us a clear vision of the ratio between the two.

How to Calculate Relative Error?

To calculate the relative Error, you have to subtract the actual value by the measured value, divide it by the actual value, and multiply by 100.

Here is the formula to help you calculate relative Error:

Relative Error Formula

RE = $\frac{AV – MV}{AV} x 100$

where; 

RE = Relative Error
AV = Actual Value
MV = Measured Value

Absolute Error vs. Relative Error: Differences and Calculations

Absolute Error gives us the quantity of Error, while Relative Error provides us with the degree of accuracy of the two quantities. Absolute Error is expressed in the unit used in the amount measured, while Relative Error has no unit since it is expressed in percentages. Dividing two values of the same unit cancels the unit used.

What is the Use of Absolute and Relative Error?

Determining the Absolute and Relative Errors can help improve the analysis of measurements. By knowing these errors, we will be guided if the number of mistakes calculated is negligible. For example, an error discrepancy can be considered insignificant if the unit used is millimeters compared to a meter. For instance, we are measuring distances from one place to another.

As seen in the photo, 1m (1 meter) distance away from the first house has a considerable value. At the same time, 1mm (1 millimeter) distance is too small of a value that can be considered negligible. Knowing this, we can evaluate if the Error computed greatly impacts the actual measurement.

Absolute and Relative Error Examples

The following are examples of word problems involving absolute Error.

Example 1:

Sham was given three of the grades of her classmates. These grades were the three closest values to Sham’s grade. If Sham’s grade is considered the actual value, what is the mean absolute error of her classmates’ grades?

Sham’s Grade: 96.3

Classmate 1: 95.6
Classmate 2: 94.8
Classmate 3: 95.2

Solution:

The absolute error of each of the classmates is equal to,

AE = AV – MV
AE1 = 96.3 – 95.6
AE1 = 0.7

AE2 = 96.3 – 94.8
AE2 = 1.5

AE3 = 96.3 – 95.2

AE3 = 1.1

The mean or average of the absolute errors is equal to,

MAE = (0.7 + 1.5 + 1.1) / 3
MAE = 1.1

Therefore, the Mean Absolute Error is 1.1

Example 2: 

Find the absolute and relative errors. The actual value is 125.68 mm, and the measured value is 119.66 mm.

Solution:

Absolute Error is given,

AE = AV – MV

AE = 125.68mm – 119.66mm

AE = 6.02mm

Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{125.68mm – 119.66mm}{125.68mm} x 100$

RE = 0.0479 x 100

RE = 4.79%

Therefore, the Absolute Value is 6.02mm with a Relative Error of 4.79%.

Example 3: 

The book’s length is 12.5cm, but Via measured only 12.4cm. Find the absolute and relative errors.

Solution:

Absolute Error is given,

AE = AV – MV

AE = 12.5cm – 12.4cm

AE = 0.1cm

Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{12.5cm – 12.4cm}{12.5cm} x 100$

RE = 0.008 x 100

RE = 0.8%

Therefore, the Absolute Value is 0.1cm with a Relative Error of 0.8%.

Example 4: 

The thermometer measures up to the nearest 2 degrees. The temperature was measured at 38° C. Find the relative Error.

Solution:

The actual value is not given, but we are given 2 degrees as the values change. That means that the value could be 37° C or 39° C. Either way, we are sure that the absolute Error is 1° C. And so the Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{AE}{AV} x 100$

RE = $\frac{1}{38} x 100$

RE = 0.0263 x 100

RE = 2.63%

Therefore, the Relative Error is 2.63%.

Example 5: 

Find the absolute and relative errors. The actual value is 56.5 mm, and the measured value is 51.2 mm.

Solution:

Absolute Error is given,

AE = AV – MV

AE = 56.5mm – 51.2mm

AE = 5.3mm

Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{56.5mm – 51.2mm}{56.5mm} x 100$

RE = 0.0938 x 100

RE = 9.38%

Therefore, the Absolute Value is 5.3mm with a Relative Error of 9.38%.

