Percent error is a fundamental measure of accuracy, quantifying the difference between an experimental value and a known, true value (the theoretical value). It assesses the quality of a measurement by expressing this deviation as a percentage of the true value. While the simple mathematical calculation can certainly yield a negative result, conventional practice in many scientific disciplines is to report the percent error as a positive number. Mathematically, percent error can be negative, but convention often dictates that it is presented without a sign.
Defining the Percent Error Calculation
The percent error calculation compares the experimental value to the theoretical value. The first step is determining the error, defined as the difference between the experimental value and the theoretical value: \(text{Error} = text{Experimental Value} – text{Theoretical Value}\).
This difference forms the numerator of the percent error formula and determines the potential for a negative result. If the experimental value is smaller than the theoretical value, the error will be negative, and if it is greater, the error will be positive.
The final step involves dividing this error by the theoretical value and multiplying by 100 to express the deviation as a percentage. This raw calculation retains the sign of the error, providing a signed percentage that reflects the direction of the deviation.
The Standard Use of Absolute Value
In many scientific and engineering contexts, the primary goal of calculating percent error is to measure the magnitude of the deviation, regardless of whether the measured value was higher or lower than expected. This focus on the distance from the true value leads to the standard convention of incorporating the absolute value function into the formula. The absolute value bars are placed around the numerator (the difference between the experimental and theoretical values), effectively removing any negative sign.
The formula is often written as: \(text{Percent Error} = frac{|text{Experimental Value} – text{Theoretical Value}|}{text{Theoretical Value}} times 100%\). Using the absolute value ensures the resulting percent error is always a non-negative number, representing the relative size of the error. When reported this way, the sign information is intentionally discarded. This positive value is useful for comparing the precision of different experiments or ensuring a measurement falls within a specified tolerance range, such as a \(pm 5%\) limit.
What a Negative Sign Indicates
When the absolute value function is intentionally omitted from the percent error calculation, the resulting sign carries valuable information about the nature of the measurement. A negative percent error specifically indicates that the experimental value obtained was less than the theoretical value. For example, if the accepted density is \(2.70 text{ g/cm}^3\) and a student measures \(2.65 text{ g/cm}^3\), the calculation yields a negative result, showing the measurement was below the true value.
This retained sign provides diagnostic insight into the experimental procedure, which is often preferred in chemistry and other sciences. If a calculated percent error for a chemical yield is negative, it suggests the measured product amount was lower than the theoretical maximum. This is expected due to factors like incomplete reactions or product loss during purification.
Conversely, a positive percent error means the experimental value was higher than the theoretical value. In the context of a yield calculation, this might suggest the presence of unreacted starting material or excess solvent, indicating a procedural flaw. Analyzing the sign allows researchers to pinpoint the likely source of error, determining if the process resulted in consistent overestimation or underestimation.