Percent Error Formula

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Percentage Error Formula

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 =

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?

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

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?

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.

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.

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: 

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.

FAQs (Frequently Asked Questions)

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

Per cent 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 per cent 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 per cent 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?

Per cent 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 per cent 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.

What is Percent Error Formula?

The percent error formula calculates the absolute value of the difference between the measured value and the actual value that is divided by the actual value and multiplied by 100. The percent error formula is about comparing the estimated value with the exact value. In mathematics, the percent error is always represented as a positive number. The Percentage or percent error formula of a given measurement can be expressed using the following expression.

Percent Error = |(Experimental Value – Theoretical value) / Theoretical value| × 100

Where, ‘| |” denotes the absolute value sign

Steps to Calculate Percent Error Formula

There are a few steps to be followed while calculating the percent error of a value, they are:

  • Step 1: Using the percent error formula, subtract the experimental value from the theoretical value. If the answer is negative, it converts into positive because of the absolute value sign.
  • Step 2: Once we have obtained the value, divide it with the actual or theoretical value. We obtain a decimal number.
  • Step 3: Convert that decimal number into a percent by multiplying it by 100 and adding the percentage sign – %

Examples Using Percent Error Formula

Example 1: The radius of a nut is measured as 0.5in. The actual value of the radius of the nut was known to be 0.47in. Find the percent error in the measurement.

Solution:  Given,

Theoritical value = 0.47in, Experimental Value = 0.5in 

Using Percent Error Formula,
Percent Error = |(Experimental Value – Theoretical value) / Theoritical value| × 100

Putting the values,
Percent Error = |{(0.5- 0.47) / 0.47}| ×100
Percent Error = |{(0.5- 0.47) / 0.47}| ×100
Percent Error = |0.06| ×100
Percent Error = 6

Therefore, the percent error in the measurement is 6%.

Example 2: The price of a car was estimated to be $12995. The actual price of the car was $13000. Find the percentage error in the estimation using the percent error formula.

Solution:  Given,

Theoritical value = $13000, Experimental Value = $12995 

Using Percent Error Formula,
Percent Error = |(Experimental Value – Theoretical value) / Theoritical value| ×100

Putting the values,
Percent Error = |{(12995 -13000) / 13000}| ×100
Percent Error = |{(-5) /13000}| × 100
Percent Error = |-0.00038461| × 100
Percent Error = 0.03

Therefore, the percent error in the estimation is 0.03%.

Example 3: In a concert, 250 people were estimated to attend but 350 people attended. Using the percent error formula, calculate the percent error in the guess value by the organizers. 

Solution: Given, 

Theoritical value = 350 , Experimental Value = 250

Using Percent Error Formula,
Percent Error = |(Experimental Value – Theoretical value) / Theoritical value| ×100

Putting the values,

Percent error = |(250 – 350) / 350| × 100

Percent error = |-100 / 350| × 100

Percent error = 0.286 × 100

Percent error = 28.6

Therefore, the percent error by the organizers is 28.6%

FAQs on Percent Error

What is Meant by Percent Error?

The percent error formula calculates the absolute value of the difference between the measured value and the actual value that is divided by the actual value and multiplied by 100. The percent error formula is about comparing the estimated value with the exact value. In mathematics, the percent error is always represented as a positive number. The formula to calculate the percent error is Percent Error = |(Experimental Value – Theoretical value) / Theoretical value| × 100

What is the Formula to Find the Percent Error?

The formula to find the percent error of a value is:

Percent Error = |(Experimental Value – Theoretical value) / Theoretical value| × 100

Where, ‘||” denotes the absolute value sign.

What are the Steps Used for Percent Error Formula?

There are a few steps to be followed while calculating the percent error of a value, they are:

  • Step 1: Subtract the experimental value from the theoretical value. If the answer is negative, it converts into positive because of the absolute value sign.
  • Step 2: Divide it with the actual or theoretical value. We obtain a decimal number.
  • Step 3: Convert that decimal number into a percent by multiplying it by 100 

Using the Percent Error Formula, Calculate the Percent Error of a Car Park Could Hold 340 cars, but we counted only 250 Parking Spaces.

Given, Estimated value = 340 and Theoretical value – 250 

Using Percent Error Formula,
Percent Error = |(Experimental Value – Theoretical value) / Theoritical value| × 100

Percent error = |(340 – 250) / 250| × 100

Percent error = 36%

Therefore, the percent error is 36%

What is Meant by Percent Error?

