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# Percentage Relative Error

The percent of error of a measurement is the relative error expressed as a percent.
Percent of error can be used to compare different measurements because, being a percent, it compare each error in terms of 100.
Relative error compares the size of the error to the size of the object being measured. When relative error given as a percent, it referred to as percent error.

Percentage Relative Error = $\frac{Absolute Error}{True Value}$ x 100

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## Relative Error and Absolute Error

### Relative Error:

Relative error is generally defined by dividing absolute error with its true value; it is then represented in terms of percentage to find out the difference between absolute error and true value.

Relative error which is calculated by dividing the absolute value by the given value, it is calculated as shown below

Relative error = |V – Vapproximate| / |V| or | (V – Vapproximate ) / V |

Absolute error:

Absolute error is used to identify the exact error that has been occurred in an approximation, This can be calculated using the formula

Absolute error = Vabsolute = | V – Vapproximate|

• V is some value which we use in the solution.
• V – Vapproximate gives us the absolute or exact value (observed - exact value).

### Solved Example

Question: Find the relative error when, True value = 200 & Approximate value = 198.5
Solution:

Given,
True value = 200
Approximate value = 198.5

Relative error = $\frac{absolute\ error}{true\ value}$

= $\frac{(true\ value – approximate\ value)}{true\ value}$

= $\frac{(200 – 198.5)}{50}$

= $\frac{1.5}{50}$

Relative error = 0.03

Percentage relative error = 0.03 X 100 = 3 %

## Relative Error Formula

The formula for Relative error can be written as,

Relative error = $\frac{Absolute error}{Accepted value}$ x 100

As we know absolute error = true value - approximate value

This is also called as relative error equation. Relative error an be calculated with the above given formula.

## Percentage Relative Error Problems

Below are some examples based on percentage relative error

### Solved Examples

Question 1: True value = 100
Approximate value = 97.5
Solution:

Relative error = $\frac{absolute\ error}{true\ value}$

= $\frac{(true\ value – approximate\ value)}{true\ value}$

= $\frac{(100 – 97.5)}{100}$

= $\frac{2.5}{100}$

Relative error = 0.025

Percentage relative error = 0.025 X 100 = 2.5%

Question 2: True value = 30
Approximate value = 29.7
Solution:

Relative error = $\frac{absolute\ error}{true\ value}$

= $\frac{(true\ value – approximate\ value)}{true\ value}$

= $\frac{(30 – 29.75)}{30}$

= $\frac{0.25}{30}$

Relative error = 0.0083

Percentage relative error = 0.0083 X 100 = 0.83%

Question 3: True value = 40
Approximate value = 38.65
Solution:

Relative error = $\frac{absolute\ error}{true\ value}$

= $\frac{(true\ value – approximate\ value)}{true\ value}$

= $\frac{(40 – 38.65)}{40}$

= $\frac{1.35}{40}$

Relative error = 0.3375

Percentage relative error = 0.3375 X 100 = 33.75%

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