Example 6: 

We are given an approximate value of π is 22/7 = 3.1428571, and the actual value is 3.1415926. Calculate the absolute and relative errors.

Absolute Error is given,

AE = AV – MV

AE = 3.1415926 – 3.14428571

AE = -0.00269311

We can observe here that the value is negative. But, since it is absolute, it will become positive, giving the value 0.00269311.

Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{3.1415926 – 3.14428571}{3.1415926} x 100$

RE = 0.00085724 x 100

RE = 0.08572436%

Therefore, the Absolute Value is 0.00269311 with a Relative Error of 0.08572436%.

Example 7: 

Let the approximate values of a number 1/3 be 0.30, 0.33, 0.34. Find out the mean absolute Error.

Solution:

The absolute Error of each approximate value is equal to,

AE = AV – MV

AE1 = 1/3 – 0.30

AE1 = 0.0333

AE2 = 1/3 – 0.33

AE2 = 0.0033

AE3 = 1/3 – 0.34

AE3 = -0.0067 = 0.0067

The mean or average of the absolute errors is equal to,

MAE = (0.033 + 0.0033 + 0.0067) / 3

MAE = 0.014

Therefore, the Mean Absolute Error is 0.014.

Example 8: 

A 12m x 8m table was measured as 11.8m x 7.9m. Find the absolute and relative errors.

Solution:

First, we need to calculate the areas with the formula L x W (Length x Width),

Actual Area = 12m x 8m

Actual Area = 96m²

Measured Area = 11.8m x 7.9m

Measured Area = 93.22m²

Absolute Error is given,

AE = AV – MV

AE = 96m – 93.2m

AE = 2.78m²

Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{96m – 93.22}{96m} x 100$

RE = 0.02895 x 100

RE = 2.895%

Therefore, the Absolute Value is 2.78m² with a Relative Error of 2.895%.

Example 9: 

Let the approximate values of a number 1/4 be 0.25, 0.249, 0.246. Find out the mean absolute Error.

Solution:

The absolute Error of each approximate value is equal to,

AE = AV – MV

AE1 = 1/4 – 0.25

AE1 = 0

AE2 = 1/4 – 0.249

AE2 = 0.001

AE3 = 1/4 – 0.246

AE3 = 0.004

The mean or average of the absolute errors is equal to,

MAE = (0 + 0.001 + 0.004) / 3

MAE = 0.0167

Therefore, the Mean Absolute Error is 0.0167.

Example 10: 

Anna was tasked to find the absolute and relative Error of the measurements of a box. She measured 4.95m x 3.64m x 2.11m of a 40m³.

Solution:

First, let us calculate the volume based on Anna’s measured values. The formula for volume is given by L x W x H (Length x Width x Height).

Volume = L x W x H

Volume = 4.95m x 3.64m x 2.11m

Volume = 38.018m³

Absolute Error is given,

AE = AV – MV

AE = 40m³ – 38.018m³

AE = 1.982m³

Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{40 – 38.018}{40} x 100$

RE = 0.0495 x 100

RE = 4.95%

Therefore, the Absolute Value is 1.982m³ with a Relative Error of 4.95%.

Example 11: 

We are given an approximate value of 1/8 is 0.1259, and the actual value is 0.125. Calculate the absolute and relative errors.

Solution:

Absolute Error is given,

AE = AV – MV

AE = 0.125 – 0.1259

AE = -0.0009

We can observe here that the value is negative. But, since it is absolute, it will become positive, giving the value 0.0009.

Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{0.125 – 0.1259}{0.125} x 100$

RE = 0.0072 x 100

RE = 0.72%

Therefore, the Absolute Value is 0.0009 with a Relative Error of 0.72%.

Example 12: 

Keith was given three sets of volumes to solve and determine which volume was the closest to the actual value 8.88m x 7.45m x 7.17m, using the absolute and relative errors.

