Percentage Error Revision Notes for Leaving Cert Mathematics

Percentage Error

What is percentage error?

When making measurements or calculations, we often need to understand how accurate our results are compared to the true value. Percentage error helps us express this accuracy as a percentage, making it easier to understand and compare different types of errors.

The concept starts with relative error, which compares the size of an error to the true value of what we're measuring. When we multiply the relative error by 100, we get the percentage error.

Key formulas

Understanding percentage error requires two important formulas:

chatImportant

Relative Error Formula: $\text{Relative Error} = \frac{\text{Error}}{\text{True Value}}$

Percentage Error Formula: $\text{Percentage Error} = \frac{\text{Error}}{\text{True Value}} \times \frac{100}{1}\%$

Critical rule: The value of the error is always taken as positive, regardless of whether our estimate was too high or too low.

lightbulbExample

Worked Example 1: Rounding Numbers

Problem: To calculate 121 + 46 + 37 + 26, Robbie makes a rough estimate by rounding each number to the nearest 10. Calculate his percentage error.

Step-by-step solution:

Step 1: Find the rough estimate

121 rounds to 120
46 rounds to 50
37 rounds to 40
26 rounds to 30
Rough estimate = 120 + 50 + 40 + 30 = 240

Step 2: Calculate the true value

True value = 121 + 46 + 37 + 26 = 230

Step 3: Find the error

Error = 240 - 230 = 10 (always positive)

Step 4: Apply the percentage error formula

$\text{Percentage Error} = \frac{10}{230} \times \frac{100}{1} = 4.347\% = \text{4.3\%}$

Tolerance and measurement bounds

In real-world measurements, we must account for the fact that no measurement is perfectly exact. Tolerance refers to the acceptable range of error in a measurement.

When we measure something to the nearest unit (like the nearest centimetre), the true value could be anywhere within a certain range. This creates upper bounds and lower bounds for our measurements.

infoNote

Key principle: The lower bound and upper bound represent the minimum and maximum values of a measurement or calculation.

For example, if Ciara measures a box width as 24 cm (to the nearest cm):

The measurement could be anywhere between 23.5 cm and 24.5 cm
Lower bound = 23.5 cm (minimum possible value)
Upper bound = 24.5 cm (maximum possible value)

lightbulbExample

Worked Example 2: Rectangle Perimeter Bounds

Problem: A rectangle has length 28 cm and breadth 16 cm, measured to the nearest centimetre. Find the maximum and minimum perimeter.

Step-by-step solution:

Step 1: Find the upper bounds

Length: 28 cm → upper bound = 28.5 cm
Breadth: 16 cm → upper bound = 16.5 cm

Step 2: Calculate maximum perimeter (using upper bounds)

Maximum perimeter = 2(length + breadth)
= 2(28.5 + 16.5) cm
= 2(45) cm = 90 cm

Step 3: Find the lower bounds

Length: 28 cm → lower bound = 27.5 cm
Breadth: 16 cm → lower bound = 15.5 cm

Step 4: Calculate minimum perimeter (using lower bounds)

Minimum perimeter = 2(27.5 + 15.5) cm
= 2(43) cm = 86 cm

infoNote

Exam Tips

Always remember that error values are positive
When finding bounds, add/subtract 0.5 of the measurement unit
Use upper bounds for maximum calculations
Use lower bounds for minimum calculations
Round your final percentage error appropriately (usually to 1 decimal place)

bookmarkSummary

Key Points to Remember:

Percentage error compares the size of an error to the true value, expressed as a percentage
Formula: Percentage Error = (Error ÷ True Value) × 100%
Error values are always positive - take the absolute difference
Tolerance creates upper and lower bounds around measurements
Upper bounds give maximum values, lower bounds give minimum values

Percentage Error (Leaving Cert Mathematics): Revision Notes