The document provides a concise guide on calculating uncertainties in measurements, explaining the concepts of absolute and relative uncertainty. It includes practical examples, such as measuring volume and reaction rates, and describes how to propagate uncertainty through calculations. The guide emphasizes the importance of knowing the uncertainty of measuring instruments to enhance accuracy in scientific experiments.

1. Calculating Uncertainties
A Quick Guide

2. What Is An Uncertainty?

No measuring instrument (be it a plastic
ruler or the world’s most accurate
thermometer) is perfectly accurate

When you make any measurement, there
always is some uncertainty as to the
exact value.

For example:


The ruler says this red line is 3.5 cm long
Due to imperfections in the design and
manufacturing of the ruler, I can’t be sure
that it is exactly 3.500 cm, just something
close to that, perhaps 3.492. or 3.521

3. Measuring Uncertainties

Most equipment manufacturers know
the level of uncertainty in their
instruments, and will tell you.

For example:



The instruction manual that came with my
ruler tells me it is accurate to +/- 0.05 cm.
This means my 3.5 cm line is actually
anywhere between 3.45 and 3.55 cm long
Importantly: we have no way of knowing
where in this range the actual length is,
unless we use a more accurate ruler

4. How Big Are The Uncertainties?

Most good apparatus will have the
uncertainty written on it, so make a
note of it.

Where this is not the case, use
half the smallest division:


For example: if a balance can
measure to two decimal places, the
uncertainty would by +/- 0.005 g
When manually measuring time,
you should round to the nearest
whole second, and decide the
uncertainty based on the nature of
your measurement.

5. Absolute
and
Relative Uncertainty

Absolute uncertainty is
the actual size of the
uncertainty in the units
used to measure it.

This is the size of the uncertainty
relative to the value measured,
and is usually expressed as a
percentage

This is what the
previous slide referred
to


In our ruler example,
the absolute uncertainty
is +/- 0.05 cm
Relative uncertainty can be
calculated by dividing the
absolute uncertainty by the
measured value and multiplying
by 100

In our ruler example, the relative
uncertainty is
 0.05 / 3.5 x 100 = 1.4%

To minimise relative uncertainty,
you should aim to make bigger
measurements

To minimise absolute
uncertainty, you should
use the most accurate
equipment possible.

6. How do uncertainties affect my calculations?

If the numbers you are putting into a calculation are
uncertain, the result of the calculation will be too

You need to be able to calculate the degree of
uncertainty

The Golden Rules:

When adding/subtracting: add the absolute uncertainty

When multiplying/dividing: add the relative uncertainty

7. Example: A Titration

In a titration, the initial reading on my burette was 0.0
cm3, and the final reading was 15.7 cm3. The burette is
accurate to +/- 0.05 cm3. What are the most and least
amounts of liquid I could have added?

The volume of liquid added is the final reading minus the
initial reading, so we need to add absolute uncertainty
in each reading.

Absolute uncertainty = 0.05 + 0.05 = 0.10 cm3

Most amount = 15.7 + 0.10 = 15.8 cm3

Least amount = 15.7 - 0.10 = 15.6 cm3

8. Example 2: A rate of reaction

In an experiment on the rate of a reaction, a student timed how long
it would take to produce 100 cm3 of gas, at a variety of different
temperatures. At 30OC, it took 26.67 seconds. The gas syringe used
was accurate to +/- 0.25 cm3. What is the average rate of reaction,
and what is the relative uncertainty in this value?

Rate = volume / time = 100 / 27 = 3.70 cm3s-1

Time is rounded to the nearest whole second as human reaction times do
not allow for 2 decimal places of accuracy

Absolute uncertainty of volume: +/- 0.25 cm3

Absolute uncertainty of time: +/- 0.5s


This is an approximation, taking into account reaction time and the
difficulty of pressing stop exactly at 100 cm3.
You should make similar approximations whenever you are manually
recording time, and should write a short sentence to justify them

9. Example 2 continued

Relative uncertainty of volume


Relative uncertainty of time


% Uncertainty = (absolute uncertainty / measured value) x 100
= 0.25/100 x 100 = 0.25%
% Uncertainty = (0.5 / 27) x 100 = 1.85%
Relative uncertainty of rate


% Uncertainty (rate) = % uncertainty (volume) + % uncertainty
(time)
= 0.25 + 1.85
= 2.10%
The relative uncertainties were added as the rate calculation
required a division calculation

10. Uncertainty propagation of averages

This is more complicated as we need to make a
choice: the uncertainty is either:
The absolute uncertainty of the measured value
OR
The standard deviation of our data

We must choose whichever is larger

11. For example

With the previous example, if I did three repeat titrations all accurate to +/0.10 cm3, what is the average titre?
Trial
Volume added
(cm3) +/- 0.10 cm3
1
2

15.9
Absolute uncertainty of measured values = +/- 0.10 cm3
Standard deviation = +/- 0.25 cm3


To calculate standard deviation: Calculate the ‘variance’ by subtracting each value
from the average value, squaring it and then averaging the squared values; now
take the quare root of the variance. See here:
http://www.mathsisfun.com/data/standard-deviation.html
Alternatively use the ‘STDEVP*’ function in Excel…. ‘=STDEVP(Range)’


15.4
3

15.7
This calculates standard deviation of a population, rather than ‘STDEV’ which calculates
standard deviation of a sample (see the link above for an explanation of the difference).
The standard deviation is larger than the absolute uncertainty so:

Average titre = (15.7+15.4+15.9)/3 = (15.7 +/- 0.30) cm3

12. Some Practice Questions
With a stopwatch you time that it takes a friend 8.5 s (+/- 0.25 s, human reaction
time) to run 50 metres (+/- 0.50 m). If speed = distance / time:
1.
a.
How fast was the friend running?
b.
What is the relative error in the speed?
c.
What are the fastest and slowest possible speeds?
Whilst doing an experiment on density, you find that a lump of material with a mass
of 1.22 g (+/- 0.0010g) has a volume of 0.65 cm3 (+/- 0.05 cm3). If density = mass /
volume:
2.
a.
What is the density of the material?
b.
What is the relative error in the density?
c.
What are the highest and lowest possible values for the density?
d.
How could you improve the experiment to reduce the uncertainty in the result?
A candle was burnt and the energy it produced measured. The initial mass of the
candle was 25.1 g (+/- 0.05) grams and the final mass was 22.7 g (+/- 0.05 g). It
was found the candle released 80.2 kJ energy (+/- 1.5 kJ).
3.
a.
Calculate the energy released per gram of wax burnt (energy released/mass of candle burnt).
b.
Calculate the absolute and relative error in the mass of candle wax burnt.
c.
Calculate the relative error in the energy released per gram.
Answers: Q1 a) 5.67 m/s, b) 3.9%, c) max: 6.13 m/s, min: 5.67 m/s; Q2 a) 1.88 g/cm3, b) 7.8%, c) max: 2.03 g/cm3, min:
1.73 g/cm3, d) measure volume more accurately, and/or use a bigger lump to reduce relative error in volume; Q3 a) 33.4 kJ/g, b)
Abs: +/- 0.10 g, Rel: +/- 4.2%, c) +/- 6.0%, d) max: 75.4 kJ/g, min: 85.0 kJ/g