Whenever an experiment is done, we get results that will match with the actual value or sometimes vary with the actual value. Error is the difference between the estimated value and the actual value. Measurement errors arise because of unavoidable faults in the measuring instrument and limitations of the human eye. Errors come in all sizes, and sometimes we need to decide if the error in our measurement is so big that it makes the measurement useless. The smaller the error indicates that we are close to the actual value. Therefore, scientists have devised a method to calculate the extent of error in estimation.

Percent Error Definition

Percent error is the difference between the actual value and the estimated value compared to the actual value and is expressed in a percentage format. In other words, you find the difference between the actual answer and the guessed answer, divide it by the actual answer, and express it as a percentage. Percent errors indicate how huge our errors are when we measure something. For example, a 5% error indicates that we got very close to the accepted value, while 60% means that we were quite far from the actual value.

Percent Error Formula

Percent error will let us know how much extent these unavoidable errors affect our experimental results. The formula for finding percent error:

Most of the time, the percentage error is expressed as a positive value. Absolute value can be some times termed as true value or theoretical value. The absolute value of the error is divided by an true value and shown as a percent.

Think Tank

  • The accepted distance to the moon is 238,855 miles on a particular day. You measure the distance as 249,200 miles. What is the percent error?
  • Ron is planning a hiking trip, and he estimated the gradient of the trail to be 210 ft/mile. After hiking and tracking the trail, he found that the trail was actually 202 ft/mile. What was his percent error? Did he overestimate or underestimate the gradient?

How do you Find the Percent Error?

Percentage error can be calculated using three simple steps:

  • Calculate the error (subtract estimated value from the actual value) ignore any negative (-) sign. i.e., take the absolute value of error.

Absolute Error = Approximate Value – Exact Value 

  • Divide the error by the actual value (sometimes, we may get a decimal number).

Relative Error = |Approximate Value – Exact Value|/Exact Value

  • Convert that to a percentage (by multiplying by 100 attach “%” sign) 

Percent Error = |Approximate Value – Exact Value|/Exact Value × 100%

Important Notes

  • Percent error is the difference between the actual value and the estimated value compared to the actual value and is expressed in a percentage format.
  • Percent Error = {(Actual Value – Estimated Value)/Actual Value} × 100
  • Percent errors indicate how huge our errors are when we measure something.

Solved Examples on Percent Error

Example 2: Harry got a traffic penalty notice for police speeding for traveling 70 mph in a 60 mph zone. Harry claimed his speedometer said 60 mph, not 70 mph. What could Harry claim as his percent error?

Solution:

Let us arrive at %error in 3 steps:

Absolute Error = |70−60| = 10

Percent Error = 10/60=0.1667

= 0.1667×100 = 16.67%

Harry can claim 16.67% as his percent error

Example 3: Helen’s math class had 24 students yesterday. She miscounted the class total and recorded it as 18 students. What is Helen’s percent error?

Solution:

The actual number of students: 24 and Recorded number of students: 18

Absolute Error = 24 – 18 = 6

Percent Error = 6/24 = 0.25

= 0.25 × 100 = 25%

Helen’s percent error is 25%

Example 4: The Handbook of Chemistry and Physics lists the density of a certain liquid to be 0.7988 units. Daniel experimentally finds this liquid to have a density of 0.7925 units. The teacher allows up to +/- 0.500% error to make an “A” in the lab. Did Daniel make an “A”? Prove your answer. 

Solution:

Given, Theoretical value of density of a liquid: 0.7988 units and Daniel’s Experimental value of the density: 0.7925 units

Percent Error = {(Actual Value – Estimated Value)/Actual Value} × 100

% Error = (0.7988−0.7925)/0.7988 × 100

= 0.0063/0.7988 × 100

= 0.788%

Daniel got a 0.788% error. But, the teacher allows only a 0.5% error.

So, Daniel could not make an “A”.

FAQs on Percent Error

Can you have a Negative Percent Error?

If the experimental value is less than the accepted value, then the percent error is negative. Generally, the error is calculated as the absolute difference to avoid the confusion of a negative error.

What does a Percent Error tell you?

Percent error tells us how much extent few unavoidable errors affect our experimental results. It is measured by taking the difference between the actual value and the observed value. Small percent errors indicate that you are close to the accepted or real value.

Percent errors tell how big your errors are when you measure something in an experiment. 

Is High Percent Error Good or Bad?

