Errors and Uncertainties in Measurement
Analogue measuring devices
Every measuring instrument has some uncertainty or error
associated with it. For an analogue measuring instrument, the
minimum uncertainty is usually considered to be ± half the
smallest graduation of the scale.
For this measuring cylinder, the smallest graduation is 0.2 mL
so the measurement uncertainty is taken to be ± 0.1 mL.
The volume of liquid in the measuring cylinder is read as 6.6 ± 0.1 mL. The actual volume of
water lies somewhere between 6.5 and 6.7 mL.
Digital measuring devices
For a digital measuring instrument, the minimum uncertainty is usually considered to be ± 1
of the smallest decimal place visible on the screen.
For this mass balance, the smallest decimal place visible
is 0.001 g so the measurement uncertainty is taken to be
± 0.001 g.
The mass of the object on the mass balance is read as
0.149 ± 0.001 g. The actual mass lies somewhere
between 0.148 and 0.150 g.
Task 1
Look at the measuring instruments around the room. State whether the device is analogue
or digital, before determining the measurement uncertainty using the rules above.
Measuring device
Analogue or digital?
Uncertainty (with unit)
Ruler
Analogue
± 0.5 mm
Mass balance
10 mL measuring cylinder
250 mL measuring cylinder
Stopwatch
Thermometer
Voltmeter (top scale)
Task 2 – Taking measurements
Use the measuring instruments around the room to measure the following quantities
together with their uncertainties and complete the table below.
Quantity (and unit)
Capacity of the cup (mL)
Height of Mr Edwards (cm)
Mass of size ‘D’ battery (g)
Time taken for ping pong
ball to drop 2m (s)
Length of the longest side of
one index card (cm)
Temperature of tap water
(˚C)
Value
Uncertainty
Repeating measurements
We can be more and more confident of our measurements if we repeat them multiple
times. When we repeat measurements multiple times we take a mean average of the
measured values. The true value is likely to be close to the mean. We calculate a mean by
summing (adding) the values and dividing by the number of values.
We state the uncertainty as ± half the range of measured values. We find the range by
taking the largest measured value and subtracting the smallest measured value.
For example, if the mass of a particular apple is measured 5 times as 160 g, 162 g, 165 g,
161 g and 166 g we can calculate our mean by summing the 5 values and dividing by 5, i.e.
mean average = [ 160 + 162 + 165 + 161 + 166 ] ÷ 5 = 162.8 g
To determine the uncertainty, we must find our range by taking the largest measured value
(166 g) and subtracting the smallest measured value (160 g). Remember the uncertainty is ±
half the range, i.e.
uncertainty = range ÷ 2 = [ 166 – 160 ] ÷ 2 = ± 3 g
In a results table we would display our data like this:
Mass of
Apple (g)
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
160
162
165
161
166
Mean
Uncertainty
average
162.8
±3
We usually round the absolute uncertainty to 1 s.f. and the value of the measurement to the
same number of decimal places.
Thus we can state the mass of the apple as: 163 ± 3 g
Task 3 – Calculating means and uncertainties from repeated measurements
Let’s share our data as a class and use the data to calculate mean averages and
uncertainties.
Quantity
(and unit)
Capacity of
the cup (mL)
Height of Mr
Edwards (cm)
Mass of size
‘D’ battery (g)
Time taken
for ping pong
ball to drop
2m (s)
Length of the
longest side
of one index
card (cm)
Temperature
of tap water
(˚C)
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Mean
average
Uncertainty