Percent Error Formula

Percent error is a measure of how far an experimental or measured value is from a known accepted, true, or theoretical value, expressed as a percentage. It is the standard way to report measurement accuracy in math, chemistry, physics, and lab reports. By scaling the error against the size of the quantity being measured, percent error makes it possible to compare measurements that come in very different sizes.

A 1 cm error means very different things on a 10 cm ruler and on a 100 m field. The first is 10% off; the second is 0.001% off. The raw difference is the same. Percent error is what makes that comparison fair.

The formula has three pieces: the absolute error in the numerator, the absolute value of the accepted value in the denominator, and a multiply-by-100 to convert the fraction into a percentage.

The vertical bars are absolute-value bars: they strip the sign from whatever is between them. The result of the whole formula is therefore zero or positive, never negative.

The arithmetic is short. Four steps get you from the two input values to the final percentage.

1. Subtract the accepted value from the experimental value to get the signed difference.
2. Take the absolute value of that difference to get the absolute error.
3. Divide the absolute error by the absolute value of the accepted value to get a fraction.
4. Multiply the fraction by 100 to convert it to a percent.

The first two steps can be done in either order. Some texts subtract accepted from experimental; others subtract experimental from accepted. It does not matter for the final answer because the absolute value drops the sign either way.

The absolute value is what makes percent error report a single positive number. Without it, the result would carry the sign of the difference. That sign tells you whether the experimental value came in high or low, which is sometimes useful, but it makes errors awkward to compare across experiments.

In a typical chemistry or physics class, the question is usually the size of the gap, not its direction. Two students who measured a quantity at 4% above and 4% below the accepted value have made errors of the same size. Reporting both as 4% percent error captures that symmetry. The absolute-value version is what the standard classroom formula uses for that reason.

Some advanced contexts (in metrology, calibration, or certain engineering settings) use signed error instead, because the direction of a systematic error is itself useful information. Both forms are mathematically valid, and you should follow whichever convention your textbook or instructor uses.

Suppose the accepted value is 50 and an experiment measures 48.

1. Signed difference: 48 − 50 = −2.
2. Absolute error: |−2| = 2.
3. Fraction: 2 / |50| = 0.04.
4. Multiply by 100: 0.04 × 100 = 4%.

The percent error is 4%. The experiment landed 2 units below the accepted value, which is 4% of the accepted value. The same arithmetic runs in the percent error calculator, which prints all four steps with your numbers in place.

A common high-school chemistry lab measures the density of a metal sample by dividing its mass by its volume. Suppose a student measures the density of an aluminum cube as 2.55 g/cm³. The accepted density of aluminum is 2.70 g/cm³.

• signed difference = 2.55 − 2.70 = −0.15
• absolute error = |−0.15| = 0.15
• fraction = 0.15 / |2.70| ≈ 0.0556
• percent error ≈ 5.56%

The measured density is about 5.56% lower than the accepted value. In a school lab that is in the reasonable-but-imperfect range; common sources of error include trapped air bubbles when measuring volume by water displacement, an unzeroed balance, or surface oxidation on the metal. Percent error sized the gap; finding the cause takes a second look at the procedure.

A classic physics lab uses a pendulum or a falling body to measure the acceleration of gravity g. The accepted value near the surface of the Earth is 9.81 m/s². Suppose a student measures 9.74 m/s².

• signed difference = 9.74 − 9.81 = −0.07
• absolute error = |−0.07| = 0.07
• fraction = 0.07 / |9.81| ≈ 0.00714
• percent error ≈ 0.71%

A measurement of g within 1% of the accepted value is excellent for a school lab and reasonable for a careful undergraduate experiment. Errors on this kind of measurement usually come from timing (especially with a stopwatch), small angle approximations in the pendulum equation, or air resistance for free-fall experiments.

A few traps that catch students:

• Forgetting the absolute value. Without it, you get signed error, which can be negative. In the standard classroom formula, percent error is always positive.
• Dividing by the experimental value. The accepted value is the reference, so it goes in the denominator. Dividing by the experimental value gives a different ratio that is not the standard percent error.
• Using percent error when there is no accepted value. If you only have repeated measurements and no theoretical or published reference, you cannot compute percent error in a meaningful way. Use standard deviation instead to describe how much the measurements vary among themselves.
• Confusing percent error with percent change. Percent error compares a measurement to a fixed reference and is reported as a positive number. Percent change compares two values that are both equally “real,” and the sign tells you the direction of the change.
• Reporting too many decimal places. Two or three significant figures is plenty for a school-lab percent error. A reading of “5.5621%” pretends to a precision the measurement does not actually have.

Percent error is undefined when the accepted value is zero, because the formula divides by the accepted value and division by zero is not defined. There is no workaround that keeps the same shape of the formula. Practical options when the accepted value is zero:

• Report the absolute error. The number |experimental − 0| is just the experimental value (in absolute terms) and still tells you how far off the measurement was from the expected value of zero.
• Use a non-zero reference if one exists. In some experiments the “zero” reference comes from a baseline calibration; the meaningful reference for percent error is the calibration standard, not the literal zero on the scale.
• Switch to a different metric. For data sets where the true value can be zero, statisticians use measures like mean absolute error or root mean square error instead.

The percent error calculator flags an accepted value of zero as undefined rather than printing a misleading number.

A few calculations look similar but answer different questions. Knowing which one you want avoids reporting the wrong number.

• Percent error compares a measurement to a fixed accepted reference, with absolute value, and is always reported as a positive percentage. |experimental − accepted| / |accepted| × 100.
• Percent change compares two values neither of which is a fixed reference. The sign matters because it tells you the direction. ((new − old) / |old|) × 100. See the percentage increase calculator.
• Standard deviation describes the spread of repeated measurements around their mean, with no reference value involved. Precision, not accuracy. See the standard deviation calculator and the Standard Deviation Formula guide.

A measurement can be both precise and inaccurate (small standard deviation, large percent error) or accurate but imprecise (small percent error on the mean, large standard deviation across the trials). Reporting both paints the fuller picture.

• Formula: percent error = |experimental − accepted| / |accepted| × 100.
• Absolute value forces the answer to be zero or positive.
• The accepted value goes in the denominator. It is the reference.
• Multiply by 100 at the end to express the result as a percent.
• Undefined when the accepted value is zero. Report the absolute error or pick a non-zero reference instead.
• Percent error is about accuracy. Standard deviation is about precision. They are not the same thing.
• The percent error calculator runs all four steps and prints the worked calculation.