1. Measurement & Error
Maths Skills in Science
Introductory Chemistry
Canadian Academy
MrT

2. Measurement & Error
• Distinguish between quantitative and qualitative data
• State SI units of measurement for mass, length, concentration,
temperature, density and other values
• Distinguish between accuracy and precision
• Determine the uncertainty of digital and analogue measuring tools
• Calculate the % error of a given or recorded set of values
• State and calculate values to appropriate numbers of significant
digits
2

3. Quantitative & Qualitative Data
Quantitative
• Can be measured, using numbers and units
• Mathematical relationships used in analysis and conclusions
Qualitative
• Observational, not numbers
• Colour, smell, texture, observed events
3

4. There’s no such word as ‘amount’
Digital balance
Mass (g)
solids
“Amount”
Volume (ml)
fluids Concentration
4

5. There’s no such word as ‘amount’
Matter is anything that has mass and volume (takes up space):
We measure mass in kilograms (kg), or grams (g).
We measure volume in litres (l) or millilitres (ml) (or cm3).
But we also describe other quantities:
• Time in seconds (s)
• Length or distance in metres (m)
• Energy in joules (J) or kilojoules (kJ)
• Heat/temperature in degrees Celcius (oC) (the SI* unit is Kelvin, K)
• Quantity of a substance in moles (mol)
• Concentration of a solution in moles per litre (M)
• Acidity or alkalinity in pH (0 is a strong acid, 7 is neutral, 14 a strong base)
• Electrical conductivity in Siemens (S)
5

6. Chemistry Lab Manual Criterion B: Communication in Science
Work in groups to practice using the lab By the end of the lesson, make sure you have
equipment correctly and safely. completed these tasks:
• Measuring volume
Think about the units and uncertainties • Measuring mass
of each piece of equipment. • Measuring temperature
• Measuring pH
• Measuring conductivity
You will also learn about: • Bunsen driving test
• Accuracy vs precision
• Random vs systematic error If you are waiting for a set or are finished,
• Significant digits there are plenty of questions to work on!
• Scientific notation
Completed Lab manuals are self-assessed,
and handed in at the end of the class.
6

7. Chemistry Lab Manual Criterion B: Communication in Science

8. Uncertainties
All measurements have uncertainties – a range of values in which the true
measurement could lie. More precise measuring tools have a smaller degree of
uncertainty.
In Science it is important for us to estimate and note the uncertainty in the
measurements that we record. Look at this example measurement:
Known Estimated
This measurement could be 1.23449 or
1.234
1.2335. Both would get rounded to 1.234.
g (±0.001) Therefore with this digital measurement
there is an uncertainty of ±0.001.
Three decimal places
(precision)
Uncertainty is ± 1 of the smallest division.
Remember the SI units!

9. 36.0ml ± 0.5ml
44ml ± 1ml
7.4ml ± 0.1ml
9

10. Uncertainty Measuring equipment that presents more decimal
places usually has a lower uncertainty.
Uncertainty = ± 0.01g Uncertainty = ± 1g

11. Precision vs Accuracy
• Accuracy describes how correct – how close to the true answer –
the results are.
• Precision describes how repeatable they are.
Results A Results B Results C
Accuracy Low
Precision High
Generally, measuring tools with a greater degree of precision have smaller divisions (more
decimal places).
Image source: Bishop, M. Precision vs Accuracy (Fig 1.12). From Preparatory Chemistry (ebook) via http://preparatorychemistry.com/Bishop_Book_1_eBook.pdf

12. Systematic Error vs Random Error
• Systematic errors are repeated in the same way throughout an investigation, such
as using a balance incorrectly in the same way for each measurement. This can be
corrected. Precision describes how repeatable they are.
• Random error cannot easily be corrected as it affects measurements differently.
Results A Results B Results C
Systematic error None No
Random Error No None
Image source: Bishop, M. Precision vs Accuracy (Fig 1.12). From Preparatory Chemistry (ebook) via http://preparatorychemistry.com/Bishop_Book_1_eBook.pdf

13. What type of error?
MrT did not calibrate (set up) the pH probe
properly, so every reading was pH 0.5 too high.
Something was left on the digital balance for every
recording, so the results were always too high.

