Percentage means out of 100 or per 100. It is expressed as a fraction (decimal) of 100. Percentage comes from the latin word Per centum, centum meaning 100.

As the measurement instruments are never intact. This will lead to lot of errors. This error is calculated using percentage error, as it is easier to estimate the possible error in measuring.

Percentage error is all about comparing a guess or estimate to an exact value. To calculate percentage error you need to know the approximate value as well as the exact value.

## Definition

Percentage error is defined as the difference between approximate value and observed value, as a percentage of the observed value.

It helps in comparing an estimate to an observed value.  The "|" symbols is used for absolute value, where negative turns to positive.

## Formula

Given below is the formula to calculate percentage error:

Percentage Error = $\frac{Approximate\ value - observed\ value|}{|Approximate\ value|}$ $\times$ 100
Percentage error can never be negative. It is always positive.

## How to Calculate

There are very simple steps to calculate percentage error. Below you can find them to calculate the percentage error:
1) Find the error occurred in a given problem (Subtracting one value from the other) ignore the minus sign, if any.

2) Divide the error obtained in first step by the exact value. The solution will be in terms of a decimal number.

3) 0.0 The answer should be in terms of percentage. So multiply the solution obtained in the above step by 100 and add a "%" sign.

## Examples

Given below are some examples based on percentage error:

Example 1: Calculate the percentage error if the approximate value and observed value is 45 and 35 respectively?

Solution:

Given: Approximate value = 45,
Observed value = 35

Percentage Error = $\frac{|Approximate\ Value - Observed\ Value|}{Observed\ Value|}$ $\times$ 100 %

= $\frac{|45 - 35|}{|35|}$

= $\frac{|10|}{|35|}$ $\times$ 100 %

= 28.57 %
Percentage error is found to be 28.57%.

Example 2: A measuring scale wrongly  measures an value as 9 cm due to some marginal errors. Calculate the percentage error if the actual measurement of the value is 15.3 cm?

Solution:
Given: Approximate value = 9cm

Observed value = 15.3 cm

Percentage Error = $\frac{|Approximate\ Value - Observed\ Value|}{Observed\ Value|}$ $\times$ 100 %

= $\frac{|9 - 15.3|}{|15.3|}$

= $\frac{|-63|}{|15.3|}$ $\times$ 100 %

= 41.18 %
Percentage error is found to be 41.18%.

Example 3: According to a study in a report it was mentioned that the car parking held 5600 cars, but after careful investigation it was found that there were 5650 parking spaces. What is the percentage error?

Solution: Given Approximate Value : 5600 cars
Observed Value = 5650 cars

Percentage Error = $\frac{|Approximate\ Value - Observed\ Value|}{Observed\ Value|}$ $\times$ 100 %

= $\frac{|5600 - 5650 |}{|5650|}$

= $\frac{|-50|}{|5650|}$ $\times$ 100 %

= 0.885 %
For the given problem, percentage error is found to be 0.885%.

## Practice Problems

Given below are questions on percentage error. Kindly solve this as this will help you in having a clear understanding of the topic.

1) For a rock concert 7400 people were expected to turn up for a concert, but in fact there were around 8100 people. Find the percentage error.

2) According to weather forecast it was predicted that in Paris there will be 30 mm of rain, inturn there was only 20 mm of rain.  Find the percentage error.