What Is The Formula For Calculating Percentage Error, And What Is The Percentage Error In The Given Metal Density Measurement?

In experimental sciences, accuracy and precision are paramount. When conducting experiments, it's crucial to evaluate the reliability of the results obtained. One common method for this evaluation is calculating the percent error. Percent error provides a quantitative measure of the difference between an experimental value and a known, accepted value. This article delves into the concept of percent error, its significance, and a step-by-step guide on how to calculate it. We will use an example related to density measurement in chemistry to illustrate the process.

Percent error is a crucial metric in various scientific disciplines, including chemistry, physics, and engineering. It helps researchers and students assess the accuracy of their experimental results. In essence, it quantifies the discrepancy between an experimental value (the value obtained during an experiment) and the accepted, or theoretical, value (the value generally recognized as the true or standard value). The formula for calculating percent error is straightforward:

% Error = |(Accepted Value - Experimental Value) / Accepted Value| × 100

Where:

• Accepted Value: This is the true or standard value, often obtained from reference materials or established scientific literature. It's the value that is widely recognized as the correct value for the property being measured.
• Experimental Value: This is the value obtained during the experiment or measurement. It's the result that the researcher or student has obtained through their experimental procedure.
• The absolute value (denoted by the vertical bars) ensures that the percent error is always a positive value, as we are interested in the magnitude of the difference, not the direction.

Why is Percent Error Important?

Percent error helps determine the reliability of experimental data. A low percent error indicates that the experimental value is close to the accepted value, suggesting that the experimental procedure was accurate and the results are reliable. Conversely, a high percent error suggests that there were significant errors in the experimental procedure or measurements, and the results may not be reliable. Identifying and understanding the source of errors is a critical part of the scientific process, as it allows for refinement of experimental techniques and improvement of future results.

To calculate percent error, follow these steps:

1. Identify the Accepted Value: Determine the accepted, or true, value for the measurement. This value is usually found in reference materials, textbooks, or scientific literature. Make sure you use the correct units for the accepted value.
2. Determine the Experimental Value: Identify the value obtained from your experiment or measurement. Again, ensure the units are consistent with the accepted value.
3. Calculate the Absolute Difference: Subtract the experimental value from the accepted value. Take the absolute value of the result. This step gives you the magnitude of the difference between the two values, regardless of the sign.
4. Divide by the Accepted Value: Divide the absolute difference by the accepted value. This step normalizes the difference, giving you the error as a fraction of the accepted value.
5. Multiply by 100: Multiply the result from the previous step by 100 to express the error as a percentage. The final result is the percent error.

Let's consider a practical example to illustrate the calculation of percent error. Suppose a student is tasked with determining the density of a metal sample in the laboratory. The metal is thought to be iron, which has an accepted density of 7.86 g/cm³. The student performs multiple trials and obtains the following experimental densities:

• Trial 1: 7.78 g/cm³
• Trial 2: 7.80 g/cm³
• Trial 3: 7.82 g/cm³

To calculate the percent error, we first need to determine the average experimental density. Then, we can apply the percent error formula.

1. Calculate the Average Experimental Density

To find the average experimental density, add the densities from each trial and divide by the number of trials:

Average Density = (7.78 g/cm³ + 7.80 g/cm³ + 7.82 g/cm³) / 3
Average Density = 23.40 g/cm³ / 3
Average Density = 7.80 g/cm³

The average experimental density is 7.80 g/cm³.

2. Apply the Percent Error Formula

Now that we have the average experimental density, we can calculate the percent error using the formula:

% Error = |(Accepted Value - Experimental Value) / Accepted Value| × 100
% Error = |(7.86 g/cm³ - 7.80 g/cm³) / 7.86 g/cm³| × 100

3. Calculate the Difference

Subtract the experimental value (average density) from the accepted value:

Difference = 7.86 g/cm³ - 7.80 g/cm³
Difference = 0.06 g/cm³

4. Divide by the Accepted Value

Divide the difference by the accepted value:

Fractional Error = 0.06 g/cm³ / 7.86 g/cm³
Fractional Error ≈ 0.00763

5. Multiply by 100

Multiply the fractional error by 100 to express it as a percentage:

% Error = 0.00763 × 100
% Error ≈ 0.763%

Therefore, the percent error in this density measurement is approximately 0.763%.

In this example, the percent error is 0.763%, which is a relatively low value. Generally, a percent error of less than 5% is considered acceptable in many experimental settings, suggesting that the experimental procedure was performed with reasonable accuracy. However, the acceptable percent error can vary depending on the specific experiment and the required level of precision. For instance, in highly sensitive experiments or industrial quality control, a much lower percent error may be required.

A high percent error, on the other hand, indicates that there might be significant discrepancies between the experimental results and the accepted value. This could be due to several factors, including:

• Systematic Errors: These are consistent errors that occur due to a flaw in the experimental setup or procedure. For example, an improperly calibrated instrument or a consistent mistake in measurement technique.
• Random Errors: These are unpredictable errors that can occur due to chance variations in the experimental conditions or the observer's judgment. Random errors can be minimized by performing multiple trials and calculating the average value.
• Human Errors: These errors occur due to mistakes made by the person conducting the experiment, such as misreading a scale or incorrectly recording data.

Identifying the source of errors is a critical part of the scientific process. If the percent error is high, the researcher needs to carefully examine the experimental procedure, equipment, and techniques to identify and correct the source of the error. This may involve recalibrating instruments, refining the experimental technique, or repeating the experiment with greater care.

Calculating percent error is an essential practice in experimental sciences. It provides a quantitative measure of the accuracy of experimental results, allowing researchers and students to assess the reliability of their data. By comparing experimental values with accepted values and calculating the percent error, scientists can identify potential sources of error and refine their experimental techniques. A low percent error indicates that the experimental results are accurate and reliable, while a high percent error suggests that there may be significant errors in the experimental procedure or measurements. Understanding and minimizing errors is a crucial part of the scientific process, leading to more reliable and accurate results.

In the example we discussed, the percent error in the density measurement of a metal sample was found to be approximately 0.763%, indicating a high level of accuracy. By following the steps outlined in this article, you can confidently calculate and interpret percent error in various experimental scenarios, ensuring the validity and reliability of your scientific findings.