Percent Error: The Formula, Two Worked Examples, and When Not to Use It

Percent error answers a narrow question: how far off was I, relative to what the answer should have been. It only works when there is a value you are willing to call correct, which is the part most people skip over.

The formula, and the value that goes on the bottom

Take the measured value, subtract the accepted value, drop the sign, divide by the accepted value, then multiply by 100. The aluminium example above gives 3.33%. Try it with a thermometer: water boiling at sea level reads 98.4°C when the accepted value is 100°C, so the difference is 1.6, divided by 100 is 0.016, and the percent error is 1.6%.

The accepted value always goes on the bottom, never the measured one. Dividing by your own measurement makes the error look different depending on which direction you missed, which defeats the point of the comparison. The accepted value is the fixed reference, so it is the fixed denominator.

The same sum outside a lab

Put a certified 150 lb calibration weight on a bathroom scale and it reads 148.5 lb. The difference is 1.5 lb, divided by 150 is 0.01, so the scale is 1% out. That is a genuinely useful number, because it tells you the error scales: the same scale would be about 1.8 lb out at 180 lb, not 1.5 lb. Percentages travel across the range in a way absolute differences do not.

Percent error versus percent difference

These get mixed up constantly. Percent error compares a measurement to a known truth. Percent difference compares two measurements when neither one is authoritative, and it divides by the average of the two rather than by either one. Take the two density readings 2.61 and 2.70: the difference is 0.09 and the average is 2.655, so the percent difference is 3.39%, slightly higher than the 3.33% percent error. Small gap here, but it widens as the two values diverge. Our percentage difference guide works through when each one applies.

When there is no accepted value

Repeat measurements of something with no published figure, such as the mass of a particular rock, and percent error has nothing to compare against. What you want then is a measure of spread, usually standard deviation across your trials, which describes precision rather than accuracy. A set of readings can be tightly clustered and still consistently wrong, and only one of those two problems shows up in percent error.

Using the calculator and where to find the inputs

Two fields. The measured or experimental value is whatever your instrument produced, entered at the precision you actually read it, including trailing zeros that carry meaning. The accepted value, which some calculators label theoretical or true, comes from a reference: a data table, a textbook constant, a manufacturer specification, or the value your calculation predicted.

Keep the units identical on both sides before you enter anything. Comparing 2.61 g/cm³ with 2,700 kg/m³ gives an enormous error that is entirely your own doing. The calculator returns the absolute percent error by default, which is what lab reports normally ask for, and it will show the signed version if you want to know whether the reading ran high or low. If your work also needs the spread across repeated trials, the standard deviation guide covers that side.

Common questions

Can percent error be negative? Only if you keep the sign, which some courses require. A negative value means the measurement came in below the accepted figure. Most lab reports want the absolute value, so check the instructions before you hand it in with a minus sign attached.

What is an acceptable percent error? It depends on the equipment and the field. Under 5% is generally fine for a school lab with basic glassware, under 1% is expected in analytical chemistry, and engineering tolerances can be tighter still. The number is only meaningful against the standard your work is held to.

Can percent error be over 100%? Yes, when the measurement is more than double the accepted value, or when the accepted value is very small. Anything above 100% is usually a unit mistake or a decimal place rather than a genuinely bad measurement.

Why does my percent error look huge for a small difference? Because the accepted value is small. Missing by 0.02 against an accepted value of 0.10 is a 20% error even though the absolute gap is tiny. That is the formula behaving correctly, and it is why percent error is a poor choice for values close to zero.