1^st Set: 6.92m x 5.96m x 4.12m
2^nd Set: 9.33m x 1.53m x 6.32m
3^rd Set: 6.88m x 7.17m x 4.49m

Solution:

Let us find the volume. The formula for volume is given by L x W x H (Length x Width x Height).

1^st Set: 

Volume1 = 6.92m x 5.96m x 4.12m
Volume1 = 169.92m³

2^nd Set:

Volume2 = 9.33m x 1.53m x 6.32m
Volume2 = 90.21m³

3^rd Set:

Volume3 = 6.88m x 7.17m x 4.49m
Volume3 = 221.49m³

True Value:

Volume = 8.88m x 7.45m x 7.17m
Volume = 474.34m³

The absolute and relative Error of each volume is equal to,

AE = AV – MV and RE = $\frac{AV – MV}{AV}$ x 100

AE1 = 474.34m³ – 169.92m³

AE1 = 304.42m³

RE1 = $\frac{AV – MV}{AV} x 100$

RE1 = $\frac{474.34 – 304.42}{474.34} x 100$

RE1 = 35.82%

AE2 = 474.34m³ – 90.21m³

AE2 = 384.13m³

RE2 = $\frac{AV – MV}{AV} x 100$

RE2 = $\frac{474.34 – 90.21}{474.34} x 100$

RE2 = 80.98%

AE3 = 474.34m³ – 221.49m³

AE3 = 252.85m³

RE3 = $\frac{AV – MV}{AV} x 100$

RE3 = $\frac{474.34 – 221.49}{474.34} x 100$

RE3 = 53.31%

Therefore, the 1^st Set has the closest value to the actual value since it has the lowest relative error percentage of 35.82%.

Example 13: 

Find the absolute and relative errors. The actual value is 100.1 mm, and the measured value is 99.1 mm.

Solution:

Absolute Error is given,

AE = AV – MV

AE = 100.1mm – 99.1mm

AE = 1mm

Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{100.1mm -99.1mm}{100.1mm} x 100$

RE = 0.00999 x 100

RE = 0.999%

Therefore, the Absolute Value is 1mm with a Relative Error of 0.999%.

Example 14: 

Let the approximate values of a number 3/4 be 0.75, 0.759, and 0.746. Find out the mean absolute Error.

Solution:

The absolute Error of each approximate value is equal to,

AE = AV – MV

AE1 = 3/4 – 0.75

AE1 = 0

AE2 = 3/4 – 0.759

AE2 = – 0.009 = 0.009

AE3 = 3/4 – 0.746

AE3 = 0.004

The mean or average of the absolute errors is equal to,

MAE = (0 + 0.009 + 0.004) / 3

MAE = 0.0043

Therefore, the Mean Absolute Error is 0.0043.

Example 15: 

The weighing scale measures up to the nearest 3 kilograms. The log was measured as 48 kilograms. Find the relative Error.

Solution:

The actual value is not given, but we are given 3 kilograms as the values change. That means that the value could be 46.5kg or 49.5kg. Either way, we are sure that the absolute Error of 3kg. And so the Relative Error is given,

RE = $\frac{AV – MV}{AV} x 100$

RE = $\frac{AE}{AV} x 100$

RE = $\frac{3}{48} x 100$

RE = 0.0625 x 100

RE = 6.25%

Therefore, the Relative Error is 6.25%.

Summary

• Definition and Formulas

Absolute Error is the variance between a quantity’s actual value and the measured value. It measures how far-off a measurement is from the true value given.

The formula for Absolute Error is,

AE = AV – MV

where; 

AE = Absolute Error
AV = Actual Value
MV = Measured Value 

It can be classified into the Absolute Accuracy Error and the Mean Absolute Error. The formula for Mean Absolute Error is,

MAE=$\frac{sum\: of\: all\: absolute\: errors}{number\: of\: absolute\: errors}$

where,

MAE = Mean Absolute Error

Relative Error compares how significant the difference between the absolute Error and the actual value is. It is expressed in percentages, giving us a clear vision of the ratio between the two.

