Percent Error

**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?"

Guessing Jelly Beans

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

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

Answer: Your percent error is **10%**.

Boiling Point of Water

At sea level water boils at $100\ °\text{C}$. Thermometer A reads $102.5\ °\text{C}$ in boiling water and thermometer B reads $97.5\ °\text{C}$. Find the percent error of each thermometer.

Absolute error of thermometer A: $|102.5 - 100| = |2.5| = 2.5$ = $2.5\ °\text{C}$ too high

Absolute error of thermometer B: $|97.5 - 100| = |-2.5| = 2.5$ = $2.5\ °\text{C}$ too low

Divide by the actual value and multiply by 100: $\frac{2.5}{100} \times 100\% = 2.5\%$ = $2.5\%$ for both

Answer: Both thermometers have a percent error of **2.5%**. One reads too high and one too low, but the absolute value makes both errors positive.

Measuring Gravity in the Lab

In a pendulum experiment you measure the acceleration due to gravity as $9.6\ \text{m/s}^2$. The accepted value is $9.81\ \text{m/s}^2$. Find the percent error, rounded to two decimal places.

Find the absolute error: $|9.6 - 9.81| = |-0.21| = 0.21$ = $0.21\ \text{m/s}^2$

Divide by the actual value (keep all digits): $\frac{0.21}{9.81} = 0.0214067\ldots$ = $0.0214067\ldots$

Multiply by 100: $0.0214067\ldots \times 100\% = 2.14067\ldots\%$ = $2.14067\ldots\%$

Round only at the end, to two decimal places: The third decimal is 0, so round down = $2.14\%$

Answer: The percent error is about **2.14%** (rounded to two decimal places).

Same Error, Different Percent Error

Mia measures a 10 cm pencil as 12 cm. Leo measures a 200 cm door as 202 cm. Both are 2 cm off. Whose measurement is better?

Absolute errors: Mia: $|12 - 10| = 2$, Leo: $|202 - 200| = 2$ = Both are 2 cm off

Mia's percent error: $\frac{2}{10} \times 100\% = 20\%$ = $20\%$

Leo's percent error: $\frac{2}{200} \times 100\% = 1\%$ = $1\%$

Compare: $20\% > 1\%$, and $20 \div 1 = 20$ = Mia's error is 20 times larger compared with the size of the object

Answer: **Leo's** measurement is much better: 1% error against Mia's 20%. Dividing by the actual value shows how big the mistake is compared with the thing being measured.

A Percent Error Above 100%

Sam estimates that 300 people are at a concert. Only 120 people are actually there. What is Sam's percent error?

Find the absolute error: $|300 - 120| = 180$ = 180 people

Divide by the actual value: $\frac{180}{120} = 1.5$ = $1.5$

Multiply by 100: $1.5 \times 100\% = 150\%$ = $150\%$

Answer: Sam's percent error is **150%**. A percent error can be more than 100% when the estimate is more than double the actual value.

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.

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

Why: 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: $\frac{|9.6 - 9.81|}{9.81} \times 100\% \approx 2.14\%$

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

Why: 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 $10\%$.

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

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

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

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

Why: 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".

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.

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 $\frac{13}{343} \times 100\% \approx 3.79\%$.

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.

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

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.

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

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

The 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

Always divide by the actual (accepted) value, never by the measured value

Percent error is never negative: too high and too low by the same amount give the same percent error

Keep all digits until the end, then round once and say how many decimal places you used

Percent error can be more than 100% when an estimate is more than twice the actual value

Q: Can percent error be negative?

A: 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.

Q: Can percent error be more than 100%?

A: 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%.

Q: What is a good percent error?

A: 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%.

Q: How is percent error different from percent change?

A: 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.

Exit Presentation

Percent Error

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Decimals & Percents / Percentages

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 min

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