Percent Error Calculator — Free Online Error Calculator

How to Use the Percent Error Calculator

1. Enter the experimental (measured) value: Type the value you obtained from your experiment, measurement, or observation into the first input field. This is the number you are testing for accuracy. It can be any real number, positive or negative, including decimals. The calculator accepts values with up to two decimal places of precision.
2. Enter the theoretical (actual) value: Type the accepted, known, or theoretical value into the second input field. This is the reference standard you are comparing your measurement against. It should be the value from a textbook, published study, or established standard. If this value is zero, the percent error will be undefined since division by zero is not possible.
3. Read the results: The calculator instantly computes and displays three related metrics: percent error (the percentage deviation from the true value), absolute error (the raw numerical difference), and relative error (the decimal fraction of deviation). The formula panel shows the exact calculation with your input values substituted in for complete transparency.

Results update in real time as you change either input value. There is no submit button — simply type and the answer appears instantly. This makes it easy to compare multiple measurements against the same theoretical value by changing only the experimental input.

Percent Error Formula

Variables Explained

• Experimental Value: The measured, observed, or estimated value obtained from an experiment or calculation. This is the value being evaluated for accuracy against a known standard.
• Theoretical Value: The accepted, known, or true value from an authoritative source such as a textbook, published study, or calibrated instrument. This value serves as the reference point for calculating error. It must not be zero for percent error.
• | | (Absolute Value): The vertical bars indicate absolute value, which means only the magnitude of the difference matters, not its direction. This ensures the percent error is always a non-negative number regardless of whether the measurement is above or below the true value.

Step-by-Step Example

A student measures the boiling point of water at 99.1 degrees Celsius. The accepted value is 100.0 degrees Celsius. Calculate the percent error:

1. Identify the values: Experimental = 99.1, Theoretical = 100.0
2. Calculate the absolute error: |99.1 - 100.0| = |-0.9| = 0.9
3. Calculate the relative error: 0.9 / |100.0| = 0.009
4. Multiply by 100 for percent error: 0.009 x 100 = 0.9%

The student's measurement has a 0.9% error, which is excellent for a typical laboratory setting. This small error could be due to altitude differences, impurities in the water, or minor thermometer calibration issues.

Practical Examples

Example 1: Marcus's Chemistry Lab Experiment

Marcus measures the density of aluminum in a chemistry lab and gets 2.81 g/cm3. The accepted density of aluminum is 2.70 g/cm3. His instructor asks for the percent error in his measurement:

• Absolute error: |2.81 - 2.70| = 0.11 g/cm3
• Relative error: 0.11 / 2.70 = 0.0407
• Percent error: 0.0407 x 100 = 4.07%

Marcus's measurement has a 4.07% error. For a general chemistry lab, this is within acceptable limits. The slight overshoot could be due to air bubbles in the water displacement method or an impure aluminum sample. Marcus documents this in his lab report alongside potential error sources.

Example 2: Elena's Physics Experiment

Elena uses a pendulum to measure gravitational acceleration and obtains 9.65 m/s2. The accepted value is 9.81 m/s2. She calculates her percent error:

• Absolute error: |9.65 - 9.81| = 0.16 m/s2
• Relative error: 0.16 / 9.81 = 0.0163
• Percent error: 0.0163 x 100 = 1.63%

Elena's 1.63% error is quite good for a pendulum experiment. The small deviation could be caused by air resistance, string elasticity, or slight timing errors. Her instructor would likely consider this measurement successful, especially for a student-level experiment.

Example 3: Tom's Manufacturing Quality Check

Tom works in a factory that produces bolts with a target diameter of 12.00 mm. A batch sample measures 12.15 mm. The quality control specification requires less than 2% error:

• Absolute error: |12.15 - 12.00| = 0.15 mm
• Percent error: (0.15 / 12.00) x 100 = 1.25%
• Quality check: 1.25% is less than the 2% tolerance

The batch passes quality control with a 1.25% error, within the 2% tolerance. However, Tom notes the consistent positive bias (oversized bolts) and recommends checking the machine calibration to prevent the error from trending upward in future batches.