14. Accuracy, Precision and Error
• Accuracy describes how correct – how close to the true answer – the results are.
• Precision describes how repeatable they are.
• Systematic errors are repeated in the same way throughout an investigation, such
as using a balance incorrectly in the same way for each measurement. This can be
corrected. Precision describes how repeatable they are.
• Random error cannot easily be corrected as it affects measurements differently.
Results A Results B Results C
Accuracy Low High Low
Precision High High Low
Systematic error High None No
Random Error No None High
Image source: Bishop, M. Precision vs Accuracy (Fig 1.12). From Preparatory Chemistry (ebook) via http://preparatorychemistry.com/Bishop_Book_1_eBook.pdf

15. Significant Digits
Use this online tutorial from Norton
Chemistry to learn about significant digits
(figures) and how to use them
appropriately in your calculations.
http://media.wwnorton.com/college/chemistry/chemtours/interfa
ce.asp?chapter=chapter_01&folder=significant_figures
How many sig. digs?
Number # sig. digits Record the following numbers to 3 sig. digits:
12420 =
53 0.03209 =
5.03 4050.0 =
01.67 0.0101010 =
1.067 Practice calculations with significant digits:
100 • 2.01 - 1.0 =
1000.0 • 123 + 456.789 =
• 1.2 x 3.45 =
0.003450 • 34.678 /3.33 =

16. The Sig Fig Song
Significant Digits
(AKA Significant Figures)
When we write measurements or the results of
calculations in Science, we need to use the correct
number of significant digits. This tells us the
acceptable level of precision in our measurements.
Significant Digits are:
• All non-zero digits
• All zeroes in-between non-zeroes
• All digits after a decimal place http://www.youtube.com/watch?v=ZuVPkBb-z2I
Leading zeroes don’t count: Try these numbers:
• Count from the left • 123.45
• SigDigs start at the first non-zero • 123000
Trailing zeroes in numbers > 1 don’t count: • 1.008
• Unless there is a line over one of them • 01.67
• This shows that it is known, not an • 1000
estimate.
• 1000
• 1000.0

17. The Sig Fig Song
Significant Digits
(AKA Significant Figures)
When we write measurements or the results of
calculations in Science, we need to use the correct
number of significant digits. This tells us the
acceptable level of precision in our measurements.
Significant Digits are:
• All non-zero digits
• All zeroes in-between non-zeroes
• All digits after a decimal place http://www.youtube.com/watch?v=ZuVPkBb-z2I
Leading zeroes don’t count: Try these numbers:
• Count from the left • 123.45 has 5 SigDigs
• SigDigs start at the first non-zero • 123000 has 3 SigDigs
Trailing zeroes in numbers > 1 don’t count: • 1.008 has 4 SigDigs
• Unless there is a line over one of them • 01.67 has 3 SigDigs
• This shows that it is known, not an • 1000 has 1 SigDig
estimate.
• 1000 has 3 SigDigs
• 1000.0 has 5 SigDigs

18. Significant Digits Practice
Use the periodic table to write your name as closely as possible.
e.g. Ra C He Lu
Challenge 1
The wall chart periodic table presents
many decimal places.
Convert each one to
3 significant digits:
Ra = 226
C = 12.011  12.0
He = 4.0062  4.01
Lu = 174.967  175
How does this periodic table compare to the IB periodic table?
Have the IB presented their table based on sigdigs or decimal places?

19. The Sig Fig Song 2: Calculations
Calculating with SigDigs
Calculations will often result in too many decimal
places or digits in an answer. We need to use
appropriate rounding and present our answers to
the correct number of SigDigs.
Rule number 1:
Carry out all the calculations first and round
6.6
the final answer – do not round the numbers
as you go, it could mess up the result.
http://www.youtube.com/watch?v=kB2szfcwu1A
Adding and Subtracting
• Identify the measurement with the smallest number of
decimal places  this is the number of d.p. to use in
the final answer.
2.01 – 1.0 =
Multiplying and Dividing
• Identify the measurement with the smallest number of
SigDigs  this is the number of SigDigs to use in the
final answer.
1.2 x 3.45 =

20. The Sig Fig Song 2: Calculations
Calculating with SigDigs
Calculations will often result in too many decimal
places or digits in an answer. We need to use
appropriate rounding and present our answers to
the correct number of SigDigs.
Rule number 1:
Carry out all the calculations first and round
6.6
the final answer – do not round the numbers
as you go, it could mess up the result.
http://www.youtube.com/watch?v=kB2szfcwu1A
Adding and Subtracting
• Identify the measurement with the smallest number of
decimal places  this is the number of d.p. to use in
the final answer.
2.01 – 1.0 = 1.01, round to 1.0
1d.p.
Multiplying and Dividing
• Identify the measurement with the smallest number of
SigDigs  this is the number of SigDigs to use in the
final answer.
1.2 x 3.45 = 4.14 round to 4.1
2sf.

