Percentage Error

As the name suggests, percent error is the difference between the exact or known value of something and its approximate or measured value, in percentage form. In scientific experiments, it is used to report the difference between the experimental value to its true or exact value. It is calculated as the percentage of the exact value. As a real-world example, if you look at a gumball machine and make an estimate of how many gumballs are there and then you actually go ahead and calculate the number of gumballs, then you will be able to measure the percent error you made in your guess. 


Percent Error Formula

Percent error lets you see how far off you are in estimating the value of something from its exact value. These errors could happen due to the imprecision of equipment, measurement (human error or tool error), or some adjustments done in calculation methods (rounding off, etc.). There is a simple and straightforward formula for calculating this percent error and is given below:


Percent error = \[\frac{\text{(Approximate or experimental Value - Exact or known Value}}{\text{Exact or known Value)}} * 100\]


If the percent error is close to 0, then your approximation is very close to the actual or true value. This formula is very important to determine the precision of your calculations. For most applications, the percent error is represented as a positive number, but for some sciences like chemistry, it is customary to express it as a negative number since a positive value in chemistry would point to a potential problem with the experiment or reactions which are not accounted for.


How to Calculate Percent Error

The following steps need to be taken in order to calculate percent error in any experiment or observation:

  • You get the “error” value by subtracting one value from another. If you are not keeping the sign, then the order does not matter, but if you are keeping a negative sign, then you get the “error” value by subtracting the exact value from the measured value. 

  • You then divide this “error” value by the known or exact value (not your measured or experimental value). This division will give you a decimal number.

  • Multiply this decimal value with 100 to convert it into a percentage value.

  • Finally, you would add a % notation in front of the calculated value to report your percent error.


Solved Examples on Percent Error

We have here a few diverse examples on calculating percent errors to delve into the concept and get more clarity:


1. In a concert, it was estimated by the organizers that 90 people would show up but in fact, 120 people came to the concert. Calculate the percent error in the guess value of organizers.


The formula for percent error = \[\frac{\text{(Approximate or experimental Value - Exact or known Value}}{\text{Exact or known Value)}} * 100\]


Putting the above values we get; % error = \[\frac{|90 - 120|}{120} * 100\] = \[\frac{30}{120} * 100\] = 25%.


2. Ole Rømer was a Danish astronomer who observed that depending on the distance of Jupiter from Earth, the periods of the Jupiter’s satellites seemed to fluctuate.  The satellites took longer to appear from behind the planet if Jupiter was further away from Earth than otherwise. He related this to the speed of light and gave an approximate value, 220,000 km/s, for the velocity of light. The accepted value of the speed of light currently is 299,800 km/s. What percent error did Rømer's observation have?


% error = \[\frac{|220,000 - 299,800|}{299,800} * 100\] = 26.62%


Conclusion

Percent error is the difference between a measured value and the exact value of any quantity under observation. It is calculated as the percentage of the exact or known value. One can calculate its value by the formula: 


\[\frac{\text{Estimated or approximate value - known or exact value}}{\text{known or exact value}} * 100\]


The sign of the percent error is not considered in most applications except in chemistry and some other sciences where it is customary to keep a negative sign. Percent error is a type of error calculation. Few other types of common error calculations are relative error and absolute error.

FAQ (Frequently Asked Questions)

1. What is the Formula for Percent Error and How is it Calculated?

Percent error is determined by the difference between the exact value and the approximate value of a quantity, divided by the exact value and then multiplied by 100 to represent it as a percentage of the exact value.


Percent error = |Approximate value – Exact Value|/Exact value * 100.

2. What is Relative Error and How is it Different From Percent Error?

The relative error is the difference between the known and measured value divided by the known value. When this is multiplied by 100 it becomes a percent error. Hence:


Relative error = |Estimated or approximate value - Exact Value|/Exact value.


Percent error = |Approximate value – Exact Value|/Exact value * 100.

3. What is Absolute Error and How is it Different From Percent Error?

Absolute error is just the difference between the known and measured values. When it is divided by the known value and then multiplied by 100, it becomes a percent error. Hence:


Absolute error = |Approximate value – Exact Value|


Percent error = |Approximate value – Exact Value|/Exact value * 100.

4. What are the Uses of Calculating Percent Error?

Percent error is a means to gauge how accurate and close the estimate is to the exact value of any given experiment or quantity. This method lets you determine if the collection of data is progressing in the right direction or not. It is mostly used by statistics experts and corporate companies. It is also of high importance to students who want to pursue economics.

5. What are Some of the Reasons for Percent Error.

There are many reasons for a difference in the measured value from the known value, some of the common reasons for the percent error is human error, an issue with experiment, calculation error (like rounding off, etc.), random error, systematic error, the precision of the tool or the instrument used for measuring, etc.