Percent Error Explained

You measured gravity at 9.8 meters per second squared, but the textbook says 9.81. How far off were you? Percent error answers that in one number: about 0.1 percent. It scales your miss by the size of the true value, so small slips on big numbers stay small.

This guide shows the formula, the signed version that keeps direction, and how percent error differs from two close cousins. You will also see why a tiny error does not prove careful work.

What Percent Error Measures

Percent error measures how far a measured value sits from the accepted value, as a share of that accepted value. The accepted value is the reference you trust, such as a textbook constant or a certified standard.

The formula has three parts. Percent error = |measured – accepted| / |accepted| x 100. The top is the size of your miss, the bottom is the true size, and the 100 turns a decimal into a percent.

Why Scale the Error at All

A raw error of 1 centimeter tells you little on its own. On a 10 centimeter part it is a 10% error. On a 200 centimeter board, the same centimeter is only 0.5%. Scaling lets you compare accuracy across tasks of very different sizes.

Science and engineering classes use it for exactly this reason. A single percentage lets a teacher compare a pendulum lab with a density lab on the same footing.

How Do You Calculate Percent Error?

You subtract, take the absolute value, divide by the accepted value, and multiply by 100. The five steps below follow the gravity example from the lead, where you measured 9.8 and the accepted value is 9.81.

1. Record both values. Write the measured value, 9.8, and the accepted value, 9.81, in the same units.
2. Find the raw error. Subtract accepted from measured: 9.8 – 9.81 = -0.01.
3. Drop the sign. Take the absolute value, so the error becomes 0.01.
4. Divide by the accepted value. 0.01 / 9.81 = 0.00102, which is the relative error.
5. Convert to a percent. Multiply by 100 to get 0.102%, then round to your data’s precision.

A chemistry example shows a bigger miss. Aluminum has an accepted density of 2.70 g/cm3, and a student measures 2.45 g/cm3. The gap is 0.25, and 0.25 / 2.70 x 100 gives a percent error of 9.26%.

Which value was the reference in the gravity case? NIST defines standard gravity as exactly 9.80665 m/s2. Against that exact figure, the 9.8 reading has a percent error of 0.068%, so always state which accepted value you used.

Signed Percent Error and the Direction of a Miss

The standard formula throws away the sign, so a reading 2 units high and one 2 units low both give 4.0%. That is fine for grading closeness. It hides a useful clue, though: which way your method leans.

The signed version keeps the sign. Signed percent error = (measured – accepted) / |accepted| x 100. A positive result means you measured high, and a negative result means you measured low.

Reading the Sign

For the aluminum sample, the signed result is (2.45 – 2.70) / 2.70 x 100 = -9.26%. The minus sign says the density came out low. Trapped air bubbles on the metal would push a volume reading up and the density down, so the sign points to a cause.

Repeated misses in one direction suggest a systematic error, such as a scale that reads 0.5 grams light. Misses that flip between plus and minus point to random error instead. Report the absolute form when your instructor asks for percent error, and add the signed form in your discussion.

When the Accepted Value Is Zero

Percent error breaks when the accepted value is zero, because you cannot divide by zero. Report the plain absolute error in the original units for readings like a zero-degree offset or a null voltage.

How Is Percent Error Different From Percent Difference and Relative Error?

Percent error compares a measurement with a trusted value, while percent difference compares two measurements when neither one is the truth. Relative error is simply percent error before you multiply by 100.

Percent difference divides the gap by the average of the two values, so it treats both numbers as equals. Two lab groups who measure 52 and 50 would report a 3.92% difference. Our guide to percentage difference vs percentage change covers that symmetric method in depth.

Percentage change is a third idea. It tracks growth from an old value to a new one, like a price rising from 50 to 52. The percentage change formula explains that directional math, which answers a different question than measurement accuracy.

Is a Low Percent Error the Same as Precise Work?

No. A low percent error shows accuracy, meaning closeness to the accepted value. Precision means repeated readings agree with each other, and a lab can have one without the other.

Set A looks careful, with a spread of only 0.02. Its 2.96% error reveals a bias, perhaps a timer started late every trial. Set B hits 0% error on average, yet its spread of 0.25 means any single reading is unreliable.

Using Percent Error in a Lab Report

Give the accepted value and its source, show one full calculation, and round to match your data. Then explain the likely causes and whether the sign suggests a bias.

Try it with your own numbers: the percent error tool checks the math and flags the direction in one step.

Percent Error: Frequently Asked Questions

Sources and Further Reading

This article is general math and science education for students and lab work. Check your course or method for its own reporting rules. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.