Percentage Error Calculator

A percentage is a fraction with the denominator as 100. When we express some fraction as a percentage, we express how many parts of the 100, the fraction denotes. In other words, a percentage indicates the quantity of something in terms of 100. The symbol for percentage is %. For example, the ratio 15/16 as a percentage is 93.75%.

The simplest way of expressing a fraction (or ratio) as a percentage is A/B = P × 100, where ‘A’ is the numerator of the fraction, ‘B’ is the denominator and ‘P’ is the percentage. In a percentage calculation, the denominator denotes the total value of a quantity, and the numerator denotes some part of the total value. The percentage simply expresses this ratio in terms of 100.

The percentage error comes up in the measurements of physical quantities. When we try to experimentally measure the value of a particular quantity, we take repeated measurements. The measured quantity might differ from the true value to a certain extent. Then, the percentage error helps to find the relative error between the true value and the observed value, and expresses it in the form of a percentage. The percentage error is a measurement of the discrepancy between the true and observed value, which might occur due to human error or limitations of the instrument being used.

Let TV = true value and OV = observed value be the two known values.

The percentage error makes use of two other errors: the absolute error which is the absolute value of the difference between the true value and observed value, and the relative error which is essentially the absolute error divided by the true value.

$$ Absolute \; error \; = \; |true \; value − observed \; value| = |TV − OV| $$

$$ Relative \; error = {{absolute \; error}\over{true \;value}} = {{|true \; value − observed \; value|}\over{true \; value}} = {{|TV − OV|}\over TV} $$

Then, the formula for percentage error is $$ Percentage \; error = relative \; error × 100 = {{|true \; value − observed \; value|}\over{true \; value}} * 100 $$

$$ or, Percentage \; error = {{|TV − OV|}\over TV} * 100 $$

Suppose an experiment was performed to calculate the acceleration due to gravity (g) at a place. The observed value of g was found to be 9.582 m/s², which deviates slightly from the true value of g = 9.8 m/s². Hence, the percentage error is,

$$ Percentage \; error = {{|true \; value − observed \; value|}\over{true \; value}} * 100 $$

$$ = {{|9.8 − 9.582|}\over{9.8}}*100 = {0.218\over9.8}*100 = 2.22 \% $$

Hence, the percentage error in this case is nearly 2%.

To find the percentage error between the true and observed values of a measurement, the steps are as follows. For this, we consider an example where the true value = 150, and observed value = 151.73

     i) First, find the difference between the true value and the observed value. Take the absolute value of this difference. This is known as the absolute error. Hence, $$ Absolute \; error = |true \; value − observed \; value| = |150 − 151.73| \\ Absolute \; error = |−1.73| = 1.73 $$

     ii) The next thing we need is the relative error. This is equal to the absolute error divided by the true value. $$ Relative \; error = {{|true \; value − observed \; value|}\over{true \; value}} = {1.73 \over 150} = 0.11533 $$

     iii) Finally, multiply the relative error by 100 to get the percentage error. $$ Percentage \; error = {{|true \; value − observed \; value|}\over{true \; value}} * 100 = {1.73 \over 150} * 100 = 0.11533 × 100 = 11.533\% $$

     iv) The resulting number is written with a percentage sign and is called the percentage error. For this example, the error is 11.533%.

Percentage growth is used to show the increase in a value by comparing the initial value and final value over some time. Percentage decrease shows the decrease in a value over a period of time by comparing the initial and final values.

$$ Percentage \; growth = {{Final \; value − initial \; value} \over {|initial \; value|}} * 100 $$

$$ Percentage \; decrease = {{initial \; value − final \; value} \over {|initial \; value|}} * 100 $$

The formula varies slightly in these two cases. Moreover, we do not consider the absolute value of the difference between the two numbers.

Percentage difference, on the other hand, is used when both the values mean the same thing. Here, one value is not older than the other value, and there is no change in the values over a period of time.

$$ Percentage \; difference, \;P \; = {{|First \; value − second \; value|}\over{({{first \; value + second \; value}\over2})}} * 100 $$

In this formula, we consider the absolute value of the difference between the two numbers. So, the position of the two values in the formula does not matter. However, a change in the position of the values in the case of percentage growth (or percentage decrease, would give a completely different interpretation altogether.

Finally, percentage error is a comparison between the true value and observed value of a measurement. Here, we consider the absolute value of the difference between the true and observed values, but we divide this by the true value. Unlike percentage growth, this is not used to measure the change in quantity over a period of time. It is a measure of the errors which creep during experiments.

$$ Percentage \; error = {{|true \; value − observed \; value|}\over{true \; value}} * 100 $$

Percentage errors are primarily used to find discrepancies between the experimental values and true values after finding the measurements of some physical quantity. Consider the real-life example below.

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