Relative Error Calculator

Quantify the precision of measurements by comparing observed values to true values.

This tool helps you calculate the relative error, which is a measure of the uncertainty of measurement compared to the size of the measurement.

Practical Examples

See how the Relative Error Calculator works with real-world data.

Chemistry Experiment

basic

A student measures the mass of a compound to be 10.2g, but the known mass is 10.5g.

True Value: 10.5

Observed Value: 10.2

Manufacturing Tolerance

real-world

A manufactured rod is supposed to be 50cm long (true value), but it measures 50.1cm.

True Value: 50

Observed Value: 50.1

Financial Forecasting

finance

A company forecasts a quarterly profit of $250,000, but the actual profit is $245,000.

True Value: 250000

Observed Value: 245000

Measuring Gravity

physics

In a lab, the acceleration due to gravity is measured as 9.7 m/s², while the accepted value is 9.81 m/s².

True Value: 9.81

Observed Value: 9.7

Other Titles
Understanding Relative Error: A Comprehensive Guide
An in-depth look at what relative error is, how to calculate it, and why it's a crucial concept in science, engineering, and data analysis.

What is Relative Error?

  • Defining the Core Concept
  • Absolute vs. Relative Error
  • The Importance of Context in Error Analysis
Relative error, also known as percentage error, is a measure of the inaccuracy of a measurement in relation to the size of that measurement. Unlike absolute error, which simply gives the magnitude of the difference between a measured value and a true value, relative error provides context. A 1-meter error is significant when measuring a 10-meter room but negligible when measuring the 400,000-kilometer distance to the moon. Relative error expresses this discrepancy as a fraction or a percentage, making it a more insightful and universally applicable metric for assessing precision.
The Formula for Relative Error
The calculation is straightforward. First, you find the absolute error, and then you divide it by the absolute value of the true value. The formula is: Relative Error = |(Observed Value - True Value) / True Value|. To express it as a percentage, you simply multiply the result by 100.

Simple Examples

  • If the true value is 50 and the observed value is 48, the absolute error is 2. The relative error is 2/50 = 0.04, or 4%.
  • If the true value is 1000 and the observed value is 998, the absolute error is also 2. However, the relative error is 2/1000 = 0.002, or 0.2%, indicating a much more precise measurement.

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

  • Inputting Your Values
  • Interpreting the Results
  • Common Pitfalls to Avoid
Our calculator simplifies the process into two easy steps.
Step 1: Enter the True and Observed Values
In the 'True Value' field, enter the accepted, theoretical, or known value. In the 'Observed Value' field, enter the value you measured or recorded. The calculator requires both inputs to be numerical values.
Step 2: Calculate and Analyze
Click the 'Calculate' button. The tool will instantly display the Absolute Error, the Relative Error (as a decimal), and the Percentage Error. A key point to remember is that the 'True Value' cannot be zero, as this would make the division undefined. Our calculator will alert you if you attempt this.

Calculation Walkthrough

  • True Value = 200, Observed Value = 205. Absolute Error = |205 - 200| = 5. Relative Error = 5 / 200 = 0.025. Percentage Error = 0.025 * 100 = 2.5%.

Real-World Applications of Relative Error

  • Scientific Research and Experimentation
  • Engineering and Manufacturing
  • Finance and Economics
In the Laboratory
In physics, chemistry, and biology, experimental results are constantly compared against theoretical values. Relative error is essential for validating a hypothesis or assessing the accuracy of an experimental method. A low relative error suggests the experimental results align closely with established theory.
In Manufacturing
Engineers use relative error to determine if a product meets design specifications. For machine parts, microchips, and other precision components, a very small relative error (tolerance) is required to ensure proper function and interoperability.

Practical Scenarios

  • An pharmaceutical company measures the amount of active ingredient in a pill. If the specified amount is 500mg and they measure 499mg, the relative error is a tiny 0.2%, which may be within acceptable limits.
  • An economist predicts a country's GDP will be $2.5 trillion. If the actual GDP is $2.4 trillion, the relative error of 4% indicates the forecast's accuracy.

Common Misconceptions and Correct Methods

  • Confusing Absolute and Relative Error
  • Ignoring the Sign of the Error
  • Averaging Percentage Errors Incorrectly
Absolute vs. Relative Error
The most common mistake is to only consider the absolute error. As discussed, a 1-unit error means very different things for a small true value versus a large one. Always use relative error for a contextual understanding of precision.
The Significance of the Sign
While the standard formula for relative error uses the absolute value, sometimes the sign of the error (before taking the absolute value) is important. A positive error means the observed value is higher than the true value, while a negative error means it's lower. This can provide insight into systematic biases in measurement.

Correction Examples

  • Mistake: Saying a 2-gram error is always small. Correction: A 2g error on a 10g measurement is a 20% error, which is large. A 2g error on a 1000g measurement is a 0.2% error, which is small.
  • Mistake: Simply averaging the percentage errors of multiple measurements. Correction: It's often better to calculate the relative error of the average of the measurements, or use more advanced statistical techniques like root mean square error.

Mathematical Derivation and Examples

  • Deriving the Formula
  • Worked Example 1: Physical Measurement
  • Worked Example 2: Statistical Polling
Formula Breakdown
1. Absolute Error (AE): This is the first step. It is the absolute difference between the observed value (Vobs) and the true value (Vtrue). Formula: AE = |Vobs - Vtrue|
2. Relative Error (RE): This step puts the absolute error into perspective. It is the Absolute Error divided by the absolute value of the True Value. Formula: RE = AE / |V_true|
3. Percentage Error (PE): This is the most common way to express relative error. Formula: PE = RE * 100%

Detailed Walkthrough

  • A poll predicts a candidate will receive 54% of the vote. They actually receive 51.5%. True Value = 51.5. Observed Value = 54. Absolute Error = |54 - 51.5| = 2.5. Relative Error = 2.5 / 51.5 ≈ 0.0485. Percentage Error ≈ 4.85%.