How to Calculate Percent Error in a Lab Report

No measurement is perfect. A ruler has a smallest mark, a stopwatch depends on your reaction time and a scale drifts with temperature. Percent error is the standard way to say how close a measurement came to the value it should have been. It is a staple of school science labs, but the same idea is used to check instruments, test estimates and compare forecasts with what really happened.

Why percent and not just the difference?

Being off by 0.1 means very different things depending on what you measured. If you weigh a 100 g sample and get 99.2 g, you are off by 0.8 g, which is only 0.8%. If you measure a 0.5 mL volume and get 0.6 mL, you are off by just 0.1 mL, but that is a 20% error. The raw difference, called the absolute error, hides that. Dividing by the accepted value scales the error to the size of what you measured, so results from completely different experiments can be compared.

The formula, step by step

With E as your experimental (measured) value and T as the theoretical (accepted) value:

1. Signed error: E − T. Positive means you measured too high, negative means too low.
2. Absolute error: the size of that difference, |E − T|.
3. Relative error: absolute error ÷ |T|.
4. Percent error: relative error × 100.

Most textbooks report the percent error as a positive number and mention the direction in words, such as "about 5.6% too high". Some courses keep the sign instead. Both are fine as long as you say which one you use.

A worked lab example

Your class measures the density of an aluminum block. You find its mass, measure its sides to get the volume, and divide. Your result is 2.85 g/cm³. The accepted density of aluminum is about 2.70 g/cm³.

1. Signed error: 2.85 − 2.70 = +0.15 g/cm³
2. Absolute error: 0.15 g/cm³
3. Relative error: 0.15 ÷ 2.70 ≈ 0.0556
4. Percent error: about 5.56%, and the measurement is too high

Your lab partner gets 2.55 g/cm³. The absolute error is also 0.15, so the percent error is the same 5.56%, but the signed percent error is −5.56%: too low. The sign is what tells the two results apart, and it is a useful clue when you look for the cause.

Interpreting your result

A percent error is only the start of the discussion. The useful part of a lab report is explaining it.

• Which way was it off? A density that is too high suggests the volume was measured too small or the mass too large. A ruler read at an angle, for example, can make the sides look shorter than they are, and a small volume gives a large density.
• Is it random or systematic? A balance that was not zeroed adds the same error every time, a systematic error. Shaky readings that scatter both ways are random errors. Systematic errors push all your results in one direction; random ones average out.
• Is it within tolerance? Compare your error with the precision of your equipment. If the ruler only reads to the nearest millimeter on a 2 cm side, that alone can cause an error of several percent.

Repeat trials help

A single measurement can be lucky or unlucky. Suppose you repeat the aluminum measurement three times and get 2.66, 2.74 and 2.71 g/cm³. Their individual percent errors are about 1.48%, 1.48% and 0.37%. The average of the three is about 2.7033 g/cm³, and its percent error is only about 0.12%. The random errors partly cancel out.

This also shows the difference between accuracy and precision. Accuracy is how close you are to the true value, which percent error measures. Precision is how close your repeated results are to each other, which you judge from their spread, for example with the standard deviation. A broken scale can give very precise, wrong results.

Percent error versus percent difference

Use percent error when one value is the reference. When you compare two results that are equally trustworthy, such as your measurement and your partner's, neither one is "correct", so use percent difference instead. It divides by the average of the two:

Percent difference = |A − B| ÷ ((A + B) ÷ 2) × 100

For 2.85 and 2.55: the difference is 0.30, the average is 2.70, and the percent difference is about 11.11%. Two results that are each only 5.56% off the accepted value are 11.11% apart from each other, because they missed in opposite directions.

How to use the Percent Error Calculator

Choose Percent error, enter your measured value and the accepted value, and press Calculate. The Percent Error Calculator shows the percent error together with the signed error, absolute error, relative error and signed percent error, and lists each step with your numbers so you can copy the working into your report. Switch to Percent difference when you are comparing two measurements and neither one is the reference.

Common mistakes

• Dividing by the measured value. The accepted value goes in the denominator. Dividing by your own result gives a slightly different and incorrect answer.
• Forgetting the absolute value of a negative reference. If the accepted value is −40 °C and you measure −38 °C, the error is 2 and the percent error is 2 ÷ 40 = 5%, not −5%.
• Using percent error with an accepted value of zero. You cannot divide by zero. Report the absolute error instead.
• Rounding too early. Keep all digits until the final step, then round the percent error to match the precision of your data, usually two or three significant figures.
• Mixing units. Convert both values to the same unit before you start.

FAQ

Is percent error the same as percent change?

The formulas look alike, but the meaning differs. Percent change measures how much a value grew or shrank from an old value to a new one, such as a price rising. Percent error measures how far a measurement is from a reference that is assumed to be correct.

How many decimal places should I give?

Match the precision of your measurements. If your data have three significant figures, a percent error of 5.56% is appropriate; 5.5555556% suggests a precision you do not have.

Can percent error be more than 100%?

Yes. If the accepted value is 2.0 and you measure 5.0, the error is 3.0 and the percent error is 150%. It usually means a big mistake, such as a unit mix-up or a misread scale.

What if there is no accepted value?

Then percent error does not apply. Use percent difference between repeated measurements or between two methods, and describe the spread of your results.