Percent Error Calculator
Calculator Use
The Percent Error Calculator calculates the difference between between an experimental or observed value and a theoretical actual value. It creates a ratio of the difference relative to the actual value and gives it as a percentage.
Answers show the work for the percent error calculation.
What is Percent Error?
Percent error is the relative size of the difference between an experimental or estimated value, and the true, accepted value. It compares the difference in values to the expected actual value and tells you how far off your experimental or observed value is.
For example, say you bought a bag of jelly beans and the label said it weighed 10 ounces. When you actually weighed the jelly beans it was 10.3 ounces. Percent error calculates the ratio of 0.3 ounces to 10 ounces, and then gives it as a percentage. In this case the percent error is 3%.
Maybe you poured yourself a serving of 30 jelly beans. Then you double-checked the nutrition label to see there are 20 jelly beans in a serving. The percent error of your original serving size compared to recommended serving size is 30 - 20 = 10. Put that into a ratio with the recommended serving of 20 gives you a percent error of 50%. So your original serving size was 50% more than the expected serving size.
The calculator also finds:
- Experimental Value - when percent error is known
- Theoretical Value - when percent error is known
- Absolute Error - the numerical difference between estimated and actual values
- Relative Error - the absolute error relative to what the actual value should be
How to Calculate Percent Error
Percent error is also known as approximation error. It equals the absolute value of the experimental value minus the theoretical value, divided by the theoretical value, multiplied by 100.
- Subtract theoretical value from experimental value
- Take the absolute value of the result
- Divide that by the theoretical value
- Multiply by 100 to get a percentage
In terms of experimental and theoretical values the percent error formula is:
\[ {\text{ Percent Error}} = \] \[ \left|{\dfrac {Experimental - Theoretical}{Theoretical}}\right| \times 100\ \]or concisely
\[ \% {\text{ error}} = \left|{\dfrac {E - T}{T}}\right| \times 100\ \]You may notice that this formula is similar to the way you would calculate percent change as in our Percentage Change Calculator.
Absolute Error Formula
The absolute error is the absolute value of the difference between measured and true values. If your scale weighs something as 1.2 pounds but the actual value is 1.5 pounds your scale is off by 0.3 pounds. The absolute error is 0.3 pounds.
\[ {\text{absolute error}} = | {E - T} | \]Relative Error Formula
The relative error is the absolute error relative to what the true value should be. If your scale is off by 0.3 pounds and the true value is 1.5 pounds, the relative error is 0.3 / 1.5 = 0.2.
\[ \text{relative error} = \left| {\dfrac {E - T}{T}} \right| \]Notes on the Percent Error Calculations
The Theoretical value in chemistry, physics or science experimentation in general, is the established ideal value you would expect as a result of an experiment. Other terms you may see to represent this value are accepted, actual, expected, exact and true. This value is in the denominator of the percent error equation.
The Experimental value is the observed result of an experiment. Other terms you may see to represent this value are measured, observed, estimated and approximate.
In the numerator of the error formula you are calculating the absolute difference between the experimental value and the theoretical value or the absolute distance between the two values on a number line. The order of the values does not matter because you are taking the absolute value.
If you encounter percent error formulas that are somewhat different from the formulas here, note the alternate terms above and remember that | a - b | = | b - a |.
Experimental Value Formula
Use the percent error formula to solve for E, the experimental or observed value. In this equation P stands for percent error. Note there are two possible solutions.
\[ \ {P} = \left|{\dfrac {E - T}{T}}\right| \times 100\ \] \[ \dfrac{P}{100} = \left|{\dfrac {E - T}{T}}\right| \] \[ \pm \dfrac{P}{100} = {\dfrac {E - T}{T}} \] \[ \pm \dfrac{P}{100}{T} = {E - T} \] \[ {T} \pm \dfrac{P}{100}{T} = {E} \] \[ {E} = {T} \pm \dfrac{P}{100}{T} \] \[ {E} = {T} \left( 1 \pm \dfrac{P}{100} \right) \]Theoretical Value Formula
You can use the experimental error formula derived above to solve for T. Here again P stands for percent error. Note there are two possible solutions.
\[ {T} = \dfrac{E}{ \left( 1 \pm \dfrac{P}{100} \right) } \]Percent Error Calculation Example
Suppose you did an experiment to measure the boiling point of water and your average result is 101.5°C. This is your experimental or measured value. Since the actual boiling point of water is 100°C this would be your theoretical value.
You want to find the % error of the average boiling point of water in your experiment, 101.5°C relative to 100°C.
Plug your numbers into the percent error formula:
\[ \% {\text{ error}} = \left|{\dfrac {E - T}{T}}\right| \times 100\ \] \[ = \dfrac{(101.5 - 100)}{|100|} \times 100\ \] \[ = \dfrac{1.5}{100} \times 100\ \] \[ = 0.015 \times 100\ \] \[ = 1.5\% \; \text{error} \]So your experimental boiling point has 1.5 percent error compared to the theoretical boiling point of water.