Percent Error

Percent error is a key concept in science and mathematics that helps measure how accurate experimental results are compared to accepted values. Scientists and mathematicians use it to check the reliability of their findings. For example, Eratosthenes used it to assess the accuracy of his Earth circumference calculation. Today, percent error is widely used in fields like engineering, physics, chemistry, and economics to evaluate the credibility of measurements and results.

Percent error is a way to measure how inaccurate a measurement is relative to its true or accepted value. It tells us how far off the measurement was, expressed as a percentage.

When calculating percent error, if the answer is 0, it is precisely correct. When the error is slightly off, we get a small percent error. When the error is very wrong, we get a significant percentage error.

We can calculate percent error using a percent error formula. The standard formula for percent error is given as,

Percent error formula:

\(\text{Percent Error} = \frac{|\text{Experimental Value} - \text{True Value}|}{\text{True Value}} \times 100 \)

Where,
Experimental value = the value we measured or estimated.
True value = the correct value
The absolute value | • | ensures the error is always positive.

When we find the percent error, we can understand the impact caused by the error. The formula for percent error is:

\(\text{Percent Error} = \frac{|\text{Experimental Value} - \text{True Value}|}{\text{True Value}} \times 100 \)

Here, the measured value is the result we get after the calculation. True value is the established value.

Cross-Disciplinary Usage: The percent error equation is used not only in math but also in other disciplines, such as physics, computer science, and logical reasoning. We can learn how close an experimental calculation comes to the standard value in physics. Similarly, in chemistry, we use the percent error calculator to compare the actual and theoretical chemical yields. The percent error calculation helps students preparing for competitive exams like GMAT, SAT, etc., by showing its relevance across various questions.

Look at this example that shows how the formula works:

The measured value of a machine is 50 cm. But the actual (true) value of the machine is 48 cm. Now let us calculate the percent error.

The formula for percent error calculation is:

\(\text{Percent Error} = \frac{|\text{Experimental Value} - \text{True Value}|}{\text{True Value}} \times 100 \)

\(\text{Percent error} = \left(\frac{50 - 48}{48}\right) \times 100\)

First, subtract the true value from the measured value:

\(50 − 48 = 2\)

Next, divide 2 by the true value:
 

\(\frac{2}{48} = 0.0416\)
 

Now, we can multiply 0.0416 by 100:
 

\(0.0416 × 100 = 4.16\)

Thus, the percent error is 4.16%.

Percent error calculation is a simple method to find the variation between two values, one is the experimental value, and the other is the actual value. Percent error is a commonly used concept in various fields to estimate the accuracy of results. 

The equation for percent error can be further understood by breaking down all the steps.

Step 1: Find the difference between the measured value and the true value.

\(\text{Difference} = |\text{Experimental Value} - \text{True Value}| \)

Step 2: Now, we find the ratio between the difference and the true value.

\(\text{Ratio} = \frac{\text{Difference}}{\text{True Value}} \)

Step 3: Multiply the ratio by 100 to get the percentage.

\(\text{Percent Error} = \text{Ratio} \times 100 \)

Average percent error

The average percentage error, or the mean percentage error (MPE), can be calculated using the formula;

\(\text{MPE} = \frac{100\%}{n} \sum_{i=1}^{n} \left|\frac{T_i - E_i}{T_i}\right|\)

Where,
‘n’ is the number of quantities in the data set.
\(T_i\) is the actual or true value.
\(E_i\) is the estimated value.
\(\sum\) is the summation of the absolute value for all predicted points in time.

When calculating data, the results can be different from the actual answers. To solve this problem, we use percent error to analyze the error and the accuracy of the answer. Students who focus on laboratory research can determine how far their experimental values match the actual values.

Percent error helps identify inconsistencies after an experiment, allowing students to correct mistakes in future trials. In the field of medical science, percent error plays a vital role. It is used to check the specific standards of products in manufacturing sectors and to assess the accuracy of measurements in engineering. 

To assess the precision of measurements in a calculation or an experiment, researchers use the key concept of percent error. It helps analyze result reliability. Understanding the various types of percent error will enable students to achieve accurate results. The different types of percent errors are explained below in detail.

Absolute percent error:

An absolute percent error calculates the difference between a measured value and the true value. It ignores the degree of measured and true value. It doesn’t consider whether the measured values are greater or lower than the actual values. To find the absolute error, we can use the formula:

Absolute Error = Measured Value − True Value

For example, if percent error = ((50 − 48) / 48) × 100

The absolute value of the percent error is (2/48) × 100 = 4.1666...

Relative percent error:

Relative percent error focuses on the error’s relative size. It compares the error to the measured value. When comparing different types of errors in calculations, relative percent error is helpful. 

For example, if percent error = ((12.4 − 12.0) / 12.0) × 100

The relative percent error = (measured value − true value) / true value 

Percent error = (12.4 − 12.0) / 12.0 = 0.033

Percent error in experimental value:

It analyzes the difference between experimental values and the expected values. This is commonly used in the field of experimental science, where professionals need to know how much the supposed value deviates from the true value. 

For example, percent error = ((experimental value − true value) / true value) × 100

Percent error in the measurements of constants:

When we measure the speed of light or the gravity acceleration, the percent error indicates how close the measured value is to the actual value.

For example,

Percent error = ((Measured constant value − Known constant value) / Known constant value) × 100.

 

Determining how close the measured value is to the actual value helps students analyze the accuracy of their calculations. Following some tips and tricks will help kids improve their knowledge of percent error calculation. Here, some of the tips and tricks are listed.
 

• Don't consider the negative sign in the calculation. If you get a negative result, consider it as a positive answer. For example, if you get −1.4, consider it as 1.4. The negative sign is ignored because percent error is always about the immensity of the error, not its direction.
   
• Be careful to apply the correct formula for percent error calculation. The denominator will be the true value. Use proper values for each term.
   
• To get the final value as a percentage, multiply the value by 100. This converts a decimal into a percentage. 
   
• Double-check the values to find the highest percent error. A large percent error indicates a major mistake in the calculation.
   
• Teachers should break the formula into 3 steps when teaching, rather than showing the full formula at once. It will reduce mistakes and build confidence.
• Parents should encourage their children to use simple and round numbers first. Start working with easy and true values like 10, 50, 100… and then introduce trickier numbers later.
   
• Parents and teachers should teach the concept involving real-life experiences, like when we try to find the percent change of a science experiment, we get small percent errors. Real measurements can never be perfectly exact.
   
• Parents should encourage their children to double-check the answers. Ask them to re-measure, recalculate, check units, and compare the answers with a friend.

• Measured value: It is the value obtained from an experiment or calculation. Sometimes the value may contain errors and it will lead to incorrect results.  
   
• True value: It is the established value or the actual value in a calculation or experiment. To ensure the accuracy of the measured value, the true value is used as a reference. 
   
• Percent error: It is the absolute difference between the estimated value and the actual value. The difference is divided by the actual value and the answer is multiplied by 100 to denote it as a percentage. 
   
• Absolute percent error: It measures the difference between a measured value and the actual value. To calculate the absolute percent error, we have to subtract the measured value from the actual value.