Teacher Guide: Percent Error

Learn how to measure how far a measurement or estimate is from the actual value, as a percent of the actual value.

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For Teachers

Learning Objectives

Calculate the absolute error between a measured and an actual value
Calculate percent error by dividing the absolute error by the actual value and multiplying by 100
Explain why percent error divides by the actual value and uses the absolute value
Round a percent error correctly and state the number of decimal places
Use percent error to judge lab results, estimates and measuring tools, including errors above 100%

Prerequisites

• Introduction to Percentages (converting decimals to percents)
• Finding Percent of a Number
• Percent Increase and Decrease (percent change)
• Absolute value of a number
• Rounding decimals

Discussion Starters

1. Two students are each 2 cm off: one measured a pencil, the other a door. Who made the bigger mistake? Why?
2. Why do we divide by the actual value and not by the measured value?
3. Is a 3% error good or bad? What does it depend on?
4. Can an estimate be wrong by more than 100%? Can it be wrong by more than 100% if the guess is too low?
5. Percent change keeps the minus sign, but percent error does not. Why do the two formulas treat the sign differently?

Common Misconceptions

Percent error and absolute error are the same thing

Remediation: Compare a 2 cm error on a 10 cm pencil (20%) with a 2 cm error on a 200 cm door (1%). Same absolute error, very different percent error. Show the bar chart of the three objects.

It does not matter whether you divide by the measured or the actual value

Remediation: Use the concert example: guess 300, actual 120. Dividing by 120 gives 150%, dividing by 300 gives 60%. Ask which number describes the mistake honestly: the reference must be the true value.

A percent error can never be larger than 100%

Remediation: Students confuse it with percent decrease, which stops at 100%. Show that an overestimate can be any size: guessing 50 when the answer is 20 gives 150%. Only an underestimate stays at or below 100%.

Differentiation Ideas

For Struggling Students:

• Start with actual values of 10, 100 or 1000 so the division is easy
• Use a three-step checklist: subtract and drop the sign, divide by the actual value, multiply by 100
• Draw both values on a number line and measure the gap before calculating

For On-Level Students:

• Calculate percent errors from real lab data with decimals and round to two decimal places
• Compare the percent errors of several guesses in an estimation contest
• Explain in words why the same absolute error can give different percent errors

For Advanced Students:

• Work backwards: given the actual value and the percent error, find both possible measured values
• Investigate why an underestimate can never have a percent error above 100%
• Research the accepted tolerance of a real measuring tool and decide whether a given result is within it

Standards Alignment

7.RP.A.3 (CCSS.MATH.CONTENT.7.RP.A.3)
Use proportional relationships to solve multistep ratio and percent problems, including percent error
7.NS.A.3 (CCSS.MATH.CONTENT.7.NS.A.3)
Solve real-world and mathematical problems involving the four operations with rational numbers
HSN.Q.A.3 (CCSS.MATH.CONTENT.HSN.Q.A.3)
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities

Lesson Resources

visualError on the Number Line
Shows the measured value, the actual value and the distance between them (the absolute error)
activityEstimation Jar
Students guess the number of items in a jar, then calculate and rank their percent errors
activityPendulum Lab
Students time a pendulum, compute g and compare it with 9.81 m/s² using percent error
worksheetPercent Error Practice
Lab, estimation and quality control problems, including rounding and errors above 100%

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Percent error tells you how far a measured or estimated value is from the actual value, written as a percent of the actual value.

\text{Percent error} = \frac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100\%

The calculation has three parts:

1.Absolute error: the distance between the two values, $|\text{measured} - \text{actual}|$ . The absolute value bars make it positive, because only the size of the mistake matters.
2.Divide by the actual value: this compares the mistake with the true size of the thing you measured.
3.Multiply by 100: this turns the decimal into a percent.

Example: You guess that a jar holds 450 jelly beans, but it really holds 500.

\frac{|450 - 500|}{500} \times 100\% = \frac{50}{500} \times 100\% = 10\%

The actual value is also called the accepted, true or theoretical value. The measured value is also called the experimental or estimated value.

In Percent Increase and Decrease you divided by the original value and kept the sign (+ for growth,

-

for a drop). Percent error is similar, but it always divides by the actual value and is never negative: it only answers "how big was the mistake?"

Worked Examples

At a school fair you guess that a jar holds 450 jelly beans. The jar actually holds 500. What is your percent error?

Find the absolute error

$|450 - 500| = |-50| = 50$ → Absolute error = 50 jelly beans

Divide by the actual value

$\frac{50}{500} = 0.1$

Multiply by 100

$0.1 \times 100\% = 10\%$

Answer: Your percent error is 10%.

Common Mistakes

Dividing by the measured value instead of the actual value: $\frac{|9.6 - 9.81|}{9.6}$

Why it's wrong: Percent error compares your mistake with the TRUE size of the quantity. The measured value is the one that may be wrong, so it cannot be the reference.

Correct: Always divide by the actual (accepted) value: $\frac{|9.6 - 9.81|}{9.81} \times 100\% \approx 2.14\%$

Forgetting the absolute value and writing a negative percent error, like $-10\%$

Why it's wrong: Percent error measures the size of the mistake, not its direction. Being 50 too low is as big a mistake as being 50 too high.

Correct: Use $|\text{measured} - \text{actual}|$ . For a guess of 450 and an actual value of 500: $|450 - 500| = 50$ , percent error $10\%$ .

