IB Chemistry on Uncertainty Calculation and significant figures

1. Tutorial on Uncertainty, Error analysis and
significant figures .
Prepared by
Lawrence Kok
http://lawrencekok.blogspot.com

2. Significant figures
Used in measurements
Degree of precision
Show digits believed to be
correct/certain + 1 estimated/uncertain
All reads 80
80
80.0
80.00
80.000
least precise
Certain
23.00
Uncertain
5
23.005g
more precise
Number sf necessary to express a measurement
• Consistent with precision of measurement
• Precise equipment = Measurement more sf
• Last digit always an estimate/uncertain
measurement
15.831g
(15.831 ± 0.001)g
(5 sig figures)

3. Significant figures
Used in measurements
Degree of precision
Show digits believed to be
correct/certain + 1 estimated/uncertain
All reads 80
80
80.0
80.00
80.000
least precise
Certain
23.00
Uncertain
5
Zeros bet
(significant)
4.109 = 4sf
902 = 3sf
5002.05 = 6sf
measurement
15.831g
23.005g
more precise
(15.831 ± 0.001)g
(5 sig figures)
Rules for significant figures
All non zero digit
(significant)
31.24 = 4 sf
563 = 3 sf
23 = 2sf
Number sf necessary to express a measurement
• Consistent with precision of measurement
• Precise equipment = Measurement more sf
• Last digit always an estimate/uncertain
Zeros after
decimal point
(significant)
4.580 = 4 sf
9.30 = 3sf
86.90000 = 7sf
3.040 = 4sf
67.030 = 5sf
Zero right of decimal point and
following a non zero digit
(significant)
0.00500 = 3sf
0.02450 = 4sf
0.04050 = 4sf
0.50 = 2sf
Deals with precision NOT accuracy!!!!!!!!
Precise measurement doesnt mean, it’s accurate
( instrument may not be accurate)
Zeros to left of digit
(NOT significant)
0.0023 = 2sf
0.000342 = 3sf
0.00003 = 1sf
Zero without decimal
(ambiguous)
80 = may have 1 or 2 sf
500 = may have 1 or 3 sf

4. Significant figures
Used in measurements
Degree of precision
Show digits believed to be
correct/certain + 1 estimated/uncertain
All reads 80
80
80.0
80.00
80.000
least precise
Certain
23.00
Uncertain
5
Zeros bet
(significant)
4.109 = 4sf
902 = 3sf
5002.05 = 6sf
measurement
15.831g
23.005g
more precise
(15.831 ± 0.001)g
(5 sig figures)
Rules for significant figures
All non zero digit
(significant)
31.24 = 4 sf
563 = 3 sf
23 = 2sf
Number sf necessary to express a measurement
• Consistent with precision of measurement
• Precise equipment = Measurement more sf
• Last digit always an estimate/uncertain
Zeros after
decimal point
(significant)
4.580 = 4 sf
9.30 = 3sf
86.90000 = 7sf
3.040 = 4sf
67.030 = 5sf
Zero right of decimal point and
following a non zero digit
(significant)
0.00500 = 3sf
0.02450 = 4sf
0.04050 = 4sf
0.50 = 2sf
Deals with precision NOT accuracy!!!!!!!!
Precise measurement doesnt mean, it’s accurate
( instrument may not be accurate)
Zeros to left of digit
(NOT significant)
0.0023 = 2sf
0.000342 = 3sf
0.00003 = 1sf
Zero without decimal
(ambiguous)
80 = may have 1 or 2 sf
500 = may have 1 or 3 sf
Click here and here for notes on sig figures

5. Significant figures
1
22
Smallest division = 0.1
22
Max = 21.63
2
Certain
21.6
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01
3
Certain = 21.6
4
Uncertain = 21.62 ±0.01
5
(21.62 ±0.01)
Measurement = Certain digits + 1 uncertain digit
Min = 21.61
Answer = 21.62 (4 sf)
21.6
(certain)
2
(uncertain)

6. Significant figures
1
22
Smallest division = 0.1
22
Max = 21.63
2
Certain
21.6
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01
3
Certain = 21.6
4
Uncertain = 21.62 ±0.01
5
(21.62 ±0.01)
Measurement = Certain digits + 1 uncertain digit
Min = 21.61
Answer = 21.62 (4 sf)
21.6
(certain)
1
Smallest division = 1
2
Uncertainty = 1/10 of smallest division.
= 1/10 of 1
= 1/10 x 1 = ±0.1
3
Certain = 36
4
Uncertain = 36.5 ±0.1
5
Measurement = Certain digits + 1 uncertain digit
2
(uncertain)
Certain
36
Max = 36.6
(36.5 ±0.1)
Min = 36.4
Answer = 36.5 (3 sf)
36.
5
(certain) (uncertain)

7. Significant figures
1
Smallest division = 10
Max = 47
2
Certain
40
Uncertainty = 1/10 of smallest division.
= 1/10 of 10
= 1/10 x 10 = ±1
3
Certain = 40
4
Uncertain = 46 ±1
5
(46 ±1)
Measurement = Certain digits + 1 uncertain digit
Min = 45
Answer = 46 (2 sf)
4
(certain)
6
(uncertain)

