Measurement Errors Revision Notes for HSC SSCE Mathematics Standard

Measurement Errors

Introduction to measurement uncertainty

When we take measurements using any instrument, there will always be some degree of error and uncertainty. Even when using very sensitive instruments, repeating the same measurement will often give slightly different results each time.

There are several sources of measurement errors that can affect our readings:

Reading errors: Mistakes made when interpreting the scale on an instrument
Parallax error: Errors caused by viewing the measurement from an angle rather than straight on
Calibration error: Inaccuracies in the instrument itself

infoNote

The accuracy of any measurement can be improved by taking multiple measurements of the same quantity using the same instrument. This helps us identify and reduce random errors.

Understanding precision

When we use a measuring instrument like a ruler or thermometer, the precision refers to the smallest division or unit that we can read on that instrument. This is also called the limit of reading.

For example, if you have a 30 cm ruler that is marked in millimetres, the smallest unit you can measure is 1 mm. Therefore, this ruler has a precision of 1 mm.

chatImportant

The precision of an instrument limits how accurately we can measure. No matter how carefully we try to read a measurement, our accuracy is restricted to plus or minus half of the precision, written as $\pm \frac{1}{2}$ .

Example: If you measure something as 10 mm on a ruler with 1 mm precision, the actual measurement could be anywhere in the range $10 \pm 0.5$ mm.

Absolute error and bounds

Every measurement we make is an approximation and contains some error. The absolute error represents the difference between the actual value and the measured value shown by the instrument.

chatImportant

The maximum value for an absolute error is always half of the precision:

$\text{Absolute error} = \pm \frac{1}{2} \times \text{precision}$

This is the single most important rule to remember when calculating measurement errors!

This absolute error creates a range of possible values for our measurement, defined by an upper bound and a lower bound.

Term	Definition	Formula
Precision	Smallest unit on measuring instrument or limit of reading	N/A
Absolute error	Measured value minus actual value	$\pm \frac{1}{2} \times \text{precision}$
Upper bound	Maximum possible value of the measurement	Measurement $+$ Absolute error
Lower bound	Minimum possible value of the measurement	Measurement $-$ Absolute error

lightbulbExample

Calculating Bounds: 10 mm Measurement

Using a measurement of 10 mm with precision 1 mm:

Step 1: Calculate the absolute error

Absolute error $= \pm \frac{1}{2} \times 1 = \pm 0.5$ mm

Step 2: Calculate the upper bound

Upper bound $= 10 + 0.5 = 10.5$ mm

Step 3: Calculate the lower bound

Lower bound $= 10 - 0.5 = 9.5$ mm

Conclusion: The true value lies somewhere between 9.5 mm and 10.5 mm.

Relative error

While the absolute error tells us the size of the error in the same units as the measurement, the relative error gives us an indication of how significant this error is compared to the size of what we're measuring.

infoNote

A 0.5 mm error might be very significant when measuring something small (like a 2 mm insect), but quite insignificant when measuring something large (like a 50 metre swimming pool). This is why relative error is useful!

The relative error is calculated by dividing the absolute error by the actual measurement:

$\text{Relative error} = \pm \left(\frac{\text{Absolute error}}{\text{Measurement}}\right)$

Example: For our measurement of 10 mm with absolute error of 0.5 mm:

$\text{Relative error} = \pm \frac{0.5}{10} = \pm 0.05$

This tells us the error is 0.05 (or 5 hundredths) of the measurement size.

Percentage error

The percentage error is simply the relative error expressed as a percentage. This makes it easier to interpret and compare errors across different measurements.

To find the percentage error, we multiply the relative error by 100%:

$\text{Percentage error} = \pm \left(\frac{\text{Absolute error}}{\text{Measurement}}\right) \times 100\%$

Example: Continuing with our 10 mm measurement:

$\text{Percentage error} = \pm \frac{0.5}{10} \times 100\% = \pm 5\%$

This means our measurement has an error of plus or minus 5%.

infoNote

Exam tip: Percentage error is often easier to interpret than absolute or relative error, and is commonly used in scientific work to compare the precision of different measurements.

Worked example: Finding measurement errors

lightbulbExample

Complete Worked Example: Ruler Measurement

A ruler is marked in millimetres.

Questions:

a) What is the length indicated by the arrow on this ruler?
b) What is the precision or limit of reading?
c) What is the upper and lower bound for each measurement?
d) Find the relative error. Answer correct to three decimal places.
e) Find the percentage error. Answer correct to one decimal place.

Solution:

a) Finding the measurement

The arrow is pointing to the 38 mm mark on the ruler.

Length $= 38 \text{ mm}$

b) Finding the precision

The precision is the smallest unit marked on the ruler. Looking at the ruler, we can see it is marked in millimetres.

Precision $= 1 \text{ mm}$

c) Finding the upper and lower bounds

First, calculate half the precision:

$\frac{1}{2} \times \text{precision} = \frac{1}{2} \times 1 = 0.5 \text{ mm}$

The lower bound is found by subtracting the absolute error from the measurement:

$\text{Lower bound} = 38 - 0.5 = 37.5 \text{ mm}$

The upper bound is found by adding the absolute error to the measurement:

$\text{Upper bound} = 38 + 0.5 = 38.5 \text{ mm}$

d) Finding the relative error

Using the formula for relative error:

$\text{Relative error} = \pm \left(\frac{\text{Absolute error}}{\text{Measurement}}\right)$

$= \pm \frac{0.5}{38}$

$= \pm 0.013 \text{ (to 3 decimal places)}$

e) Finding the percentage error

Using the formula for percentage error:

$\text{Percentage error} = \pm \left(\frac{\text{Absolute error}}{\text{Measurement}}\right) \times 100\%$

$= \pm \frac{0.5}{38} \times 100\%$

$\approx \pm 1.3\% \text{ (to 1 decimal place)}$

Remember!

bookmarkSummary

Key Points to Remember:

Precision is the smallest unit on your measuring instrument and determines the limit of your reading accuracy
Absolute error is always $\pm \frac{1}{2}$ of the precision, regardless of what you're measuring
Upper bound = Measurement + Absolute error; Lower bound = Measurement - Absolute error
Relative error shows how significant the error is compared to the measurement size: $\pm \frac{\text{Absolute error}}{\text{Measurement}}$
Percentage error is relative error expressed as a percentage: multiply relative error by 100%

Measurement Errors (HSC SSCE Mathematics Standard): Revision Notes