Error Types Reference Table

Tips and Complete Guide

Understanding Systematic vs. Random Error

Errors in measurement fall into two categories. Systematic errors consistently shift measurements in one direction due to flawed equipment, incorrect calibration, or a consistent environmental factor. For example, a scale that reads 0.5 grams too high will cause every measurement to be inflated. Random errors fluctuate unpredictably due to uncontrollable variations like air currents, vibrations, or human reaction time. To minimize systematic error, calibrate instruments against known standards. To minimize random error, take many measurements and calculate the average. Our standard deviation calculator can help quantify the spread of your random errors.

Percent Error in Different Scientific Fields

Acceptable percent error varies significantly by discipline. In analytical chemistry, errors below 1% are expected for well-established methods. In organic chemistry synthesis, yields within 5% of theoretical are considered good. Physics experiments often target 2-3% error, though precision experiments at NIST achieve errors of parts per billion. Medical laboratory tests typically require less than 5% error for diagnostic reliability. Engineering tolerances depend on the application: aerospace components demand sub-0.1% precision, while residential construction may allow 1-2% variation. Always check the specific standards for your field.

Error Propagation in Multi-Step Calculations

When a final result depends on multiple measured values, errors from each measurement combine and propagate through the calculation. For addition and subtraction, absolute errors add. For multiplication and division, relative errors add. For example, if you multiply two values with 2% and 3% relative errors, the result has approximately 5% relative error. Understanding error propagation helps you identify which measurement contributes the most to the overall uncertainty, so you can focus improvement efforts where they matter most.

Significant Figures and Percent Error

The number of significant figures in your answer should reflect the precision of your measurements. If your experimental value has three significant figures, reporting a percent error to six decimal places is misleading — it implies precision that does not exist. Generally, match the number of significant figures in your percent error to those in your least precise measurement. For example, if you measured 9.8 m/s2 (two significant figures) versus a theoretical 9.81 m/s2, reporting 1% error is more appropriate than 1.0194%. Use our rounding calculator to round results to the correct number of significant figures.

Reducing Percent Error Through Better Experimental Design

Beyond using better instruments, you can reduce percent error through improved experimental methodology. Controlling environmental variables is critical: temperature, humidity, air currents, and vibrations can all introduce measurement error. Perform your measurements in a stable environment whenever possible. Taking multiple independent measurements and averaging them reduces random error by a factor of 1/sqrt(n), where n is the number of measurements. Five measurements reduce random error by about 55% compared to a single measurement, while ten measurements reduce it by about 68%. Blinding the experimenter, when possible, prevents unconscious bias from affecting readings. Calibrating your instruments against known standards before each session eliminates drift-related systematic errors. Recording all environmental conditions alongside your measurements allows you to identify and correct for systematic patterns after the fact. Finally, designing your experiment to measure quantities that are as large as possible relative to the instrument resolution maximizes the signal-to-noise ratio and directly reduces percent error. A thermometer with 0.1-degree resolution produces 1% error on a 10-degree measurement but only 0.1% error on a 100-degree measurement.

Common Mistakes to Avoid

• Swapping experimental and theoretical values: The theoretical value goes in the denominator. Putting the experimental value in the denominator gives a different (incorrect) result. Always ask which value is the accepted reference.
• Forgetting absolute value: Without absolute value, you get a signed error that can be misleading. The standard percent error formula always produces a non-negative result. Use signed error only if your instructor specifically requests it.
• Confusing percent error with percent difference: Percent error requires a known true value as reference. Percent difference is for comparing two experimental values with no known standard. Using the wrong formula leads to incorrect conclusions.
• Dividing by zero: When the theoretical value is zero, percent error is undefined. Use absolute error or an alternative metric instead. Some students incorrectly report 0% error in this case.
• Over-reporting precision: Reporting percent error to many decimal places when your instruments only have limited precision creates a false sense of accuracy. Round your result to match the precision of your input data.

Frequently Asked Questions