Percent Error

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

Advanced25 min18 problems
Practice Problems

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.
Percent error=∣measured−actual∣actual×100%\text{Percent error} = \frac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100\%
The calculation has three parts:
  1. 1.Absolute error: the distance between the two values, ∣measured−actual∣|\text{measured} - \text{actual}|. The absolute value bars make it positive, because only the size of the mistake matters.
  2. 2.Divide by the actual value: this compares the mistake with the true size of the thing you measured.
  3. 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.
∣450−500∣500×100%=50500×100%=10%\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?"

Try it now

Let MM be the measured value and AA the actual value. Which formula gives the percent error?

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?
1

Find the absolute error

∣450−500∣=∣−50∣=50|450 - 500| = |-50| = 50 → Absolute error = 50 jelly beans
2

Divide by the actual value

50500=0.1\frac{50}{500} = 0.1
3

Multiply by 100

0.1×100%=10%0.1 \times 100\% = 10\%

Common Mistakes

Dividing by the measured value instead of the actual value: ∣9.6−9.81∣9.6\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: ∣9.6−9.81∣9.81×100%≈2.14%\frac{|9.6 - 9.81|}{9.81} \times 100\% \approx 2.14\%

Forgetting the absolute value and writing a negative percent error, like −10%-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 ∣measured−actual∣|\text{measured} - \text{actual}|. For a guess of 450 and an actual value of 500: ∣450−500∣=50|450 - 500| = 50, percent error 10%10\%.

Forgetting to multiply by 100 and giving 0.10.1 as the percent error

Why it's wrong: 50500=0.1\frac{50}{500} = 0.1 is a decimal, not a percent. 0.1%0.1\% would mean a mistake 100 times smaller.

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

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

Why it's wrong: Rounding 0.219.81\frac{0.21}{9.81} to 0.020.02 first gives 2%2\% instead of 2.14%2.14\%. And an answer like 2.1%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%2.14\%, rounded to two decimal places".

Interactive Visual

to

Click on numbers to select them. Adjust the range to explore different values.

Bar Chart

Same error of 2 cm, different percent error

Pencil, 10 cm(20%)
Table, 50 cm(4%)
Door, 200 cm(1%)

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Let MM be the measured value and AA the actual value. Which formula gives the percent error?

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 m/s29.6\ \text{m/s}^2, but the accepted value is 9.81 m/s29.81\ \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 m/s330\ \text{m/s}. The accepted value is 343 m/s343\ \text{m/s}, so the percent error is 13343×100%≈3.79%\frac{13}{343} \times 100\% \approx 3.79\%.

1Try It Yourself

You measure the density of an aluminum block as 2.646 g/cm32.646\ \text{g/cm}^3. The accepted density of aluminum is 2.70 g/cm32.70\ \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 10100×100%=10%\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 1100×100%=1%\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 =∣measured−actual∣actual×100%= \frac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100\%
  • 2The absolute error ∣measured−actual∣|\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

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.
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.
Yes. If the actual value is 20 and you guess 50, the absolute error is 30, and 3020×100%=150%\frac{30}{20} \times 100\% = 150\%. Any guess that is more than double the actual value gives a percent error above 100%.
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%.
Percent change compares a new value with the original value and keeps the sign: +25%+25\% is growth, −15%-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, ∣measured−actual∣|\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|-3| = 3

Formula Card

Percent error

Percent error=∣measured−actual∣actual×100%\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

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

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

Working backwards

measured=actual±actual×percent error100\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

Today's challengeOne problem, three tries

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Percent Increase and Decrease

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