Percent Error Formula
- Introduction
- What is Percent Error?
- Percent Error Formula
- How to calculate Percent Error?
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
Percent error measures how much a measurement differs from the true or accepted value. When we measure something, the result might not be exactly correct due to various reasons like calculation mistakes, adjustments done in a calculation like rounding off, or limitations of the tools we use. So, we need to calculate the percentage error to understand how far off our measurement is.
For instance, if we're estimating the average height of an object, a `20%` error indicates that we missed the mark by a significant margin. On the other hand, a `5%` error indicates that we're much closer to the actual value. So, percent error helps us understand the accuracy of our measurements or estimates. Likewise, a percent error close to `0%` means that our estimation or approximation is very close to the actual value.
What is Percent Error?
Percent error is a measure that tells us how far off our estimate or measurement is from the actual value, expressed as a percentage. It's like figuring out how much we missed the mark. It is the difference between the actual value and the estimated value compared to the actual value expressed as a percentage.
To do this, we first find the absolute error by subtracting the measured value from the true value. Then, we divide this absolute error by the true value and multiply it by `100` to get the percentage error. This helps us understand the accuracy of our measurement and how much it deviates from what it should be.
Percent Error Formula
Percent error is a useful tool to understand how mistakes affect our experimental findings. It's calculated by comparing the actual value with the measured value, then expressing the difference as a percentage of the actual value. The formula for percent error is:
\(\text{Percent Error} = \left|\frac{{\text{Measured value} - \text{Actual value}}}{{\text{Actual value}}}\right| \times 100\)
How to Calculate Percent Error?
Steps to calculate the percent error are:
Step `1`: Find the absolute error by subtracting the measured value from the actual value and disregarding any negative signs. This means we need to consider the absolute value of the difference.
Step `2`: Calculate the relative error by dividing the absolute error by the actual value.
Step `3`: Multiply the relative error by `100` and attach the “`%`” sign.
Solved Examples
Example `1`: Chemistry Experiment:
A student measures the density of a substance to be \(8.5 \, \text{g/cm}^3\). The accepted value for the density is \(8.3 \, \text{g/cm}^3\). What is the percent error in the student's measurement?
Solution:
Actual value `=` \(8.3 \, \text{g/cm}^3\)
Measured value `=` \(8.5 \, \text{g/cm}^3\)
\(\text{Percent error} = \left|\frac{{\text{Measured value} - \text{Actual value}}}{{\text{Actual value}}}\right| \times 100\% \)
\(= \left|\frac{{8.5 - 8.3}}{{8.3}}\right| \times 100\% \)
\(= \frac{{0.2}}{{8.3}} \times 100\% \)
\(\approx 2.41\%\)
The percent error is approximately \(2.41\%\).
Example `2`: Physics Experiment:
During an experiment, a physics student measures the speed of light to be \(295,000 \, \text{km/s}\). The accepted value for the speed of light is \(300,000 \, \text{km/s}\). Calculate the percent error in the student's measurement.
Solution:
Actual value `=` \(300,000 \, \text{km/s}\)
Measured value `=` \(295,000 \, \text{km/s}\)
\(\text{Percent error} = \left|\frac{{\text{Measured value} - \text{Actual value}}}{{\text{Actual value}}}\right| \times 100\% \)
\(= \left|\frac{295,000 - 300,000}{300,000}\right| \times 100\% \)
\(= \frac{5,000}{300,000} \times 100\% \)
\( \approx 1.67\% \)
The percent error is approximately \(1.67\%\).
Example `3`: Manufacturing Defects:
A factory produces \(2000\) widgets, but upon inspection, it is found that only \(1800\) of them are of acceptable quality. What is the percent error in the production?
Solution:
Actual production `=` \(2000\) widgets
Acceptable production `=` \(1800\) widgets
\( \text{Percent error} = \left|\frac{\text{Actual production} - \text{Acceptable production}}{\text{Actual production}}\right| \times 100\% \)
\( = \left|\frac{2000 - 1800}{2000}\right| \times 100\% \)
\( = \frac{200}{2000} \times 100\% \)
\(= 10\% \)
The percent error in production is \(10\%\).
Example `4`: Population Estimate:
After concluding his survey, a demographer estimates the population of a town to be \(50,000\) people. The actual population, however, is \(48,000\) people. What is the percent error in the demographer's estimate
Solution:
Actual population `=` \(48,000\) people
Estimated population `=` \(50,000\) people
\( \text{Percent error} = \left| \frac{\text{Estimated population} - \text{Actual population}}{\text{Actual population}} \right| \times 100\% \)
\( = \left| \frac{50,000 - 48,000}{48,000} \right| \times 100\% \)
\( = \frac{2,000}{48,000} \times 100\% \)
\( \approx 4.17\% \)
The percent error in the estimate is approximately \(4.17\%\).
