Percentage Error

As the name suggests, per cent 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 per cent error you made in your guess. 

Per cent 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 per cent error and is given below:

Per cent error = (Approximate or experimental Value - Exact or known Value/Exact or known Value)∗100

If the per cent 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 per cent 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?

The following steps need to be taken in order to calculate per cent 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 per cent error.

Solved Examples on Percent Error

We have here a few diverse examples on calculating per cent 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 per cent error in the guess value of organizers.

The formula for percent error =

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

Putting the above values we get; % error = \[\frac{\mid 90-120\mid }{120}\ast 100 = \frac{30}{120}\ast100\] = 25%

2. Ole Rømer was a Danish astronomer who observed that depending on the distance of Jupiter from Earth, the periods of 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 = \[\frac{\mid2,20,000-299,800 \mid }{299,800}\ast \] 100 = 26.62% 

Method for Finding Per Cent Error

It is quite simple to find per cent error. Students need to know a few important things for finding a per cent error. They must know the estimated value and original value to find the per cent error.

First, they have to find the difference between the estimated value and the original value. The value could be negative or positive. Students can ignore the negative sign. They have to subtract the original value from the estimated value.

After finding the difference students will divide the difference with the original value and multiply with one hundred to get the per cent value. This is the way to find the per cent error for any experiment. 

It is quite useful for students in different fields. Therefore, students must understand the formula and the method of calculating the per cent error. Vedantu provides the best information on per cent error. Students can visit the Vedantu website to get the required definition, formula and examples related to the per cent error. This can help students to prepare well for their exams.

Solved Examples for Percent Error

A few solved examples are given here that can help students to understand the type of questions asked in the exam related to the per cent error and they can also understand the method for finding the per cent error.

1. A man installed a stall and thought that daily 100 people would visit the stall but only 80 turned up every day. Calculate the per cent error.

Solutions:  Students have to apply the formula:

Estimated value:  100

Original value: 80

Percent error= Estimated value- original value

      Original value 

        100-80

              80

                                ¼ x 100= 100 /4 = 25%

The Benefits of Finding Per Cent Errors

There are numerous benefits of finding the per cent errors. A few benefits of finding per cent errors are given here:

Per cent error is important to know the accuracy. Accuracy means the degree of closeness of a measured value to its original value. The per cent error is calculated by dividing the difference of the estimated value and the original value by the original value and multiplying it with 100. 

The most important benefit of finding a per cent error is to know how close you are to the true value. The per cent error could be as low as negligible or could be very high depending on your observations. Thus, if the per cent error is very low you can neglect it but if the per cent error is high you have to calculate or measure the things again to get the absolute value. 

Few Worked Examples:

1. It is estimated that the distance to the moon is 235,755 miles on a particular day. But, the actual distance is found to be 250,655 miles. Calculate the per cent error.

Ans. The percent error can be calculated as:

235,755 - 250,655/ 250,655 = 0.059 x100 = 5.9%

2. John was planning a hiking trip with friends. He estimated the height of the hiking trail to be 215 ft/mile. But, when he went with his friends he found the actual height of the trail was 230 ft/mile. What was the per cent error in  John’s calculation?

Ans. \[\frac{215-230}{230} =\frac{15}{230}\] = 0.065 x 100 = 6.5%

3. A school organized a fest that was open for all. Teachers and students estimated that 1000 people will visit every day. But, the actual number of people visiting the fest was 1050. Calculate the per cent error.

Ans. \[\frac{1000-1050}{1050} =\frac{50}{1050}\] = 0.047 x 100 = 4.7%

4. A man wanted to prepare a square lawn in front of his house. He estimated it to cover 450 square meters of area. But, when he started digging for the garden the actual area to be covered was 470 meters square. Calculate the per cent error.

Ans.  \[\frac{450-470}{470} =\frac{20}{470}\] = 0.042 x 100 = 4.2%

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: 

\[\frac{\text{Estimated or approximate value - known or exact value}}{\text{known or exact value}}\ast\] 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.

When we do the analysis we can make errors. Per cent errors help us to determine our errors when we measure something. If the percent error is small it means that we have calculated close to the exact value. For example, if the percent error is only 2% it means that we are very close to the original value but if the percent error is big that is up to 30% it means we are very far off from the original value. Measurement errors are common due to different reasons.  Some of the reasons for percent errors are given here:

Percent errors can occur due to imprecise materials available. Sometimes, people doing an experiment do not have proper materials available with them that can lead to percent error. 

Errors can also occur due to improper instruments that are available for calculations as the instrument available may not have the capacity to measure a particular item exactly.