8. Significant figures
1
Smallest division = 10
Max = 47
2
Certain
40
Uncertainty = 1/10 of smallest division.
= 1/10 of 10
= 1/10 x 10 = ±1
3
Certain = 40
4
Uncertain = 46 ±1
5
(46 ±1)
Measurement = Certain digits + 1 uncertain digit
Min = 45
Answer = 46 (2 sf)
4
(certain)
1
Certain
3.4
Smallest division = 0.1
2
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01
3
Certain = 3.4
4
Uncertain = 3.41±0.01
5
Measurement = Certain digits + 1 uncertain digit
6
(uncertain)
Max = 3.42
(3.41 ±0.01)
Min = 3.40
Answer = 3.41 (3sf)
3.4
(certain)
1
(uncertain)

9. Significant figures
1
Smallest division = 0.05
Max = 0.48
0.1
2
0.2
0.3
0.4
0.5
Certain
0.45
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.05
= 1/10 x 0.05 = ±0.005 (±0.01)
3
Certain = 0.45
4
Uncertain = 0.47 ± 0.01
5
(0.47 ±0.01)
Measurement = Certain digits + 1 uncertain digit
Min = 0.46
Answer = 0.47 (2 sf)
0.4
(certain)
7
(uncertain)

10. Significant figures
1
Smallest division = 0.05
Max = 0.48
0.1
2
0.2
0.3
Certain
0.45
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.05
= 1/10 x 0.05 = ±0.005 (±0.01)
3
Certain = 0.45
4
Uncertain = 0.47 ± 0.01
5
0.4
(0.47 ±0.01)
Measurement = Certain digits + 1 uncertain digit
Min = 0.46
0.5
Answer = 0.47 (2 sf)
0.4
(certain)
7
(uncertain)
Measurement
1
Smallest division = 0.1
2
Uncertainty = 1/10 of smallest division.
= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01
3
Certain = 5.7
4
Uncertain = 5.72 ± 0.01
(5.72 ±0.01)
Answer = 5.72 (3sf)
5.7
(certain)
2
(uncertain)
1
Smallest division = 1
2
Uncertainty = 1/10 of smallest division.
= 1/10 of 1
= 1/10 x 1 = ±0.1
3
Certain = 3
4
Uncertain = 3.0 ± 0.1
(3.0 ±0.1)
Answer =3.0 (2 sf)
3
0
(certain) (uncertain)

11. Rules for sig figures addition /subtraction:
• Last digit retained is set by the first doubtful digit.
• Number decimal places be the same as least number of decimal places in any numbers being added/subtracted
23.112233
1.3324
+ 0.25
24.694633
uncertain
least number
decimal places
4.2
2.32
+ 0.6157
7.1357
least number
decimal places
1.367
- 1.34
0.027
uncertain
least number
decimal places
uncertain
4.7832
1.234
+ 2.02
8.0372
12.587
4.25
+ 0.12
16.957
uncertain
least number
decimal places
uncertain
least number
decimal places
2.300 x 103
+ 4.59 x 103
6.890 x 103
least number
decimal places
1247
134.5
450
+ 78
1909.5
68.7
- 68.42
0.28
uncertain
least number
decimal places
least number
decimal places
uncertain
1.0236
- 0.97268
0.05092
7.987
- 0.54
7.447
Convert to same exponent
x 104
476.8
47.68
+ 23.2 x 103
x 103
+ 23.2 x 103
500.0 x 103
least number
decimal places
uncertain
uncertain
least number
decimal places
least number
decimal places

12. Rules for sig figures addition /subtraction:
• Last digit retained is set by the first doubtful digit.
• Number decimal places be the same as least number of decimal places in any numbers being added/subtracted
23.112233
1.3324
+ 0.25
24.694633
uncertain
least number
decimal places
round down
4.7832
1.234
+ 2.02
8.0372
uncertain
least number
decimal places
round down
1247
134.5
450
+ 78
1909.5
uncertain
least number
decimal places
1.0236
- 0.97268
0.05092
4.2
2.32
+ 0.6157
7.1357
8.04
least number
decimal places
uncertain
round down
round up
0.03
uncertain
least number
decimal places
68.7
- 68.42
0.28
0.0509
least number
decimal places
uncertain
7.987
- 0.54
7.447
uncertain
least number
decimal places
round up
round down
round up
0.3
16.96
7.1
1.367
- 1.34
0.027
1910
12.587
4.25
+ 0.12
16.957
uncertain
round down
round up
24.69
least number
decimal places
uncertain
least number
decimal places
2.300 x 103
+ 4.59 x 103
6.890 x 103
least number
decimal places
7.45
Convert to same exponent
x 104
476.8
47.68
+ 23.2 x 103
x 103
+ 23.2 x 103
500.0 x 103
round up
6.89 x 103
500.0 x 103
5.000 x 105
least number
decimal places

