Percent Error Calculator

Calculate the difference between an observed and a true value as a percentage.

Enter the observed (measured) value and the true (accepted) value below to compute the percent error.

Please enter a valid number.

Please enter a valid number.

Practical Examples

See how the Percent Error Calculator works with real-world data. Click on an example to load it.

Physics Experiment: Gravity

example1

A student measures the acceleration due to gravity as 9.5 m/s², while the accepted value is 9.8 m/s².

Observed: 9.5

True: 9.8

Chemistry Titration

example2

In a titration, a student finds the concentration of an acid to be 0.102 M, but the known concentration is 0.100 M.

Observed: 0.102

True: 0.100

Manufacturing Tolerance

example3

A machine part is manufactured to be 50.1 mm long, but the required specification is 50.0 mm.

Observed: 50.1

True: 50.0

Financial Forecasting

example4

An analyst predicted a company's earnings per share to be $2.55, but the actual earnings were $2.50.

Observed: 2.55

True: 2.50

Other Titles
Understanding Percent Error: A Comprehensive Guide
A deep dive into what percent error is, how to calculate it, and why it's a crucial concept in science, engineering, and data analysis.

What is Percent Error?

  • Defining Percent Error
  • The Importance of Accuracy vs. Precision
  • The Formula Explained
Defining Percent Error
Percent error is a measure of how inaccurate a measurement is, expressed as a percentage of the accepted or true value. It's a fundamental concept in any field that relies on data and measurement, from scientific experiments to financial analysis. A low percent error indicates that the observed value is very close to the true value, signifying high accuracy.
The Importance of Accuracy vs. Precision
It's crucial to distinguish between accuracy and precision. Accuracy refers to how close a measurement is to the true or accepted value. Percent error directly measures accuracy. Precision, on the other hand, refers to how close multiple measurements are to each other, regardless of their proximity to the true value. You can be precise without being accurate, but the goal is always to be both.
The Formula Explained
The formula for percent error is: Percent Error = [|Observed Value - True Value| / |True Value|] × 100%. The absolute value in the numerator ensures the error is a positive quantity, as the direction of the error (whether it's higher or lower) doesn't matter for this calculation. The result is then multiplied by 100 to express it as a percentage.

Step-by-Step Guide to Using the Percent Error Calculator

  • Inputting Your Values
  • Interpreting the Result
  • Common Pitfalls to Avoid
Inputting Your Values
To use the calculator, you need two pieces of information: the 'Observed Value' and the 'True Value'. The Observed Value is the result you obtained from your measurement. The True Value is the standard, theoretical, or accepted value that you are comparing against.
Interpreting the Result
After clicking 'Calculate', the tool will display the percent error. For example, a result of 2% means your observed value is 2% different from the true value. The context determines whether this is an acceptable level of error. In high-precision engineering, a 2% error might be significant, while in some social science surveys, it could be negligible.
Common Pitfalls to Avoid
A frequent mistake is swapping the observed and true values. While the absolute difference in the numerator remains the same, it's crucial for conceptual clarity. Another major error is entering a 'True Value' of zero, as this leads to an undefined result (division by zero). Our calculator will alert you if this happens.

Real-World Applications of Percent Error

  • Scientific Research
  • Engineering and Manufacturing
  • Finance and Economics
Scientific Research
In laboratories, percent error is used to validate experimental results. If a chemistry student calculates the yield of a reaction, they will compare their result to the theoretical yield to determine the percent error, which reflects the experiment's success and potential sources of error.
Engineering and Manufacturing
Engineers rely on percent error to assess the quality of products. If a component must have a specific dimension, the manufacturer will measure the actual dimension and calculate the percent error to ensure it falls within acceptable tolerance levels. This is critical for the safety and functionality of machines, buildings, and electronics.
Finance and Economics
Financial analysts and economists use percent error to evaluate the accuracy of their models. For instance, an analyst might predict a company's revenue. After the actual revenue is reported, they can calculate the percent error of their prediction to refine their forecasting models for the future.

Common Misconceptions and Correct Methods

  • Error vs. Mistake
  • Absolute Error vs. Relative Error
  • Negative Percent Error
Error vs. Mistake
In a scientific context, an 'error' is an inherent uncertainty in a measurement, not a 'mistake' or blunder. Mistakes, like misreading a scale, can be avoided. Errors, such as limitations of the instrument, are unavoidable and are quantified by percent error.
Absolute Error vs. Relative Error
The absolute error is simply |Observed - True|. However, this value lacks context. An error of 1 cm is small when measuring a room but large when measuring a pinhead. Percent error is a form of 'relative error' because it relates the error to the size of the true value, providing a more meaningful measure of accuracy.
Negative Percent Error
Some definitions of percent error omit the absolute value to indicate the direction of the error (e.g., a negative percent error means the observed value was lower than the true value). However, the standard definition uses the absolute value to purely quantify the magnitude of the error, which is what our calculator does.

Mathematical Derivation and Examples

  • The Core Formula
  • Worked Example 1: Boiling Point
  • Worked Example 2: Survey Data
The Core Formula
The derivation is straightforward. First, calculate the absolute difference between the observed and true values: Absolute Error = |Observed Value - True Value|. Next, to understand this error in context, you create a ratio of this error to the true value: Relative Error = Absolute Error / |True Value|. Finally, to express this as a percentage, you multiply by 100.
Worked Example 1: Boiling Point
A student measures the boiling point of water at 98.5°C. The true boiling point at standard pressure is 100°C. 1. Absolute Error = |98.5 - 100| = 1.5. 2. Relative Error = 1.5 / 100 = 0.015. 3. Percent Error = 0.015 × 100% = 1.5%.
Worked Example 2: Survey Data
A poll predicted a candidate would receive 45% of the vote. They actually received 48%. The 'observed' value is the poll's prediction, and the 'true' value is the actual outcome. 1. Absolute Error = |45 - 48| = 3. 2. Relative Error = 3 / 48 = 0.0625. 3. Percent Error = 0.0625 × 100% = 6.25%.