Measurements are never exact. This can be for a variety of reasons, including the following:
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.
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).
Absolute error | $=$= | $\frac{1}{2}\times\text{precision }$12×precision |
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.
The limits of accuracy for a measurement are the possible upper and lower bounds of the measurement.
Upper bound | $=$= | $\text{measurement }+\text{absolute error }$measurement +absolute error |
Lower bound | $=$= | $\text{measurement }-\text{absolute error }$measurement −absolute error |
Sometimes, the limits of accuracy of a measurement will be expressed in the form:
$\text{measurement }\pm\text{absolute error }$measurement ±absolute error
A person's height is measured to be $1.68$1.68 metres rounded to the nearest centimetre.
Calculate,
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).
Absolute error | $=$= | $\frac{1}{2}\times\text{precision }$12×precision |
$=$= | $\frac{0.01}{2}$0.012 | |
$=$= | $0.005$0.005 m |
Upper bound | $=$= | $\text{measurement }+\text{absolute error }$measurement +absolute error |
$=$= | $1.68+0.005$1.68+0.005 | |
$=$= | $1.685$1.685 m |
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.
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:
Maximum height | $=$= | $2040+5$2040+5 mm |
$=$= | $2045$2045 mm | |
$=$= | $2.045$2.045 m |
The minimum height is the lower bound:
Maximum height | $=$= | $2040-5$2040−5 mm |
$=$= | $2035$2035 mm | |
$=$= | $2.035$2.035 m |
Between what limits does the cost of a CD lie if it is known to be $\$50$$50 correct to the nearest $\$5$$5?
Upper bound = $\$$$$\editable{}$
Lower bound = $\$$$$\editable{}$
State the limits of accuracy for a distance measured to be $13.45$13.45 km.
Upper bound = $\editable{}$ km
Lower bound = $\editable{}$ km
The length of a piece of rope is measured to be $19.99$19.99 m using a ruler. What is the upper bound of the largest possible length of this rope?
In calculations involving areas and volumes, each measured side length will have an upper and lower bound.
A field has dimensions $15.4$15.4 $\text{m }\times17.6$m ×17.6 $\text{m }$m , to the nearest $10$10 cm.
What is the upper bound of the area of the field?
What is the lower bound of the area of the field?
What is the upper bound of the perimeter of the field?
What is the lower bound of the perimeter of the field?