Example `5`: Distance Measurement:
A surveyor measures a field to be \(200\) meters long, but upon re-measurement, it is found to be \(205\) meters long. Calculate the percent error in the surveyor's measurement.
Solution:
Actual length `=` \(205\) meters
Measured length `=` \(200\) meters
\( \text{Percent error} = \left|\frac{{\text{Measured length} - \text{Actual length}}}{{\text{Actual length}}}\right| \times 100\% \)
\(= \left|\frac{{200 - 205}}{{205}}\right| \times 100\% \)
\( = \frac{{5}}{{205}} \times 100\% \)
\( \approx 2.44\% \)
The percent error in the measurement is approximately \(2.44\%\).
Example `6`: A student’s estimation was found to have an error of `18%`. His experimental measurement of the volume was `15` ml. What are the two possible values for the actual measurement?
Solution:
Estimation error: \(18\%\)
Measured value: \(15\, \text{ml} \)
Step `1`: Calculate the absolute error:
\( \text{Absolute Error} = 18\% \times 15 \, \text{ml} \)
\( = 0.18 \times 15 \, \text{ml} \)
\(= 2.7 \, \text{ml} \)
Step `2`: Determine the range of possible values for the actual measurement:
a) For the maximum possible actual measurement:
\( x_{\text{max}} = 15 \, \text{ml} + \text{Absolute Error} \)
\( x_{\text{max}} = 15 \, \text{ml} + 2.7 \, \text{ml} \)
\( x_{\text{max}} = 17.7 \, \text{ml} \)
b) For the minimum possible actual measurement:
\( x_{\text{min}} = 15 \, \text{ml} - \text{Absolute Error} \)
\( x_{\text{min}} = 15 \, \text{ml} - 2.7 \, \text{ml} \)
\( x_{\text{min}} = 12.3 \, \text{ml} \)
So, the two possible values for the actual measurement are \(12.3\, \text{ml} \) and \(17.7\, \text{ml} \).
Practice Problems
Q`1`. A student measures the melting point of a substance to be \(98^\circ \text{C}\), but the accepted value is \(100^\circ \text{C}\). What is the percent error in the student's measurement?
- `2%`
- `4%`
- `5%`
- `10%`
Answer: a
Q`2`. A scientist measures the acceleration due to gravity as \(9.5 \, \text{m/s}^2\), but the accepted value is \(9.8 \, \text{m/s}^2\). What is the percent error in the scientist's measurement?
- `2.04%`
- `3.06%`
- `4.08%`
- `5.10%`
Answer: b
Q`3`. A factory produces \(500\) laptops, but upon inspection, it is found that only \(450\) of them are free from defects. What is the percent error in the production?
- `8%`
- `9%`
- `10%`
- `12%`
Answer: c
Q`4`. A demographer estimates the population of a city to be \(150,000\) people. The actual population, however, is \(140,000\) people. What is the percent error in the demographer's estimate?
- `4.76%`
- `5.71%`
- `6.67%`
- `7.14%`
Answer: d
Q`5`. A surveyor measures the height of a building to be \(80\) meters, but upon re-measurement, it is found to be \(85\) meters. Calculate the percent error in the surveyor's measurement.
- `5.88%`
- `6.25%`
- `6.67%`
- `7.10%`
Answer: a
Frequently Asked Questions
Q`1`. What is percent error and why is it used?
Answer: Percent error is a measure of the accuracy of a measurement or estimate compared to the actual or accepted value. It is used to quantify how much a measurement deviates from the true value, providing insight into the precision of experimental or estimated data.
Q`2`. How do you calculate percent error?
Answer: Percent error is calculated using the formula:
\[\text{Percent error} = \left| \frac{{\text{Measured value} - \text{Actual value}}}{{\text{Actual value}}} \right| \times 100\%\]
Q`3`. How is percent error used in scientific experiments?
Answer: In scientific experiments, percent error helps researchers assess the accuracy and reliability of their measurements by comparing them to known or accepted values. It allows scientists to identify systematic errors and uncertainties in their experimental procedures.
Q`4`. Can percent error be negative?
Answer: No, the percent error cannot be negative. The absolute value operation in the percent error formula ensures that the result is always non-negative. A negative percent error would imply an inconsistency with the formula. Thus, percent error is always expressed as a non-negative value.
Q`5`. Is it possible for a percent error to be greater than `100%`?
Answer: Yes, percent error can be greater than `100%` when the measured or estimated value significantly exceeds the actual or accepted value. This typically occurs when there is a large discrepancy between the two values.