Find Percent Error Calculator

Easily calculate the percent error between an observed (measured or experimental) value and a true (accepted or theoretical) value. Enter your values below to find the percent error.

Visualizing the Error

What is Percent Error?

The percent error is a measure of the discrepancy between an observed value and a true, accepted, or theoretical value, expressed as a percentage of the true value. It’s a common way to report the difference between an experimental result and a known or expected result in science, engineering, and various other fields. A smaller percent error indicates that the observed value is close to the true value, suggesting higher accuracy in the measurement or observation.

Anyone conducting experiments, taking measurements, or comparing estimated values against actual values should use the percent error calculation. This includes students, scientists, engineers, and analysts.

A common misconception is that a negative percent error is possible with the standard formula, but because the formula typically uses the absolute difference, the percent error is usually expressed as a non-negative value. However, some conventions might omit the absolute value to indicate the direction of the error (whether the observed value is higher or lower than the true value).

Percent Error Formula and Mathematical Explanation

The formula to calculate the percent error is:

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

Where:

• Observed Value (O): The value obtained from measurement or experiment.
• True Value (T): The accepted, theoretical, or known value.

The steps to calculate percent error are:

1. Calculate the absolute difference (Absolute Error) between the observed and true values: |O – T|.
2. Divide the absolute difference by the absolute value of the true value: |O – T| / |T|. This is the Relative Error.
3. Multiply the result by 100 to express it as a percentage.

It’s important that the True Value is not zero, as division by zero is undefined.

Variables Table

Variable	Symbol	Meaning	Unit	Typical Range
Observed Value	O	The measured or experimental value.	Varies (e.g., meters, kg, seconds)	Any real number
True Value	T	The accepted or theoretical value.	Same as Observed Value	Any non-zero real number
Absolute Error	|O – T|	The magnitude of the difference between O and T.	Same as Observed Value	Non-negative real number
Relative Error	|O – T| / |T|	The absolute error relative to the true value.	Dimensionless	Non-negative real number
Percent Error	PE	The relative error expressed as a percentage.	%	Non-negative real number (%)

Table of variables used in percent error calculation.

Practical Examples (Real-World Use Cases)

Example 1: Chemistry Experiment

A student measures the boiling point of water to be 99.5 °C. The accepted boiling point of water at standard pressure is 100.0 °C.

• Observed Value (O) = 99.5 °C
• True Value (T) = 100.0 °C

Absolute Error = |99.5 – 100.0| = |-0.5| = 0.5 °C

Relative Error = 0.5 / |100.0| = 0.005

Percent Error = 0.005 * 100% = 0.5%

The student’s measurement has a percent error of 0.5%, indicating a reasonably accurate measurement.

Example 2: Manufacturing Process

A machine is supposed to cut metal rods to a length of 50.0 cm. A quality control check measures a sample rod to be 50.3 cm long.

• Observed Value (O) = 50.3 cm
• True Value (T) = 50.0 cm

Absolute Error = |50.3 – 50.0| = 0.3 cm

Relative Error = 0.3 / |50.0| = 0.006

Percent Error = 0.006 * 100% = 0.6%

The measured rod has a percent error of 0.6% in its length compared to the specification.

How to Use This Percent Error Calculator

1. Enter Observed Value: In the “Observed Value (O)” field, type the value you obtained through measurement or experiment.
2. Enter True Value: In the “True Value (T)” field, type the accepted, theoretical, or standard value you are comparing against. Ensure this value is not zero.
3. Calculate: The calculator will automatically update the results as you type, or you can click the “Calculate” button.
4. Read Results: The “Percent Error (%)” will be displayed prominently, along with the “Absolute Error” and “Relative Error”.
5. Reset: Click “Reset” to clear the fields and results for a new calculation.
6. Copy: Click “Copy Results” to copy the main result and intermediate values to your clipboard.

The displayed percent error tells you how much your observed value deviates from the true value, relative to the true value, as a percentage. A lower percent error generally means your observed value is closer to the true value.

Key Factors That Affect Percent Error Results

Several factors can influence the percent error in measurements and experiments:

1. Instrument Precision and Calibration: The quality, precision, and calibration of the measuring instrument significantly impact the observed value. Uncalibrated or low-precision instruments lead to larger errors.
2. Experimental Method/Procedure: The method used to conduct the measurement can introduce systematic or random errors, affecting the observed value and thus the percent error.
3. Environmental Conditions: Temperature, pressure, humidity, and other environmental factors can affect both the system being measured and the instruments, leading to variations in the observed value.
4. Human Error: Mistakes in reading instruments, recording data, or performing calculations can contribute to the difference between the observed and true values, increasing the percent error. Explore our experimental error analysis steps for more info.
5. Parallax Error: This occurs when the position of the observer affects the reading of an instrument (like a ruler or a needle on a gauge).
6. Purity of Substances (in Chemistry/Physics): If the substances used are not pure, the true value expected might differ, or the observed value might deviate due to contaminants.
7. Assumptions in the Model: If the true value is derived from a theoretical model, any simplifications or assumptions made in that model can contribute to a discrepancy when compared with an experimental value, affecting the percent error. Understanding accuracy vs precision is crucial here.

Understanding these factors is key to minimizing percent error and improving the reliability of experimental results. For those needing precise calculations, consider our significant figures calculator.

Frequently Asked Questions (FAQ)

What is an acceptable percent error?

The acceptable percent error varies greatly depending on the field and application. In precise scientific experiments, it might be very low (e.g., <1%), while in other areas, 5-10% might be acceptable.

Can percent error be negative?

If the absolute value is NOT used in the numerator (|O – T|), then the error can be negative (if O < T) or positive (if O > T). However, the standard percent error formula uses the absolute value, making it non-negative. A negative value without the absolute would indicate the observed value is less than the true value.

What if the true value is zero?

The percent error formula involves division by the true value, so it is undefined if the true value is zero. In such cases, only the absolute error (difference) is meaningful.

What is the difference between percent error and percent difference?

Percent error compares an observed value to a known true value. Percent difference is used when comparing two experimental values to each other when neither is considered the “true” value; it’s often calculated based on the average of the two values.

How can I reduce percent error?

Use more precise instruments, ensure proper calibration, refine experimental techniques, control environmental conditions, and minimize human error through careful observation and repetition.

Is a high percent error always bad?

Not necessarily. It depends on the context. A high percent error might be expected in very complex systems or with instruments of limited precision. The key is to understand the sources of error.

What is the difference between absolute error and relative error?

Absolute error is the simple difference |Observed – True|, having the same units as the measurement. Relative error is the absolute error divided by the true value, and it’s dimensionless or expressed as a percentage (percent error). Check out our absolute error calculator and relative error calculator.

Why is it important to calculate percent error?

Calculating percent error helps assess the accuracy of a measurement or experiment, identify potential problems with methods or instruments, and validate results against known standards. See our guide on measurement uncertainty.