Forgetting to multiply by 100 and giving $0.1$ as the percent error

Why it's wrong: $\frac{50}{500} = 0.1$ is a decimal, not a percent. $0.1\%$ would mean a mistake 100 times smaller.

Correct: Multiply by 100: $0.1 \times 100\% = 10\%$ .

Rounding too early, or not saying how many decimal places you rounded to

Why it's wrong: Rounding $\frac{0.21}{9.81}$ to $0.02$ first gives $2\%$ instead of $2.14\%$ . And an answer like $2.1\%$ without a rounding rule cannot be checked.

Correct: Keep all digits until the last step, then round once and say how: " $2.14\%$ , rounded to two decimal places".

Why It Matters

Every measurement has some error, and percent error tells you whether that error is small or serious:

Science labs: You measure gravity as $9.6\ \text{m/s}^2$ , but the accepted value is $9.81\ \text{m/s}^2$ . Is your experiment good? Percent error says it is off by about 2%.
Estimation: Guessing contests, crowd sizes and project time plans can all be judged with percent error.
Quality control: Factories check that scales, rulers and machines stay within a set percent error, for example 1%.
Medicine and cooking: A small dose or a small amount of salt needs a much more accurate measurement than a big one.

The same absolute error can be tiny or huge: being 2 cm off when measuring a door is fine, but being 2 cm off when measuring a pencil is a big mistake. Percent error makes this fair to compare.

Real World Applications

Science Lab: Density of Aluminum

Scientists judge an experiment by comparing their result with the accepted value. A percent error under about 5% is usually a good school lab result.

Example:

A class measures the speed of sound as $330\ \text{m/s}$ . The accepted value is $343\ \text{m/s}$ , so the percent error is $\frac{13}{343} \times 100\% \approx 3.79\%$ .

1Try It Yourself

You measure the density of an aluminum block as $2.646\ \text{g/cm}^3$ . The accepted density of aluminum is $2.70\ \text{g/cm}^3$ .

What is the percent error of your measurement?

Step 1: Write the mathematical expression

Write the calculation or the result (in %):

Estimation Contests

Guessing games, crowd counts and time plans are all estimates. Percent error shows how good an estimate was, whatever the size of the numbers.

Example:

If you guess 90 candies and there are 100, your percent error is $\frac{10}{100} \times 100\% = 10\%$ .

2Try It Yourself

A big jar at the fair holds 1,250 marbles. You guess 1,000 marbles.

What is your percent error?

Step 1: Write the mathematical expression

Write the calculation or the result (in %):

Checking a Kitchen Scale

Shops, pharmacies and factories test their scales with calibration weights whose mass is known exactly. The percent error tells them if the scale can still be used.

Example:

A pharmacy scale shows 101 g for a 100 g test weight: the percent error is $\frac{1}{100} \times 100\% = 1\%$ .

3Try It Yourself

You put a calibration weight of exactly 500 g on a kitchen scale. The scale shows 508 g.

What is the percent error of the scale?

Step 1: Write the mathematical expression

Write the calculation or the result (in %):

Key Takeaways

1Percent error $= \frac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100\%$
2The absolute error $|\text{measured} - \text{actual}|$ is the size of the mistake in the original units; the percent error compares it with the actual value
3Always divide by the actual (accepted) value, never by the measured value
4Percent error is never negative: too high and too low by the same amount give the same percent error
5Keep all digits until the end, then round once and say how many decimal places you used
6Percent error can be more than 100% when an estimate is more than twice the actual value

Frequently Asked Questions

Can percent error be negative?

Not with the formula in this lesson: the absolute value makes it 0 or positive. Some science teachers drop the absolute value to show whether a result was too high (+) or too low (

-

). If you see a signed percent error, the size is still the same, only the direction is added.

Can percent error be more than 100%?

Yes. If the actual value is 20 and you guess 50, the absolute error is 30, and

\frac{30}{20} \times 100\% = 150\%

. Any guess that is more than double the actual value gives a percent error above 100%.

What is a good percent error?

It depends on the task. In a school lab, under 5% is usually good and under 1% is excellent. A guessing game within 10% is great, while a medicine dose or a precision machine may need under 0.1%.

How is percent error different from percent change?

Percent change compares a new value with the original value and keeps the sign:

+25\%

is growth,

-15\%

is a drop. Percent error compares a measurement with the actual value and uses the absolute value, so it is never negative. Both divide by the reference value and multiply by 100.

Glossary

Measured value: The value you got from a measurement, an experiment or an estimate (also called the experimental or estimated value)
Actual value: The true or accepted value that the measurement is compared with (also called the accepted, true or theoretical value)
Absolute error: The distance between the measured and the actual value, $|\text{measured} - \text{actual}|$ , in the original units
Percent error: The absolute error divided by the actual value, times 100%
Absolute value: The distance of a number from zero, always 0 or positive: $|-3| = 3$

Formula Card

Percent error

\text{Percent error} = \frac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100\%

How big the mistake is, as a percent of the actual value

Absolute error

\text{Absolute error} = |\text{measured} - \text{actual}|

The size of the mistake in the original units (cm, g, people)

Working backwards

\text{measured} = \text{actual} \pm \text{actual} \times \frac{\text{percent error}}{100}

Use + if the measurement was too high and $-$ if it was too low

Teacher Guide: Percent Error

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Science Lab: Density of Aluminum

Estimation Contests

Checking a Kitchen Scale

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Percent error

Absolute error

Working backwards

More in This Topic

Introduction to Percentages

Finding Percent of a Number

Percent Increase and Decrease

Percent Proportion