Percent Error Calculator | Calculate Percentage Error Online

What is Percent Error?

Percent error is a measure of the discrepancy between an observed value and a true, or accepted, value. It is expressed as a percentage of the true value and is a common way to evaluate the accuracy of a measurement or calculation. A low percent error indicates that the observed value is close to the true value, signifying high accuracy.

The formula for percent error is:

Percent Error = (|Observed Value - True Value| / True Value) * 100%

Where: 'Observed Value' is the value you measured, and 'True Value' (also known as the theoretical or accepted value) is the actual value. The absolute value in the numerator ensures that the error is always positive, as we are typically interested in the magnitude of the error, not its direction (i.e., whether the observed value was higher or lower than the true value).

Calculation Example

Let's say you are conducting a chemistry experiment to determine the boiling point of water. The accepted true value is 100°C. You measure it in your lab and get 98.5°C.
Observed Value = 98.5°C
True Value = 100°C
Percent Error = (|98.5 - 100| / 100) * 100% = (|-1.5| / 100) * 100% = (1.5 / 100) * 100% = 1.5%
Your measurement has a 1.5% error.

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

Our calculator simplifies the process. Here's how to use it effectively:

Input the Values:

Calculate and Interpret the Result:

Usage Tips

Ensure both values are in the same units before calculating.
The 'True Value' cannot be zero, as this would result in an undefined value (division by zero).
Use the 'Reset' button to clear the input fields and start a new calculation.

Real-World Applications of Percent Error Calculations

Percent error is not just an academic exercise; it has vital applications in many professional fields:

Science and Engineering:

Finance and Economics:

Application Examples

A machinist produces a part that is supposed to be 10.00 cm long. They measure it to be 10.02 cm. The percent error is (|10.02 - 10.00| / 10.00) * 100% = 0.2%.
A pollster predicts a candidate will get 45% of the vote. The candidate actually gets 48%. The percent error of the poll is (|45 - 48| / 48) * 100% = 6.25%.
A GPS calculates your distance to a destination as 5.5 miles. The actual, verified distance is 5.8 miles. The GPS has a percent error of (|5.5 - 5.8| / 5.8) * 100% ≈ 5.17%.

Common Misconceptions and Correct Methods in Percent Error

Misconception 1: Percent Error vs. Absolute Error

Absolute error is simply the absolute difference between the observed and true values (|Observed - True|). Percent error provides more context by relating this error to the size of the true value. An absolute error of 1 cm is significant for something 10 cm long (10% error), but negligible for something 1 km long (0.0001% error).

Misconception 2: Negative Percent Error

By convention, percent error is almost always reported as a positive value, hence the use of the absolute value in the formula. The goal is to express the magnitude of the error. If the direction of the error is important, one might use 'percent difference' or simply state whether the observed value was an overestimate or underestimate, but the 'percent error' itself remains positive.

Misconception 3: Using the Observed Value in the Denominator

A common mistake is to divide the absolute error by the observed value instead of the true value. The correct method is always to use the true, accepted, or theoretical value as the reference point in the denominator. This ensures a consistent standard for comparison.

Key Takeaways

Always use the absolute value of the difference.
Always divide by the true (accepted) value.
Express the final result as a percentage.

Mathematical Derivation and Examples

The derivation of the percent error formula is straightforward and rooted in the concept of relative difference.

Step 1: Calculate the Absolute Error

First, we find the simple difference between the two values and take its absolute value to quantify the magnitude of the error, regardless of direction. This is the 'Absolute Error'.

Absolute Error = |Observed Value - True Value|

Step 2: Calculate the Relative Error

Next, to understand the error in context, we compare it to the true value. This is the 'Relative Error', which is a fractional value.

Relative Error = Absolute Error / True Value

Step 3: Convert to a Percentage

Finally, to make the relative error more intuitive and easily comparable, we multiply it by 100 to express it as a percentage.

Percent Error = Relative Error * 100%

Comprehensive Example

Imagine you are trying to measure the acceleration due to gravity, g. The accepted true value is approximately 9.81 m/s².
In your physics experiment using a pendulum, you calculate g to be 9.65 m/s².
1. **Absolute Error:** |9.65 - 9.81| = |-0.16| = 0.16 m/s².
2. **Relative Error:** 0.16 / 9.81 ≈ 0.0163.
3. **Percent Error:** 0.0163 * 100% = 1.63%.
Your experimental result has a percent error of 1.63%.