The formula for Absolute Error is,

RE = $\frac{AV – MV}{AV} x 100$

where; 

RE = Relative Error
AV = Actual Value
MV = Measured Value

• Absolute Error vs. Relative Error

Absolute Error gives us the quantity of Error, while Relative Error provides us with the degree of accuracy of the two quantities. Absolute Error is expressed in the unit used in the amount measured, while Relative Error has no unit since it is expressed in percentages. Dividing two values of the same unit cancels the unit used.

• The Uses of Absolute and Relative Error

Determining the Absolute and Relative Errors can help improve the analysis of measurements. By knowing these errors, we will be guided if the number of mistakes calculated is negligible.

Frequently Asked Questions (FAQs)

1. What is Absolute Error?

– Absolute Error is the variance between a quantity’s actual value and the measured value. It measures how far-off a measurement is from the actual value given.

2. What is a Relative Error?

– Relative Error compares how significant the difference between the absolute Error and the actual value is. It is expressed in percentages, giving us a clear vision of the ratio between the two.

3. What is the formula for Absolute Error?

– AE = AV – MV

where; 

AE = Absolute Error
AV = Actual Value
MV = Measured Value 

4. What is the formula for Relative Error?

– RE = $\frac{AV – MV}{AV} x 100$

where; 

RE = Relative Error
AV = Actual Value
MV = Measured Value

5. What are the two classifications of Absolute Error?

– The two classifications of Absolute Error are the Absolute Accuracy Error and the Mean Absolute Error

6. What is the formula for Mean Absolute Error?

– MAE=$\frac{sum\: of\: all\: absolute\: errors}{number\: of\: absolute\: errors}$

where,

MAE = Mean Absolute Error

7. What are the uses of Absolute Error and Relative Error?

– Determining the Absolute and Relative Errors can help improve the analysis of measurements. By knowing these errors, we will be guided if the number of errors calculated is negligible.

8. What is the difference between Absolute Error and Relative Error?

– Absolute Error gives us the quantity of Error, while Relative Error provides us with the degree of accuracy of the two quantities. Absolute Error is expressed in the unit used in the amount measured, while Relative Error has no unit since it is expressed in percentages. Dividing two values of the same unit cancels the unit used.

9. Why is relative Error better than absolute Error?

– The absolute Error gives how significant the Error is, while the relative Error indicates how critical the Error is to the correct value.

10. Is relative Error always smaller than absolute Error?

– The relative Error using the ruler is smaller than the relative Error using the measuring tape because the absolute Error of the ruler is smaller than the absolute Error of the measuring tape.

11. Why is absolute error always positive?

– Absolute Error is always positive. Absolute Error is the magnitude of the difference between the measured value while doing the experiment measurement and the actual value. Since it is a magnitude, it will always be positive.

12. Can the mean absolute error be negative?

– Absolute Error may be negative or positive.

13. Does Absolute Error have units?

– Yes, Absolute errors have the same units as the quantities measured.

14. Does Relative Error have units?

– No, Relative Error has no units since the units will be canceled when you divide two quantities of the same units.

15. What is the purpose of mean absolute Error?

– Mean Absolute Error allows us to compare forecasts of different series on different scales.

16. How do you interpret absolute Error?

– It is the difference between the measured value and the “true” value.

17. How do you find relative Error when the real value is zero?

– Relative Error is undefined when the actual value is zero. Also, relative Error only makes sense when a measurement scale starts at a true zero.

18. How can errors be minimized?

– Systematic errors can be minimized by improving experimental techniques, selecting better instruments, and removing personal bias as far as possible.

19. What are the most common reasons for Errors in numerical analysis

– The most common reasons for Errors in numerical analysis are rounding off and truncation.

20. How is the absolute Error related to accuracy?

– Absolute Error is the absolute value of the difference between the measured value and the actual value of a measurement. It is usually given as the maximum possible error given a measuring tool’s degree of accuracy.

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