13. Rules for sig figures - multiplication/division
• Answer contains no more significant figures than the least accurately known number.
12.34
3.22
x 1.8
71.52264
16.235
0.217
x
5
17.614975
923
÷ 20312
0.045441
least sf (2sf)
least sf (1sf)
least sf (3sf)
23.123123
x
1.3344
30.855495
4.52
÷ 6.3578
7.1093775
1300
x 57240
74412000
least sf (5sf)
least sf (3sf)
21.45
x 0.023
0.49335
0.00435
x
4.6
0.02001
least sf (2sf)
Scientific notation
2.8723
x
I.6
4.59568
least sf (2sf)
least sf (2sf)
6305
÷ 0.010
630500
least sf (2sf)
least sf (2sf)
I.3*103
x 5.724*104
7.4412 x 107
Click here for practice notes on sig figures

14. Rules for sig figures - multiplication/division
• Answer contains no more significant figures than the least accurately known number.
12.34
3.22
x 1.8
71.52264
least sf (2sf)
round up
23.123123
x
1.3344
30.855495
least sf (5sf)
21.45
x 0.023
0.49335
round down
round down
30.855
72
16.235
0.217
x
5
17.614975
least sf (1sf)
round up
4.52
÷ 6.3578
7.1093775
least sf (3sf)
923
÷ 20312
0.045441
least sf (3sf)
round down
0.0454
1300
x 57240
74412000
4.6
0.00435
x
4.6
0.02001
least sf (2sf)
round down
7.11
0.020
least sf (2sf)
Scientific notation
least sf (2sf)
round up
0.49
round up
20
2.8723
x
I.6
4.59568
least sf (2sf)
6305
÷ 0.010
630500
least sf (2sf)
round down
63000
6.3 x 105
I.3*103
x 5.724*104
7.4412 x 107
round down
74000000
7.4 x 107
Click here for practice notes on sig figures

15. Scientific notation
How many significant figures
Written as
a=1-9
Number too big/small
b = integer
3 sf
Scientific - notation = a ´10b
6,720,000,000
Size sand
= 6.72 ´109
4 sf
0.0000000001254
=1.254 ´10-10
3 sf
Speed of light
300000000
How many significant figures
4.66 x 10 6
4.660 x 10 6
4 sf
4.6600 x 10 6
4660000
3 sf
5 sf
Click here practice scientific notation
Click here practice scientific notation
= 3.00 ´108

16. Scientific notation
How many significant figures
Written as
a=1-9
Number too big/small
b = integer
3 sf
Scientific - notation = a ´10b
6,720,000,000
Size sand
= 6.72 ´109
4 sf
0.0000000001254
=1.254 ´10-10
3 sf
Speed of light
= 3.00 ´108
300000000
Scientific notation
80
3 ways to write 80
How many significant figures
4.66 x
4660000
10 6
3 sf
4.660 x 10 6
5 sf
80 – 8 x 101 – (1sf)
Digit 8 uncertain
It can be 70 to 90
80.
80. – 8.0 x 101 – (2sf)
Digit 8 is certain
It can be 79 to 81
80.0
80.0 – 8.00 x 101 – (3sf)
Digit 80 is certain
It can be 79.9 or 80.1
4 sf
4.6600 x 10 6
80
90 or 9 x 101
80 or 8 x 101
70 or 7 x 101
81 or 8.1 x 101
80 or 8.0 x 101
79 or 7.9 x 101
80.1 or 8.01 x 101
80.0 or 8.00 x 101
79.9 or 7.99 x 101
More prcise
Click here practice scientific notation
Click here practice scientific notation
✔

17. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Radius, r = 2.15 cm
Volume, V = 4/3πr3
V = 4/3 x π x (2.15)3
= 4/3 x 3.14 x 2.15 x 2.15 x 2.15
= 41.60
round down
41.6
4/3 – constant
π – constant
Their sf is not taken
(not a measurement)
least sf (3sf)

18. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Radius, r = 2.15 cm
Volume, V = 4/3πr3
V = 4/3 x π x (2.15)3
= 4/3 x 3.14 x 2.15 x 2.15 x 2.15
= 41.60
4/3 – constant
π – constant
Their sf is not taken
(not a measurement)
round down
41.6
Recording measurement using
uncertainty of equipment
Radius, r = (2.15 ±0.02) cm
4
Volume = p r 3
3
4
Volume = ´3.14 ´ 2.153 = 41.60
3
least sf (3sf)

19. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Radius, r = 2.15 cm
Volume, V = 4/3πr3
V = 4/3 x π x (2.15)3
= 4/3 x 3.14 x 2.15 x 2.15 x 2.15
= 41.60
4/3 – constant
π – constant
Their sf is not taken
(not a measurement)
least sf (3sf)
round down
41.6
Recording measurement using
uncertainty of equipment
Radius, r = (2.15 ±0.02) cm
Treatment of Uncertainty
Multiplying or dividing measured quantities
4
Volume = p r 3
3
% uncertainty = sum of % uncertainty of individual quantities
Radius, r = (2.15 ±0.02)
%uncertainty radius (%Δr) = 0.02 x 100 = 0.93%
2.15
% uncertainty V = 3 x % uncertainty r
% ΔV = 3 x % Δr
* For measurement raised to power of n, multiply % uncertainty by n
* Constant, pure/counting number has no uncertainty and sf not taken
4
Volume = p r 3
3
4
Volume = ´3.14 ´ 2.153 = 41.60
3
0.02
´100% = 0.93%
2.15
Measurement raised to power of 3,
multiply % uncertainty by 3
%DV = 3´ %Dr
%DV = 3´ 0.93 = 2.79%
Volume = (41.60 ± 2.79%)
%Dr =
AbsoluteDV =
2.79
´ 41.60 =1.16
100
Volume = (41.60 ±1.16)
Volume = (42 ±1)

20. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Radius, r = 3.0 cm
Circumference, C = 2πr
C = 2 x π x (3.0)
= 2 x 3.14 x 3.0
= 18.8495
round up
19
2 – constant
π – constant
Their sf is not taken
(not a measurement)
least sf (2sf)

21. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Radius, r = 3.0 cm
Circumference, C = 2πr
C = 2 x π x (3.0)
= 2 x 3.14 x 3.0
= 18.8495
2 – constant
π – constant
Their sf is not taken
(not a measurement)
least sf (2sf)
round up
19
Recording measurement using
uncertainty of equipment
Radius, r = (3.0 ±0.2) cm
Circumference = 2p r
Circumference = 2´3.14´3.0 =18.8495

22. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Radius, r = 3.0 cm
Circumference, C = 2πr
C = 2 x π x (3.0)
= 2 x 3.14 x 3.0
= 18.8495
2 – constant
π – constant
Their sf is not taken
(not a measurement)
least sf (2sf)
round up
19
Recording measurement using
uncertainty of equipment
Radius, r = (3.0 ±0.2) cm
Treatment of Uncertainty
Multiplying or dividing measured quantities
Circumference = 2p r
% uncertainty = sum of % uncertainty of individual quantities
Radius, r = (3.0 ±0.2)
%uncertainty radius (%Δr) = 0.2 x 100 = 6.67%
3.0
% uncertainty C = % uncertainty r
% ΔC = % Δr
* Constant, pure/counting number has no uncertainty and sf not taken
Circumference = 2p r
Circumference = 2´3.14´3.0 =18.8495
0.2
´100% = 6.67%
3.0
%Dc = %Dr
%Dc = 6.67%
Circumference = (18.8495 ± 6.67%)
%Dr =
AbsoluteDC =
6.67
´18.8495 =1.25
100
Circumference = (18.8495 ±1.25)
Circumference = (19 ±1)

23. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Time, t = 2.25 s
Displacement, s = ½ gt2
s = 1/2 x 9.8 x (2.25)2
= 24.80625
g and ½ – constant
Their sf is not taken
(not a measurement)
least sf (3sf)
round down
24.8
1
Displacement, s = ´ 9.8x(2.25)
2

24. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Time, t = 2.25 s
Displacement, s = ½ gt2
s = 1/2 x 9.8 x (2.25)2
= 24.80625
g and ½ – constant
Their sf is not taken
(not a measurement)
least sf (3sf)
round down
24.8
Recording measurement using
uncertainty of equipment
Time, t = (2.25 ±0.01) cm
1
Displacement, s = ´ 9.8x(2.25)
2
1
Displacement, s = gt 2
2
1
Displacement, s = ´ 9.8x2.25x2.25 = 24.80625
2

25. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Time, t = 2.25 s
Displacement, s = ½ gt2
g and ½ – constant
Their sf is not taken
(not a measurement)
s = 1/2 x 9.8 x (2.25)2
= 24.80625
least sf (3sf)
round down
24.8
Recording measurement using
uncertainty of equipment
Time, t = (2.25 ±0.01) cm
1
Displacement, s = ´ 9.8x(2.25)
2
1
Displacement, s = gt 2
2
1
Displacement, s = ´ 9.8x2.25x2.25 = 24.80625
2
0.01
´100% = 0.4%
2.25
Measurement raised to power of 2,
multiply % uncertainty by 2
%Ds = 2 ´ %Dt
%Ds = 2 ´ 0.4% = 0.8%
Displacement = (24.80 ± 0.8%)
%Dt =
Treatment of Uncertainty
1 2
Multiplying or dividing measured quantities Displacement, s = gt
2
% uncertainty = sum of % uncertainty of individual quantities
Time, t = (2.25 ±0.01)
%uncertainty time (%Δt) = 0.01 x 100 = 0.4%
2.25
% uncertainty s = 2 x % uncertainty t
% Δs = 2 x % Δt
* For measurement raised to power of n, multiply % uncertainty by n
AbsoluteDs =
0.4
´ 24.80 = 0.198
100
Displacement = (24.80 ± 0.198)
Displacement = (24.8 ± 0.2)

26. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Length, I = 1.25 m
L
g
T = 2p
T = 2 x π x √(1.25/9.8)
= 2 x 3.14 x 0.35714
= 2.24399
round down
2.24
least sf (3sf)
2, π and g – constant
Their sf is not taken
(not a measurement)

27. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Length, I = 1.25 m
L
g
T = 2p
least sf (3sf)
T = 2 x π x √(1.25/9.8)
= 2 x 3.14 x 0.35714
= 2.24399
round down
2.24
Recording measurement using
uncertainty of equipment
T = 2p
Length, I = (1.25 ±0.05) m
T = 2p
L
g
1.25
= 2.24
9.8
2, π and g – constant
Their sf is not taken
(not a measurement)

28. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Length, I = 1.25 m
L
g
T = 2p
least sf (3sf)
T = 2 x π x √(1.25/9.8)
= 2 x 3.14 x 0.35714
= 2.24399
2, π and g – constant
Their sf is not taken
(not a measurement)
round down
2.24
Recording measurement using
uncertainty of equipment
T = 2p
Length, I = (1.25 ±0.05) m
T = 2p
L
g
1.25
= 2.24
9.8
0.05
´100% = 4%
1.25
Measurement raised to power of 1/2,
1
%DT = ´ %Dl multiply % uncertainty by 1/2
2
%DT = 2%
T = (2.24 ± 2%)
%Dl =
Treatment of Uncertainty
Multiplying or dividing measured quantities
T = 2p
L
g
% uncertainty = sum of % uncertainty of individual quantities
Length, I = (1.25 ±0.05)
%uncertainty length (%ΔI) = 0.05 x 100 = 4%
1.25
% uncertainty T = ½ x % uncertainty I
% ΔT = ½ x % ΔI
* For measurement raised to power of n, multiply % uncertainty by n
AbsoluteDT =
2
´ 2.24 = 0.044
100
T = (2.24 ± 0.044)
T = (2.24 ± 0.04)

29. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Area, A = I x h
Length, I = 4.52 cm
Height, h = 2.0 cm
4.52
2.0
9.04
x
least sf (2sf)
round down
9.0

30. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Area, A = I x h
Length, I = 4.52 cm
Height, h = 2.0 cm
4.52
2.0
9.04
x
least sf (2sf)
round down
9.0
Recording measurement using
uncertainty of equipment
Length, I = (4.52 ±0.02) cm
Height, h = (2.0 ±0.2)cm3
Area, A = Length,l ´ height, h
Area = 4.52 ´ 2.0 = 9.04

31. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Area, A = I x h
Length, I = 4.52 cm
Height, h = 2.0 cm
4.52
2.0
9.04
x
least sf (2sf)
round down
9.0
Recording measurement using
uncertainty of equipment
Length, I = (4.52 ±0.02) cm
Height, h = (2.0 ±0.2)cm3
Area, A = Length,l ´ height, h
Area = 4.52 ´ 2.0 = 9.04
0.02
´100% = 0.442%
4.52
0.2
%Dh =
´100% = 10%
2.0
%DA = %Dl + %Dh
%DA = 0.442% +10% = 10.442%
Area = (9.04 ±10.442%)
%Dl =
Treatment of Uncertainty
Multiplying or dividing measured quantities
Area, A = Length,l ´height,h
% uncertainty = sum of % uncertainty of individual quantities
Length, l = (4.52 ±0.02)
%uncertainty length (%Δl) = 0.02 x 100 = 0.442%
4.52
Height, h = (2.0 ±0.2)
%uncertainty height (%Δh) = 0.2 x 100 = 10%
2.0
% uncertainty A = % uncertainty length + % uncertainty height
% ΔA =
% ΔI
+
%Δh
AbsoluteDA =
Area = (9.0 ± 0.9)
10.442
´ 9.04 = 0.9
100

32. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Moles, n = Conc x Vol
Conc, c
= 2.00 M
Volume, v = 2.0 dm3
2.00
2.0
4.00
x
least sf (2sf)
round down
4.0

33. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Moles, n = Conc x Vol
Conc, c
= 2.00 M
Volume, v = 2.0 dm3
2.00
2.0
4.00
x
least sf (2sf)
round down
4.0
Recording measurement using
uncertainty of equipment
Conc, c
= (2.00 ±0.02) cm
Volume, v = (2.0 ±0.1)dm3
Mole, n = Conc, c ´Volume, v
Mole = 2.00 ´ 2.0 = 4.00

34. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Moles, n = Conc x Vol
Conc, c
= 2.00 M
Volume, v = 2.0 dm3
2.00
2.0
4.00
x
least sf (2sf)
round down
4.0
Recording measurement using
uncertainty of equipment
Conc, c
= (2.00 ±0.02) cm
Volume, v = (2.0 ±0.1)dm3
Mole, n = Conc, c ´Volume, v
Mole = 2.00 ´ 2.0 = 4.00
0.02
´100% = 1%
2.00
0.1
%Dv =
´100% = 5%
2.0
%Dn = %Dc + %Dv
%Dc =
Treatment of Uncertainty
Multiplying or dividing measured quantities
Mole, n = Conc, c ´Vol, v
% uncertainty = sum of % uncertainty of individual quantities
Conc, c = (2.00 ±0.02)
%uncertainty conc (%Δc) = 0.02 x 100 = 1%
2.00
Volume, v = (2.0 ±0.1)
%uncertainty volume (%Δv) = 0.1 x 100 = 5%
2.0
% uncertainty n = % uncertainty conc + % uncertainty volume
% Δn =
% Δc
+
%Δv
%Dn = 1% + 5% = 6%
Mole = (4.00 ± 6%)
AbsoluteDn =
6
´ 4.00 = 0.24
100
Mole = (4.00 ± 0.24)
Mole = (4.0 ± 0.2)

35. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Mass, m = 482.63g
Volume, v = 258 cm3
Density = Mass
Volume
482.63
÷
258
1.870658
round down
1.87
least sf (3sf)

36. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Mass, m = 482.63g
Volume, v = 258 cm3
Density = Mass
Volume
482.63
÷
258
1.870658
least sf (3sf)
round down
1.87
Recording measurement using
uncertainty of equipment
Mass, m = (482.63 ±1)g
Volume, v = (258 ±5)cm3
Density, D =
Density, D =
Mass
Volume
482.63
=1.870658
258

37. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Mass, m = 482.63g
Volume, v = 258 cm3
Density = Mass
Volume
482.63
÷
258
1.870658
least sf (3sf)
round down
1.87
Recording measurement using
uncertainty of equipment
Mass, m = (482.63 ±1)g
Volume, v = (258 ±5)cm3
Treatment of Uncertainty
Multiplying or dividing measured quantities
Density, D =
Mass
Volume
% uncertainty = sum of % uncertainty of individual quantities
Mass, m = (482.63 ±1)
%uncertainty mass (%Δm) = 1
x 100 = 0.21%
482.63
Volume, V = (258 ±5)
%uncertainty vol (%ΔV) = 5 x 100 = 1.93%
258
% uncertainty density = % uncertainty mass + % uncertainty volume
% ΔD =
% Δm
+
%ΔV
Density, D =
Density, D =
Mass
Volume
482.63
=1.870658
258
1
´100% = 0.21%
482.63
5
%DV =
´100% = 1.93%
258
%DD = %Dm + %DV
%DD = 0.21% +1.93% = 2.14%
Density = (1.87 ± 2.14%)
%Dm =
AbsoluteDD =
2.14
´1.87 = 0.04
100
Density = (1.87 ± 0.04)

38. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Mass water = 2.00 g
ΔTemp
= 2.0 C
Enthalpy, H = mcΔT
x
2.00
4.18
2.0
16.72
c – constant
sf is not taken
(not a measurement)
least sf (2sf)
round up
17

39. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Mass water = 2.00 g
ΔTemp
= 2.0 C
Enthalpy, H = mcΔT
x
2.00
4.18
2.0
16.72
c – constant
sf is not taken
(not a measurement)
least sf (2sf)
round up
17
Recording measurement using
uncertainty of equipment
Enthalpy, H = m ´ c ´ DT
Mass water = (2.00 ±0.02)g
ΔTemp
= (2.0 ±0.4) C
Enthalpy, H = 2.00 ´ 4.18´ 2.0 =16.72

40. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Mass water = 2.00 g
ΔTemp
= 2.0 C
Enthalpy, H = mcΔT
x
2.00
4.18
2.0
16.72
c – constant
sf is not taken
(not a measurement)
least sf (2sf)
round up
17
Recording measurement using
uncertainty of equipment
Mass water = (2.00 ±0.02)g
ΔTemp
= (2.0 ±0.4) C
Treatment of Uncertainty
Multiplying or dividing measured quantities
Enthalpy, H = m ´ c ´ DT
Enthalpy, H = 2.00 ´ 4.18´ 2.0 =16.72
Enthalpy, H = m ´ c ´ DT
% uncertainty = sum of % uncertainty of individual quantities
Mass, m = (2.00 ±0.02)
%uncertainty mass (%Δm) = 0.02 x 100 = 1%
2.00
ΔTemp = (2.0 ±0.4)
%uncertainty temp (%ΔT) = 0.4 x 100 = 20%
2.0
% uncertainty H = % uncertainty mass + % uncertainty temp
% ΔH =
% Δm
+
%ΔT
0.02
´100% = 1%
2.00
0.4
%DT =
´100% = 20%
2.0
%DH = %Dm + %DT
%Dm =
%DH = 1% + 20% = 21%
Enthalpy = (16.72 ± 21%)
AbsoluteDH =
21
´16.72 = 3.51
100
Enthalpy = (16.72 ± 3.51)
Enthalpy = (17 ± 4)

