Percent error measures how far an experimental result is from an accepted or actual value. It is useful in science, engineering, and everyday measurement because every measurement can have some error. A small percent error usually means the result is close to the true value, while a large percent error means the result may be inaccurate.
Thinking of the actual value as the bullseye and the experimental value as a point on a target helps show the size of the miss.
Understanding Math: Percent Error
The important idea is scale. A difference of two grams can be serious when a sample should weigh three grams. The same two gram difference is minor when a suitcase should weigh fifty kilograms.
Absolute error gives the raw size of the gap, in the original unit. Percent error turns that gap into a share of the accepted amount.
This makes results with very different sizes easier to compare. A thermometer that misses by one degree and a distance measurement that misses by one metre cannot be judged fairly from the raw differences alone.
To calculate carefully, first find the difference between the measured result and the accepted value. Use the positive size of that difference, since percent error describes distance from the target rather than direction. Next compare that difference with the size of the accepted value.
Finally change the resulting decimal into a percent by multiplying by one hundred. Keep units consistent before subtracting.
For example, convert centimetres to metres or grams to kilograms first. A unit mistake can produce an answer that looks mathematical but has no useful meaning.
Percent error tells how close a result is, but it does not explain why it missed. Random error comes from small unpredictable changes. A student may read a ruler from a slightly different angle each time, or a balance may fluctuate.
Repeating trials can reduce the effect of random error when the results are averaged. Systematic error pushes results in one direction. A scale that is not zeroed may make every measured mass too high.
Repeating those measurements will not fix the problem. Checking equipment, using a control measurement, and following the same method each time help reveal systematic error.
Students often confuse accuracy with precision. Accuracy means closeness to the accepted value. Precision means repeated measurements are close to each other.
A set of tightly grouped results can be precise yet inaccurate if a faulty instrument shifted them all. In a lab report, record enough digits during calculation, then round only at the end to match the measuring tool. Do not claim a very tiny percent error from a ruler marked only in whole millimetres.
Sometimes there is no known actual value, such as when measuring a newly discovered quantity. In that case, percent error is not the right measure.
Scientists may compare trials, report uncertainty, or use percent difference between two measurements instead. When the accepted value is zero, percent error cannot be calculated because there is no nonzero reference amount for comparison.
Key Facts
- Percent error = |experimental value - actual value| / |actual value| x 100%
- Absolute error = |experimental value - actual value|
- Percent error compares the size of the error to the actual value.
- A percent error of 0% means the experimental value equals the actual value.
- Example: experimental = 48 g, actual = 50 g, percent error = |48 - 50| / 50 x 100% = 4%
- Percent error is undefined when the actual value is 0 because division by zero is not allowed.
Vocabulary
- Experimental value
- The value found from a measurement, experiment, calculation, or observation.
- Actual value
- The accepted, true, or reference value used for comparison.
- Absolute error
- The positive difference between the experimental value and the actual value.
- Percent error
- The error written as a percentage of the actual value.
- Accuracy
- How close a measured or calculated value is to the actual value.
Common Mistakes to Avoid
- Forgetting the absolute value, which can give a negative percent error even though percent error is usually reported as a positive size of error.
- Dividing by the experimental value instead of the actual value, which changes the meaning of the comparison and gives the wrong percent error.
- Forgetting to multiply by 100, which leaves the answer as a decimal instead of a percent.
- Rounding too early, which can make the final percent error less accurate than it should be.
Practice Questions
- 1 A student measures the length of a table as 1.92 m. The actual length is 2.00 m. Find the absolute error and percent error.
- 2 A lab group finds the density of a metal to be 7.4 g/cm^3. The accepted density is 7.8 g/cm^3. Calculate the percent error to the nearest tenth of a percent.
- 3 Two students measure the same object. Student A has an absolute error of 2 cm on an actual length of 20 cm, and Student B has an absolute error of 2 cm on an actual length of 200 cm. Explain which measurement has the smaller percent error and why.