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 [katex] 7 [/katex]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 [katex] 100℃ [/katex].

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 [katex] 5.5 [/katex] pounds. Blu’s actual weight is [katex] 4.75 [/katex] pounds. Find the percent error.

[katex] \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} [/katex]

Using the formula,

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

Common Core State Standards

How does this apply to [katex] 7 [/katex]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 [katex] 190 [/katex] people are going to attend the spring concert. The actual number of people attending was [katex] 214. [/katex] Find the percent error.

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

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

2Use the formula to calculate.

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

The percent error is [katex] 11.2 \% [/katex]

Example 2: finding percent error

If the actual value is [katex] 74 [/katex] and the estimated value is [katex] 70, [/katex] what is the percent error?

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

Use the formula to calculate.

Example 3: finding percent error

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

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

Use the formula to calculate.

Example 4: finding percent error

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

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

Use the formula to calculate.

Example 5: finding percent error

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

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

Use the formula to calculate.

Ginny wants to buy a new gaming device for her nephew. She estimates the cost to be [katex] \$250. [/katex] If the percent error is [katex] 8 \% [/katex] above the actual value, what is the actual price of the device? (round to the nearest hundredth). Find the positive difference between the actual value and the estimated value. Use the formula to calculate. 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 [katex] \textbf{100} [/katex] 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 [katex] 100. [/katex] For example, find the percent error if the estimated amount is [katex] 56 [/katex] and the actual amount is [katex] 68. [/katex] [katex] \text {Percent error }=0.176 [/katex] — cannot leave the answer as a decimal. The answer must be converted to a percent, so multiplying by [katex] 100 [/katex] is essential. [katex] \text {Percent error }=0.176 × 100=17.6 \% [/katex] • 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 [katex] 14 [/katex] and an actual value of [katex] 19. [/katex] You can find the difference by: [katex] 19-14 [/katex] OR [katex] 14-19 [/katex], in either case, be sure to take the absolute value because it’s the positive difference between the values. Practice percent error questions 1. Tim estimated the weight of his dog to be [katex] 20 [/katex] pounds. When he took his dog to the vet, the dog weighed [katex] 18 [/katex] pounds. What is the percent error? [katex] 0.11 \% [/katex] [katex] 1.11 \% [/katex] [katex] 11.1 \% [/katex] [katex] 12 \% [/katex] To find percent error, first find the positive difference between the estimated value and the actual value. [katex] 20-18=2 [/katex] Then use the formula to calculate. 2. Billy wants to buy a dozen doughnuts for his soccer team. He estimates that it will be [katex] \$ 15.[/katex] The actual cost of the dozen doughnuts is [katex] \\$ 16.25. [/katex] What is Billy’s percent error?

[katex] 0.769 \% [/katex]

[katex] 6.9 \% [/katex]

[katex] 7.9 \% [/katex]

[katex] 7.69 \% [/katex]

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

[katex] 16.25-15=1.25 [/katex]

Then use the formula to calculate.

3. The principal of Lakewood Heights High School estimated that [katex] 380 [/katex] people will come to the championship game. There were actually [katex] 432 [/katex] people at the game. What is the principal’s percent error?

[katex] 0.012 \% [/katex]

[katex] 1.20 \% [/katex]

[katex] 12 \% [/katex]

[katex] 0.12 \% [/katex]

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

[katex] 432-380=52 [/katex]

Then use the percent error formula to calculate.

4. Lena planted a tomato plant. She estimated that the plant would be [katex] 7 [/katex] inches after the first month. The plant was actually [katex] 9.9 [/katex] inches after the first month. What is the percent error?

[katex] 0.293 \% [/katex]

[katex] 29.3 \% [/katex]

[katex] 2.93 \% [/katex]

[katex] 293 \% [/katex]

Find the positive difference between the values.

[katex] 9.9-7=2.9 [/katex]

Then use the percent error formula to calculate.

Lena’s percent error is [katex] 29.3 \% [/katex]

5. Kyle estimated he would run [katex] 15 [/katex] miles this week. He actually ran [katex] 33 [/katex] miles. What is the percent error?

[katex] 0.545 \% [/katex]

[katex] 5.45 \% [/katex]

[katex] 545 \% [/katex]

[katex] 54.5 \% [/katex]

Find the positive difference between the values.

[katex] 33-15=18 [/katex]

Then use the percent error formula to calculate.

Kyle’s percent error is [katex] 54.5 \%. [/katex]

6. Kelly estimated that she would score [katex] 30 [/katex] points at her basketball game. If her percent of error is [katex] 20 \% [/katex] above the actual amount, what is the actual amount of points she scored in the game?

[katex] 25 \, points [/katex]

[katex] 26 \, points [/katex]

[katex] 30 \, points [/katex]

[katex] 20 \, points [/katex]

The estimated value is [katex] 30 [/katex] points. The actual value is unknown, label it as [katex] x. [/katex]

The percent error is [katex] 20 \% [/katex]

Use the formula to calculate.

Kelly actually scored [katex] 25 [/katex] points in the game.

Percent error FAQs

Can a decimal number be a percent error?

When calculating percent error, you will get a decimal number before converting it to a percent.

Can a percent error be negative?

Sometimes a negative sign is used to describe a percent error when the estimated value is less than the actual value, making the percent a negative value.

Is there a calculator that can quickly find the percent error?

Yes, there are digital calculators online that can do quick calculations such as a standard deviation calculator, a percent error calculator, and a percent calculator (percentage calculator). Although technology is there to help us, be sure to understand what the calculations mean.

Is a percent error of 5% good?

The smaller the percent error, the better. 5% error suggests that the initial estimate or the observed value was close to the actual value. In different contexts, a different percent error is acceptable.

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