41. Treatment of uncertainty in measurement
•
Adding or subtracting
Max absolute uncertainty is the SUM of individual uncertainties
Initial mass beaker, M1
= (20.00 ±0.01) g
Final mass beaker + water, M2 = (22.00 ±0.01)g
Initial Temp, T1 = (21.2 ±0.2)C
Final Temp, T2 = (23.2 ±0.2)C
Mass water, m = (M2 –M1)
Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Diff Temp ΔT = (T2 –T1)
Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Mass water, m = (22.00 –20.00) = 2.00
Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Mass water, m = (2.00 ±0.02)g
Diff Temp ΔT = (23.2 –21.2) = 2.0
Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Diff Temp, ΔT = (2.0 ±0.4)
Mass water, m = (2.00 ±0.02)g
ΔTemp = (2.0 ±0.4) C
Addition/Subtraction/Multiply/Divide
•
Multiplying or dividing
Max %uncertainty is the SUM of individual %uncertainties

42. Treatment of uncertainty in measurement
•
Adding or subtracting
Max absolute uncertainty is the SUM of individual uncertainties
Initial mass beaker, M1
= (20.00 ±0.01) g
Final mass beaker + water, M2 = (22.00 ±0.01)g
Initial Temp, T1 = (21.2 ±0.2)C
Final Temp, T2 = (23.2 ±0.2)C
Addition/Subtraction/Multiply/Divide
•
Multiplying or dividing
Max %uncertainty is the SUM of individual %uncertainties
Addition/Subtraction
Add absolute uncertainty
Enthalpy, H = (M2-M1) x c x (T2-T1)
Mass water, m = (M2 –M1)
Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Diff Temp ΔT = (T2 –T1)
Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Multiplication
Add % uncertainty
Mass water, m = (22.00 –20.00) = 2.00
Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Mass water, m = (2.00 ±0.02)g
Mass water, m = (2.00 ±0.02)g
Diff Temp ΔT = (23.2 –21.2) = 2.0
Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Diff Temp, ΔT = (2.0 ±0.4)
ΔTemp = (2.0 ±0.4) C
Enthalpy, H = m ´ c ´ DT
Enthalpy, H = 2.00 ´ 4.18´ 2.0 =16.72

43. Treatment of uncertainty in measurement
•
Adding or subtracting
Max absolute uncertainty is the SUM of individual uncertainties
Initial mass beaker, M1
= (20.00 ±0.01) g
Final mass beaker + water, M2 = (22.00 ±0.01)g
Addition/Subtraction/Multiply/Divide
•
Multiplying or dividing
Max %uncertainty is the SUM of individual %uncertainties
Addition/Subtraction
Add absolute uncertainty
Initial Temp, T1 = (21.2 ±0.2)C
Final Temp, T2 = (23.2 ±0.2)C
Enthalpy, H = (M2-M1) x c x (T2-T1)
Mass water, m = (M2 –M1)
Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Diff Temp ΔT = (T2 –T1)
Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Multiplication
Add % uncertainty
Mass water, m = (22.00 –20.00) = 2.00
Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02
Mass water, m = (2.00 ±0.02)g
Mass water, m = (2.00 ±0.02)g
Treatment of Uncertainty
Multiplying or dividing measured quantities
Diff Temp ΔT = (23.2 –21.2) = 2.0
Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4
Diff Temp, ΔT = (2.0 ±0.4)
ΔTemp = (2.0 ±0.4) C
Enthalpy, H = m ´ c ´ DT
% uncertainty = sum of % uncertainty of individual quantities
Mass, m = (2.00 ±0.02)
%uncertainty mass (%Δm) = 0.02 x 100 = 1%
2.00
ΔTemp = (2.0 ±0.4)
%uncertainty temp (%ΔT) = 0.4 x 100 = 20%
2.0
% uncertainty H = % uncertainty mass + % uncertainty temp
% ΔH =
% Δm
+
%ΔT
Enthalpy, H = m ´ c ´ DT
Enthalpy, H = 2.00 ´ 4.18´ 2.0 =16.72
0.02
´100% = 1%
2.00
0.4
%DT =
´100% = 20%
2.0
%DH = %Dm + %DT
%Dm =
%DH = 1% + 20% = 21%
Enthalpy = (16.72 ± 21%)
AbsoluteDH =
21
´16.72 = 3.51
100
Enthalpy = (16.72 ± 3.51)
Enthalpy = (17 ± 4)

44. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Volt, v = 2.0 V
Current, I = 3.0A
Time, t = 4.52s
t ´ I2
v1/2
4.52 x 3.0 x 3.0 = 40.68
÷ 1.414
28.769
Energy =
round up
29
least sf (2sf)

45. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
Volt, v = 2.0 V
Current, I = 3.0A
Time, t = 4.52s
t ´ I2
v1/2
4.52 x 3.0 x 3.0 = 40.68
÷ 1.414
28.769
Energy =
least sf (2sf)
round up
29
Recording measurement using
uncertainty of equipment
Volt, v = (2.0 ± 0.2)
Current, I = ( 3.0 ± 0.6)
Time, t = (4.52 ± 0.02)
t ´ I2
Energy, E = 1/2
v
4.52(3.0)2
Energy, E =
= 28.638
2.01/2

