Percentage Error Calculator

How to Use the Percentage Error Calculator

Percent error tells you how close an experimental or measured value is to an accepted, true or theoretical value. This calculator performs the arithmetic for you and also reports the absolute error and the signed deviation, so you can judge both the size and the direction of the gap.

1. Enter the accepted value. This is the trusted reference your result is compared against, such as the accepted acceleration due to gravity, 9.81 m/s², or the density of water, 1.000 g/mL. Type it into the second field.

2. Enter the measured value. This is your experimental reading, for example 9.55 m/s² from a pendulum experiment or 1.03 g/mL from a density test. Type it into the first field.

3. Read the percent error. With an accepted value of 9.81 and a measured value of 9.55, the absolute error is |9.55 − 9.81| = 0.26, and the percent error is 0.26 ÷ 9.81 × 100 = 2.65%, a typical laboratory result.

4. Check the direction. The signed deviation here is 9.55 − 9.81 = −0.26, so the measurement fell below the accepted value. Had you measured 10.05 instead, the deviation would be +0.24 and the percent error 2.45%.

5. Compare with the reference table. The built-in examples show what small (under 1%) and large (over 10%) errors look like, so you can judge whether your own figure is reasonable for the type of measurement.

6. Copy the answer. Press Copy result to paste the percent error, absolute error and deviation into a lab report, spreadsheet or message.

What Is Percent Error?

Percent error, also called percentage error, measures how far a measured or experimental value sits from an accepted, true or theoretical value, expressed as a percentage of that accepted value. Its formula is percent error = |measured − accepted| ÷ |accepted| × 100. The numerator is the absolute error — the raw size of the gap, in the same units as the measurement — and the absolute-value bars ensure the result is never negative, because we care about the size of the discrepancy rather than its direction. Percent error is the standard way science courses and quality laboratories report the accuracy of a reading, and it appears whenever a calculated or instrument-measured number is compared against a known reference, such as the true density of a metal or the rated output of a device. Percent error is easily confused with percent difference, but the denominator is the key distinction. Percent difference compares two values that are both measured, with neither treated as the truth, and therefore divides by their average: |a − b| ÷ ((a + b) ÷ 2) × 100. Percent error always divides by the accepted value because one number is the benchmark. If two readings are 4.9 and 5.1 while 5.0 is the true value, the percent error of 4.9 is 2%, whereas the percent difference between 4.9 and 5.1 is about 4%. The distinction matters when you report results, since using the wrong denominator can roughly double or halve the figure. Percent change, by contrast, tracks how a single quantity moves over time and uses the starting value as the denominator: (new − old) ÷ old × 100. A price rising from 50 to 55 is a 10% change, not a percent error. Percent error only makes sense when an authoritative value exists to compare against. In practice an error below 1% signals excellent agreement, 1% to 5% is typical for a school laboratory, and anything over 10% usually points to a measurement, calibration or unit problem that deserves investigation before the result is trusted.

Common Applications

1. Physics laboratories: a pendulum experiment measures the acceleration due to gravity as 9.55 m/s² against the accepted 9.81 m/s². The percent error is |9.55 − 9.81| ÷ 9.81 × 100 = 2.65%, a perfectly normal result for the apparatus.

2. Chemistry and density: a student calculates the density of aluminium as 2.60 g/cm³ while the accepted value is 2.70 g/cm³, giving an error of 0.10 ÷ 2.70 × 100 = 3.70%.

3. Manufacturing quality control: a batch of nominal 500 g packets averages 495 g, so against the 500 g target the percent error is 5 ÷ 500 × 100 = 1.0%, which sits within many packaging tolerances.

4. Forecasting and data science: a model predicts 1,050 sales in a month when 1,000 actually occurred, a percent error of 50 ÷ 1,000 × 100 = 5%, which lets you compare competing models on one consistent metric.

5. Engineering and calibration: a sensor rated at 100.0 °C reads 101.2 °C in a reference bath, an error of 1.2 ÷ 100 × 100 = 1.2%, telling a technician whether recalibration is due.

Common Mistakes When Calculating Percent Error

By far the most frequent error is dividing by the measured value instead of the accepted value. With a measured 47 against an accepted 50, the correct result is 3 ÷ 50 × 100 = 6%, but the wrong denominator gives 3 ÷ 47 × 100 = 6.38%. Always divide by the accepted value. A second mistake is dropping the absolute-value brackets and presenting a negative percent error as if it described accuracy; the sign belongs to the deviation, not to the percentage. Third, mixing units breaks the result — comparing a measurement in grams with an accepted value in kilograms gives a meaningless figure, so convert units first. Finally, do not confuse percent error with percent uncertainty: uncertainty describes the spread of repeated measurements, whereas percent error compares a single result against a reference.

Significant Figures and Scientific Rounding

Report percent error with the precision your data actually supports. A common convention is to round to two significant figures, or to one decimal place for small errors, because the absolute error itself is usually limited to one or two significant figures. If the accepted value is 9.81 m/s² (three significant figures) and the measured value is 9.55 m/s² (three), the absolute error 0.26 carries two significant figures, so a percent error of 2.65% is defensible while 2.6491% falsely implies precision the inputs cannot justify. When the accepted value is a defined constant, such as 100 °C for the boiling point of water at one atmosphere, you may legitimately keep more digits. Round only at the final step, because rounding intermediate absolute errors can shift the percentage noticeably.

When to Use Percent Error, Difference or Change

Choose the metric by asking what role each number plays. Use percent error when one value is an accepted standard and the other is your own result, for example a measured 9.55 m/s² against the accepted 9.81 m/s². Use percent difference when both numbers are measurements and neither is privileged, such as comparing two thermometers that read 21.4 °C and 21.8 °C. Use percent change when you are tracking one quantity across time, such as a salary rising from 2,000 to 2,150, a change of 7.5%. Whenever a true value exists, percent error is the correct and most informative choice because it anchors the comparison to the standard rather than to the average of two arbitrary readings.

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