Percent error calculator with the formula and working

Percent error measures how far a measured value is from the true (accepted) value, as a percentage of the true value: percent error = |measured − true| ÷ |true| × 100%. Measure g as 9.6 m/s² when the accepted value is 9.81 m/s² and the percent error is 2.14%. This percent error calculator works it out with the steps, the absolute and relative error, and whether you measured too high or too low.

It opens on that g measurement. Change the two values, or open the full percent error calculator over this page.

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Percent Error Calculator

Compare measured and true values as a percentage error

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How to use the percent error calculator

1. Under Calculate, choose Percent error. The other two modes are Difference and Tolerance, covered below.
2. Type your result in Measured value and the accepted value in True value. Use the same units for both.
3. Read the percent error, with the absolute error, the relative error and the direction (measured too high or too low) beside it.
4. The steps show each stage, and the graph plots percent error against the measured value, with your measurement marked. The further from the true value, the higher the V.
5. To practise, press Random problem in the Practice section.

The percent error formula

US textbooks often say experimental value for the measured one and theoretical or accepted value for the true one. The formula builds up in three steps, and the calculator shows all three:

The absolute error is in the same units as the measurement (here m/s²). The relative and percent errors have no units, which is why they let you compare the accuracy of measurements of very different sizes.

Signed percent error

Some courses leave off the absolute value bars, so a measurement below the true value gives a negative percent error. The size is the same either way. This calculator gives the size and tells you the direction separately, so for the g example it shows 2.14% and "measured too low". If your teacher wants the sign, that is −2.14%.

Worked examples you can check in the calculator

Example 1: a pendulum measurement of g

A class times a pendulum and works out g as 9.6 m/s², against a book value of 9.81 m/s².

The percent error is 2.14%, and the result was too low.

Example 2: density of a metal block

A student finds the density of a block as 2.85 g/cm³. The accepted value for that metal is 2.70 g/cm³. The absolute error is 0.15 g/cm³, so:

The percent error is 5.56%, too high. A high density can come from a volume measured too small, such as reading a displacement cylinder from above the line.

Example 3: why the true value goes on the bottom

Measuring 1.2 when the true value is 1.0 is a 20% error: 0.2 ÷ 1.0 × 100. Swap them by mistake and you get 0.2 ÷ 1.2 × 100 = 16.7%, a different answer. Always divide by the true value.

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Percent difference: when neither value is the true one

If you are comparing two measurements and neither is the accepted value, use percent difference instead. It divides by the mean of the two, so it comes out the same whichever you list first. Choose Difference under Calculate:

Two groups measure the same length as 48.2 cm and 47.6 cm. The difference is 0.6, the mean is 47.9, and the percent difference is 0.6 ÷ 47.9 × 100 = 1.25%.

Tolerance: the range a percentage allows

Tolerance works backwards from a percentage to a range of values. A 100 Ω resistor with a 5% tolerance can be anywhere from 100 − 5 = 95 Ω to 100 + 5 = 105 Ω. Choose Tolerance, type the true value and the percent allowed, and the calculator gives the largest and smallest values.

What counts as a good percent error?

There is no single cut-off: it depends on the equipment and the experiment. A school pendulum timed by hand can easily be a few percent out, while a careful titration should be much closer. A small percent error means your result is accurate, close to the true value. It doesn't say whether it is precise: repeated results that agree closely with each other can all be off by the same amount, which points to a systematic error such as a balance that wasn't zeroed. The NIST handbook calls this kind of error bias, or systematic error, and notes that it can often be found and removed by calibrating against a reference standard.

To work with a whole table of repeat readings, the practical data lab finds means and plots your results.

Common mistakes

• Dividing by the measured value. The true value always goes on the bottom.
• Forgetting to multiply by 100. 0.0214 is the relative error. The percent error is 2.14%.
• Mixing units. Convert both values to the same unit before you start.
• Using percent error to compare two measurements. When neither is the true value, use percent difference.
• Rounding too early. Keep the absolute error unrounded until the last step, especially when it is small.

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