A 5% error indicates that we got very close to the accepted value, while 60% means that we were quite far from the actual value. So, a high percent error is bad.

How do you Decrease the Percent Error?

By increasing accuracy, precision and taking the measurements under controlled conditions, we can decrease the percent error.

What is the Difference Between Percent Error and Percent Difference?

Percentage difference is the measurement of the percentage change in the initial and final quantities in an experiment, while percent error shows us the measurement of the discrepancy between an observed and a true or accepted value.

The percent error is the absolute value of the difference divided by the “correct” value times 100.

Is it Possible to have a Percent Error over 100?

Yes, a percent error of over 100% is possible. A percent error of 100% is obtained when the experimental value is twice the true value’s value. In experiments, it is always possible to get values that are way greater or lesser than the true value due to human or experimental errors.

How do you Calculate Percent Error and Absolute Error?

The computation of percentage error involves the absolute error, which is simply the difference between the observed and the true value. The absolute error is then divided by the true value, resulting in the relative error multiplied by 100 to obtain the percentage error.

Solved Example

Example 1: A scale measures wrongly a value as 8 cm due to some marginal errors. Calculate the percentage error if the actual measurement of the value is 12 cm.

Solution:

Given,
Approximate value = 8 cm
Exact value = 12 cm

Percentage Error = (Approximate Value – Exact Value / Exact Value × 100)

Percentage Error = (8 – 12)/12 × 100

= -33.3%

Comparing Approximate to Exact

First find the Error:
Subtract one value from the other. Ignore any minus sign.

Example: I estimated 260 people, but 325 came.
260 − 325 = −65, ignore the “−” sign, so my error is 65

Then find the Percentage Error:
Show the error as a percent of the exact value, so divide by the exact value and make it a percentage:

Example continued: 65/325 = 0.2 = 20%

Percentage Error is all about comparing a guess or estimate to an exact value. See percentage change, difference and error for other options.

How to Calculate

Follow these steps:

Step 1: Calculate the error (subtract one value from the other) ignore any minus sign.
Step 2: Divide the error by the exact value (we get a decimal number)
Step 3: Convert that to a percentage (by multiplying by 100 and adding a “%” sign)

As A Formula

This is the formula for “Percentage Error”:

We can also use a theoretical value (when it is well known) instead of an exact value.

Without “Absolute Value”

We can also use the formula without “Absolute Value”. This can give a positive or negative result, which may be useful to know.

Percent Error Example Calculation

In a lab, you are given a block of aluminum. You measure the dimensions of the block and its displacement in a container of a known volume of water. You calculate the density of the block of aluminum to be 2.68 g/cm3. You look up the density of a block of aluminum at room temperature and find it to be 2.70 g/cm3. Calculate the percent error of your measurement.

  1. Subtract one value from the other:2.68 – 2.70 = -0.02
  2. Depending on what you need, you may discard any negative sign (take the absolute value): 0.02This is the error.
  3. Divide the error by the true value:0.02/2.70 = 0.0074074
  4. Multiply this value by 100% to obtain the percent error:
    0.0074074 x 100% = 0.74% (expressed using 2 significant figures).Significant figures are important in science. If you report an answer using too many or too few, it may be considered incorrect, even if you set up the problem properly.

Percent Error Formula

When you calculate results that are aiming for known values, the percent error formula is useful tool for determining the precision of your calculations.  The formula is given by:

The experimental value is your calculated value, and the theoretical value is your known value.  A percentage very close to zero means you are very close to your targeted value, which is good.  It is always necessary to understand the cause of the error, such as whether it is due to the imprecision of your equipment, your own estimations, or a mistake in your experiment.

Example:  

The 17th century Danish astronomer, Ole Rømer, observed that the periods of the satellites of Jupiter would appear to fluctuate depending on the distance of Jupiter from Earth.  The further away Jupiter was, the longer the satellites would take to appear from behind the planet.  In 1676, he determined that this phenomenon was due to the fact that the speed of light was finite, and subsequently estimated its velocity to be approximately 220,000 km/s.  The current accepted value of the speed of light is almost 299,800 km/s.  What was the percent error of Rømer’s estimate?

Solution:

experimental value = 220,000 km/s = 2.2 x 108 m/s

theoretical value = 299,800 km/s 2.998 x 10m/s

So Rømer was quite a bit off by our standards today, but considering he came up with this estimate at a time when a majority of respected astronomers, like Cassini, still believed that the speed of light was infinite, his conclusion was an outstanding contribution to the field of astronomy.

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