3.02 Identifying error in measurement

Measurements are never exact. This can be for a variety of reasons, including the following:

• Measurement devices have varying degrees of accuracy or precision (i.e. a device measuring mass, may only be capable of measuring to the nearest kilogram, whereas a more accurate device could measure to the nearest gram)
• Human errors in reading the measuring device
• Calibration errors in the device (i.e. a scale not set to zero before making a measurement)

Calibration errors can usually be avoided by calibrating the device before measurements are made. Human error can be reduced by taking multiple readings of the same measurement, and averaging the results.

If we can avoid human and calibration errors then the main source of error comes from the precision of the measuring device. The term 'error', in this case, can seem misleading because it does not mean a mistake has been made. The error comes from limitations in the measuring device itself.

In this section, we look at how we can account for errors in measurement that arise due to the precision of the device.

Absolute error

The absolute error of a measurement is half of the smallest unit on the measuring device. The smallest unit is called the precision of the device.

For example, the ruler below a precision of $0.5$0.5 cm, because the smallest scale markings are $0.5$0.5 cm apart. The absolute error, of any measurement made with the ruler, will be $0.25$0.25 cm (i.e. half of $0.5$0.5 cm).

The limits of accuracy

Because every measurement is prone to error, we can use the absolute error of the measuring device to determine the interval, within which the true measurement will lie.

This interval is defined by two values, a lower bound and an upper bound, that form what is known as the limits of accuracy. The lower bound is the smallest possible value that the true measurement could be. The upper bound is the largest possible value of the true measurement.

Worked example

Example 1

A person's height is measured to be $1.68$1.68 metres rounded to the nearest centimetre.

Calculate,

1. The absolute error of the measuring device
2. The upper bound of their height
3. The lower bound of their height

Solution

Because the measurement has been made to the nearest centimetre, the precision of the measuring device is $0.01$0.01 m (i.e. $1$1 cm).

1. The absolute error is half the precision,

   Absolute error	$=$=	$\frac{1}{2}\times\text{precision }$12​×precision
   	$=$=	$\frac{0.01}{2}$0.012​
   	$=$=	$0.005$0.005 m

   
    
2. The upper bound is the largest possible value of their height.

   Upper bound	$=$=	$\text{measurement }+\text{absolute error }$measurement +absolute error
   	$=$=	$1.68+0.005$1.68+0.005
   	$=$=	$1.685$1.685 m

   
    
3. The lower bound is the smallest possible value of their height.

   Lower bound	$=$=	$\text{measurement }-\text{absolute error }$measurement −absolute error
   	$=$=	$1.68-0.005$1.68−0.005
   	$=$=	$1.675$1.675 m

This means that the person's true height lies somewhere between $1.675$1.675 and $1.685$1.685 metres.

Example 2

On an architectural drawing, the height of a door is indicated to be $2040\pm5$2040±5 mm.

Write down the maximum and minimum possible heights of the door in metres.

Solution

The maximum height is the upper bound:

The minimum height is the lower bound:

Practice Questions

Question 1

Question 2

question 3

Percentage error

To compare the sizes of errors across different measurements and situations, it is more useful to use the percentage error, rather than the absolute error.

The percentage error of a measurement is the absolute error expressed as a percentage of the measurement.

Note that the absolute error and the measurement must be in the same units before calculating the percentage error.

Worked example

Example 3

The capacity of a bottle is measured as $1.25$1.25 litres, correct to the nearest $10$10 millilitres.

Calculate the percentage error for this measurement.

Solution

The precision of the measuring device is $10$10 mL, so the absolute error is half of this value, which is $5$5 mL.

Note that the measurement is in litres and the absolute error is in millilitres. To convert the measurement into millilitres we multiply by $1000$1000:

Now, we can calculate the percentage error:

Practice question

Question 4

Relative error and absolute difference

Sometimes the error in measurement is not due to the precision of the measuring tool. The person measuring may have just taken a bad measurement. This difference between the actual value and the measured value is known as the absolute difference. We take the absolute value of this difference to make sure it is always positive.

The relative error of a measurement is the absolute difference expressed as a percentage of the measurement.

Note that the absolute difference and the measurement must be in the same units before calculating the relative error.

Practice question

Question 5

Area and volume calculations

In calculations involving areas and volumes, each measured side length will have an upper and lower bound.

• To find the maximum possible area or volume we multiply the upper bound values for each side.
• To find the minimum possible area of volume we multiply the lower bound values of each side.

Practice Question

Question 6