Percent error

Here you will learn about percent error, including how to calculate it and use it to solve problems.

Students first learn about percent error in the 7 th grade and expand this knowledge as they move through middle school and high school statistics.

What is percent error?

Percent error is a type of percent problem that describes the difference between an estimated, observed, or experimental value, and an actual, accepted, or real value as a percent of the actual value. In other words, percent error is how big your error is when you estimate a measurement. In experimentation, it can tell you how far off your observed value is from the true, accepted value. This can help in evaluations and adjustments in various scientific and statistical applications including science, engineering, and finance.

The way you calculate a percent error is similar to how you calculate a percent change (percentage change).

To calculate the percent error you can apply the following formula:

In chemistry, physics and other sciences, the actual value or theoretical value can be the established value you would expect as a result of an experiment. For example, the actual value of the boiling point of water is 100℃ .

The actual value is important for the following three reasons:

1. Baseline for comparison: Theoretical values act as a standard or expected result that experimental values are compared against.
2. Accuracy check: By comparing an experimental result to the theoretical value, scientists can calculate the percent error. This calculation highlights the accuracy and precision of the experiment.
3. Error measurement: Theoretical value is used at the base of the percent error formula (in the denominator), crucial for quantifying the deviation of the experimental result from the expected one.

The difference in values is the positive difference between the estimated value and the actual value (or the absolute value of the error). This is also known as the absolute error.

The ratio of the difference in values to the actual value, , is known as the relative error.

Let’s look at an example where percent error can be applied in real-life scenarios.

Lucas estimated the weight of his pet rabbit, Blu, to be 5.5 pounds. Blu’s actual weight is 4.75 pounds. Find the percent error.

\begin{aligned}& \text {estimated weight }=5.5 \; pounds \\\\ & \text {actual weight }=4.75 \; pounds \\\\ & \text {Difference in values also known as the absolute error: } 5.5 - 4.75=0.75 \end{aligned}

Using the formula,

Lucas’s percent error is 15.8 \% meaning his estimate was about 15.8 \% off from the actual weight measurement.

What is percent error?

Common Core State Standards

How does this apply to 7 th grade math?

• Grade 7 – Ratios and Proportional Relationships (7.RP.A.3)
  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error

How to find percent error

In order to find the percent error:

1. Find the positive difference between the actual value and the estimated value.
2. Use the formula to calculate.

Percent error examples

Example 1: finding percent error

Elba estimated that 190 people are going to attend the spring concert. The actual number of people attending was 214. Find the percent error.

1. Find the positive difference between the actual value and the estimated value.

\begin{aligned}& \text {Estimated value }=190 \\\\ & \text {Actual value }=214 \\\\ & \text {Difference in values: } 214-190=24 \end{aligned}

2Use the formula to calculate.

(you may have to round the division to three decimal places)

The percent error is 11.2 \%

Example 2: finding percent error

If the actual value is 74 and the estimated value is 70, what is the percent error?

Use the formula to calculate.

(you may have to round the division to three decimal places)

The percent error is 5.4 \%

Example 3: finding percent error

Sophia is working at a vet clinic. She estimated the weight of a cat to be 10 pounds. When she weighed the cat, the actual weight was 8.5 pounds. What is the percent error?

Use the formula to calculate.

(you may have to round the division to three decimal places)

The percent error is 17.6 \%

Example 4: finding percent error

Ryan thinks the cost of a video game is \$ 65. However, the actual cost is \$ 58. What is the percent error?

Use the formula to calculate.

Ryan’s percent error is 12.1 \%

Example 5: finding percent error

Johnny estimated the population of his small town to be 3,225 people. The actual population of his town is 4,057. What is the percent error?

Use the formula to calculate.

Johnny’s percent error is 25.8 \%.

Example 6: finding percent error

Ginny wants to buy a new gaming device for her nephew. She estimates the cost to be \$ 250. If the percent error is 8 \% above the actual value, what is the actual price of the device? (round to the nearest hundredth).

Use the formula to calculate.

The actual price of the gaming device is \$ 231.48

Teaching tips for percent error

• Incorporate discovery based learning activities so students can develop the formula for finding percent error.

• Although worksheets have their place, consider using real world examples that are relevant to students to find percent error or percent change.

• Students have access to percent error calculators online. However, having them demonstrate and explain their work will build understanding as opposed to just typing numbers into a digital calculator.

Easy mistakes to make

• Forgetting to multiply the answer by \textbf{100}
  Finding percent error means to actually find a percent. After doing the calculations of finding the difference between the values and then dividing by the actual value, students forget to multiply by 100.
  
  For example, find the percent error if the estimated amount is 56 and the actual amount is 68.
  
  \text {Percent error }=0.176 — cannot leave the answer as a decimal.
  The answer must be converted to a percent, so multiplying by 100 is essential.
  \text {Percent error }=0.176 × 100=17.6 \%

• Finding the negative difference
  When finding percent error, it technically does not matter which way you find the difference. For example, finding the percent error between an estimated value of 14 and an actual value of 19.
  
  You can find the difference by: 19-14 OR 14-19 , in either case, be sure to take the absolute value because it’s the positive difference between the values.

Related percent lessons

• Percent increase
• Percent decrease
• Percent increase and decrease
• Simple interest
• Exponential decay
• Compound interest formula

The next lessons are

• Compound measures
• Converting fractions, decimals and percents
• Ratio

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