Calculating Uncertainties

A Quick Guide

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

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

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.

Absolute and Relative Uncertainty Absolute uncertainty is

the actual size of the uncertainty in the units used to measure it.

This is what the previous slide referred to

In our ruler example, the absolute uncertainty is +/- 0.05 cm

To minimise absolute uncertainty, you should use the most accurate equipment possible.

This is the size of the uncertainty relative to the value measured, and is usually expressed as a percentage

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

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

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

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

Example 2 continued Relative uncertainty of volume

% Uncertainty = (absolute uncertainty / measured value) x 100 = 0.25/100 x 100 = 0.25%

Relative uncertainty of time % 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

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

For example With the previous example, if I did three repeat titrations all accurate to +/-

0.10 cm3, what is the average titre?

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)’ 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

Trial

Volume added(cm3) +/- 0.10

cm3

1 15.7

2 15.4

3 15.9

Some Practice Questions1. 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: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?

2. 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:

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?

3. 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).

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.

d. Calculate the highest and lowest possible values for 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