46. Significant figures and Uncertainty in measurement
t ´ I2
v1/2
4.52 x 3.0 x 3.0 = 40.68
÷ 1.414
28.769
Energy =
Recording measurement
using significant figures
Volt, v = 2.0 V
Current, I = 3.0A
Time, t = 4.52s
least sf (2sf)
round up
29
Recording measurement using
uncertainty of equipment
Volt, v = (2.0 ± 0.2)
Current, I = ( 3.0 ± 0.6)
Time, t = (4.52 ± 0.02)
Treatment of Uncertainty
Multiplying or dividing measured quantities
Energy, E =
t ´ I2
v1/2
% uncertainty = sum of % uncertainty of individual quantities
Time, t = (4.52 ±0.02)
%uncertainty time (%Δt) = 0.02 x 100 = 0.442%
4.52
Current, I = (3.0 ±0.6)
%uncertainty current (%ΔI) = 0.6 x 100 = 20%
3.0
Volt, v = (2.0±0.2)
%uncertainty volt (%Δv) = 0.2 x 100 = 10%
2.0
% ΔE = % Δt + 2 %ΔI + ½ %ΔV
* For measurement raised to power of n, multiply % uncertainty by n
t ´ I2
Energy, E = 1/2
v
4.52(3.0)2
Energy, E =
= 28.638
2.01/2
0.02
%Dt =
´100% = 0.442%
4.52
0.6
%DI =
´100% = 20%
3.0
0.2
%Dv =
´100% = 10%
2.0
1
%DE = %Dt + 2 ´%I + ´%Dv
2
%DE = (
0.02
0.6
1 0.2
´100% ) + ( 2 ´
´100% ) + ( ´
´100%
4.52
3.0
2 2.0
%DE = 0.442%+ 40%+ 5% = 45.442% = 45%
Energy, E = (28.638± 45%)
AbsoluteDE =
Energy, E = (29 ±13)
45
´ 28.638 =13
100
)

47. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
G = (20 )
H = (16 )
Z = (106)
(G + H )
Z
20 + 16 = 36
÷ 106
0.339
Speed, s =
round down
0.34
least sf (2sf)

48. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
(G + H )
Z
20 + 16 = 36
÷ 106
0.339
Speed, s =
G = (20 )
H = (16 )
Z = (106)
least sf (2sf)
round down
0.34
Recording measurement using
uncertainty of equipment
G = (20 ± 0.5)
H = (16 ± 0.5)
Z = (106 ± 1.0)
✔
Addition
add absolute uncertainty
G+H = (36 ± 1)
Z = (106 ± 1.0)
Speed, s =
(G + H )
Z
Speed, s =
(20 +16)
= 0.339
106

49. Significant figures and Uncertainty in measurement
Recording measurement
using significant figures
(G + H )
Z
20 + 16 = 36
÷ 106
0.339
Speed, s =
G = (20 )
H = (16 )
Z = (106)
least sf (2sf)
round down
0.34
Speed, s =
Recording measurement using
uncertainty of equipment
G = (20 ± 0.5)
H = (16 ± 0.5)
Z = (106 ± 1.0)
✔
Addition
add absolute uncertainty
G+H = (36 ± 1)
Z = (106 ± 1.0)
(G + H )
Z
Speed, s =
(20 +16)
= 0.339
106
%D(G + H ) =
Treatment of Uncertainty
Multiplying or dividing measured quantities
(G + H )
Speed, s =
Z
% uncertainty = sum of % uncertainty of individual quantities
(G + H) = (36 ±1)
%uncertainty (G+H) (%ΔG+H) = 1 x 100 = 2.77%
36
Z = (106 ±1.0)
%uncertainty Z (%Δz) = 1.0 x 100 = 0.94%
106
%uncertainty s = %uncertainty(G+H) + %uncertainty(Z)
% Δs = % Δ(G+H)
+
%Δz
*Adding or subtracting
Max absolute uncertainty is the SUM of individual uncertainties
%DZ =
1.0
´100% = 2.77%
36
1.0
´100% = 0.94%
106
%DS = %D(G + H)+%DZ
%DS = 2.77%+ 0.94% = 3.71%
Speed, s = (0.339 ± 3.71%)
AbsoluteDS =
3.71
´ 0.339 = 0.012
100
Speed, s = (0.339 ± 0.012)
ScientificNotation = a ´10

50. Acknowledgements
Thanks to source of pictures and video used in this presentation
http://crescentok.com/staff/jaskew/isr/tigerchem/econfig/electron4.htm
http://pureinfotech.com/wp-content/uploads/2012/09/periodicTable_20120926101018.png
http://www.wikihow.com/Find-the-Circumference-and-Area-of-a-Circle
Thanks to Creative Commons for excellent contribution on licenses
http://creativecommons.org/licenses/
Prepared by Lawrence Kok
Check out more video tutorials from my site and hope you enjoy this tutorial
http://lawrencekok.blogspot.com