21. Significant Digits are:
• All non-zero digits
• All zeroes in-between non-zeroes
• All digits after a decimal place
Leading zeroes don’t count: Trailing zeroes in numbers > 1 don’t count:
• Count from the left • Unless there is a line over one of them
• SigDigs start at the first non-zero • This shows that it is known, not an estimate.
Adding and Subtracting
• Identify the measurement with the smallest number of
decimal places  this is the number of d.p. to use in
the final answer.
2.01 – 1.0 = 1.01, round to 1.0
1d.p.
Multiplying and Dividing Rule number 1:
• Identify the measurement with the smallest number of Carry out all the calculations first
SigDigs  this is the number of SigDigs to use in the and round the final answer – do
not round the numbers as you
final answer.
go, it could mess up the result.
1.2 x 3.45 = 4.14 round to 4.1
2sf.

22. When it gets more complicated…
(5.00 / 1.235) + 3.000 + (6.35 / 4.0) = 8.630829...
= 8.6
Try these out: smallest number of SigDigs
1. Complete the calculation
2. Identify the value with the fewest sigdigs
3. Round & present the answer to that many sigdigs
1. (12.67 - 2.2) x 89.00 / 33.1 =
=
1. ( 14.2 + 23.00 + 4.566 + 20.00) =
( 12.1 - 1.22) =
Tutorial: http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs7.html

23. When it gets more complicated…
(5.00 / 1.235) + 3.000 + (6.35 / 4.0) = 8.630829...
= 8.6
Try these out: smallest number of SigDigs
1. Complete the calculation
2. Identify the value with the fewest sigdigs
3. Round & present the answer to that many sigdigs
1. (12.67 - 2.2) x 89.00 / 33.1 = 28.1519637462
= 28
1. ( 14.2 + 23.00 + 4.566 + 20.00) = 5.6770220588
( 12.1 - 1.22) = 5.68
Tutorial: http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs7.html

24. Scientific Notation How can we write very small or very large
numbers in a clear manner?
3450000000000 = 0.0000000000345 =
? ?
3.45 x 10 3.45 x 10
Always one digit before the
decimal in scientific notation
Expand the following notations:
1.0 X 103 =
1.0 X 10-6 =
4.56 X 105 =
7.01 X 10-4 =
Record the following in scientific
notation to 3 significant digits:
• 12340000 =
• 00012340 =
• 10101010 =
http://media.wwnorton.com/college/chemistry/chemtours/interface.asp?chapter=chapter_01&folder=scientific_notation

25. Scientific Notation How can we write very small or very large
numbers in a clear manner?
3450000000000 = 0.0000000000345 =
12 -11
3.45 x 10 3.45 x 10
Always one digit before the
decimal in scientific notation 3 SigDigs
Expand the following notations:
1.0 X 103 =
1.0 X 10-6 =
4.56 X 105 =
7.01 X 10-4 =
Record the following in scientific
notation to 3 significant digits:
• 12340000 =
• 00012340 =
• 10101010 =
http://media.wwnorton.com/college/chemistry/chemtours/interface.asp?chapter=chapter_01&folder=scientific_notation

26. Scientific Notation & Significant Figures
Compare the following measurements:
3.04 x 104 kJ 3.040 x 104 kJ 3.0400 x 104 kJ
Write them out in full. Which is larger?
Which is written to the greatest number of significant figures?
Which was recorded using equipment with the least uncertainty?

27. Scientific Notation & Significant Digits
Compare the following measurements:
3.04 x 104 kJ 3.040 x 104 kJ 3.0400 x 104 kJ
3 SigDigs 4 SigDigs 5 SigDigs
12340000 to 3 sigdigs could be 12300000 or 1.23 x 107
Present these numbers to 3 sigdigs and in scientific notation.
1. 00.010101010101
2. 12.34567
Calculate the following, presenting the answer to the most appropriate sigdigs & notation:
1. 1000 x 3.4
2. 8.0 x 8.0
3. 8 x 8

28. How do we learn? and why do we need to practice?
Working Memory Long-term Memory
Storage…
Perception Filter
Interpreting
Events Rearranging sometimes
Observation Comparing branched,
s sometimes as
Storage
Instructions separate
Preparation
fragments
Feedback loop for perception filter
So…
• Review, ask questions, practice
• Come back to it later, has it stuck?
• Think about measurement and error in all your lab work
From “Cultured Pearls of Tasty Truffles: Teaching chemistry for the 21st Century”, by Bill Byers 28
http://www3.ul.ie/~childsp/CinA/Issue66/TOC20_truffles.htm

29. For more resources.
Please consider a donation to charity via Biology4Good.
Click here for more information about Biology4Good charity donations.
This is a Creative Commons presentation. It may be linked and embedded but not sold or re-hosted.