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1. Chapter 03
Statistics and Data Handling
in Analytical Chemistry
Dr. Naveen Kosar
Assistant Professor
Department of Chemistry
School of Science, UMT

2. Significant Figures
The number of digits necessary to express the results of a
measurement consistent with the measured precision.
At the most basic level, Analytical Chemistry relies upon experimentation;
experimentation in turn requires numerical measurements and
measurements are always taken from instruments made by other workers.
2

3. Some information about measurements
Examples we will study include the metric ruler, the graduated cylinder, and
the scale.
2) Because of the involvement of human beings, NO measurement is exact;
some error is always involved. This means that every answer in science has
some uncertainty associated with it. We might be fairly confident we have the
correct answer, but we can never be 100% certain we have the EXACT
correct answer.
3

4. 3) Measurements always have two parts - a numerical part
(sometimes called a factor) and a dimension (a unit).
Significant figures are concerned with measurements not exact
counting.
4

5. Identifying significant digits
The following rules are helpful in identifying significant digits
1. Digits other than zero are significant.
e.g., 42.1m has 3 sig figs.
2. Zeroes are sometimes significant, and sometimes they are
not.
3. Zeroes at the beginning of a number (used just to position
the decimal point) are not significant.
e.g., 0.025m has 2 sig figs. In scientific notation, this can be
written as 2.5*10-2m 5

6. 4. Zeroes between nonzero digits are significant.
e.g., 40.1m has 3 sig figs
5. Zeroes at the end of a number that contains a decimal point are
significant.
e.g., 41.0m has 3 sig figs, while 441.20m has 5. In scientific
notation, these can be written respectively as
4.10*101 and 4.4120*102
6. Zeroes at the end of a number that does not contain a decimal
point may or may not be significant. If we wish to indicate the
number of significant figures in such numbers, it is common to
use the scientific notation.
6

7. e.g., The quantity 52800 km could be having 3, 4, or 5 sig figs—the
information is insufficient for decision. If both of the zeroes are used just to
position the decimal point (i.e., the number was measured with estimation
±100), the number is 5.28×104 km (3 sig figs) in scientific notation. If only
one of the zeroes is used to position the decimal point (i.e., the number was
measured ±10), the number is 5.280×104 km (4 sig figs). If the number is
52800±1 km , it implies 5.2800×104 km (5 sig figs).
7

8. Exact Numbers
Exact numbers can be considered as having an unlimited number of significant
figures. This applies to defined quantities too. e.g.,
 The rules of significant figures do not apply to (a) the count of 47 people in a
hall, or (b) the equivalence: 1 inch = 2.54 centimeters.
 In addition, the power of 10 used in scientific notation is an exact number,
i.e. the number 103 is exact, but the number 1000 has 1 sig fig.
It makes a lot of sense to write numbers derived from measurements in
scientific notation, since the notation clearly indicates the number of
significant digits in the number.
8

9. Look at the following example
9

10. The length is greater than 2.6 cm but less than 2.7 cm, and so the estimated
value is 2.62 cm. The measurement can be written as 2.62±0.01 cm or
26.2±0.1 mm. The number 26.2mm contains three significant figures.
10

11. How do you read it?
How do you read it?
11

12. 12

13. 13

14. 14

15. 15

16. Lecture 3
Data Handling
Significant Figures, continued….
Math using significant figures
16

17. Math With Significant Figures
Addition and Subtraction
In mathematical operations involving significant figures, the answer is reported in
such a way that it reflects the reliability of the least precise operation. Let's state
that another way: a chain is no stronger than its weakest link. An answer is no
more precise that the least precise number used to get the answer. Let's do it one
more time: imagine a team race where you and your team must finish together.
Who dictates the speed of the team? Of course, the slowest member of the team.
Your answer cannot be MORE precise than the least precise measurement.
17

18. For addition and subtraction, look at the decimal portion (i.e., to the right of the
decimal point) of the numbers ONLY. Here is what to do:
1) Count the number of significant figures in the decimal portion of each number
in the problem. (The digits to the left of the decimal place are not used to
determine the number of decimal places in the final answer.)
2) Add or subtract in the normal fashion.
3) Round the answer to the LEAST number of places in the decimal portion of
any number in the problem.
18

19. Find the formula weight for Ag2MoO4 given the following atomic weights: Ag
= 107.870, Mo = 95.94, O = 15.9994.
The number with the least number of digits after the decimal point is 95.94
which has two digits for expression of precision. Also, it is the number with the
highest uncertainty. The atomic weights for Ag and O have 3 and 4 digits after
the decimal point. Therefore, if we calculate the formula weight, we will get
375.6776. However, the answer should be reported as 375.68 ( i.e. to the same
uncertainty of the least precise value.
19

20. Multiplication and Division
• The number having the least number of significant figures is called the KEY
NUMBER. The LEAST number of significant figures in any number of the
problem determines the number of significant figures in the answer.
• In case where two or more numbers have the same number of significant
figures, the key number is determined as the number of the lowest value
regardless of decimal point.
• 95.94
• 95.96
20

21. Note that: When the uncertainty of a number is not known, the
uncertainty is assumed to be +1 of the last digit to the right
21

22. 2.5 x 3.42 = ?
The answer to this problem would be 8.6 (which was rounded
from the calculator reading of 8.55). Why?
2.5 is the key number which has two significant figures while
3.42 has three. Two significant figures is less precise than
three, so the answer has two significant figures.
22
95.94
959.4

23. Relative uncertainty in key number = ( +1/25) = +0.04
Now find the absolute uncertainty in answer:
(8.55/25) = 0.342
Therefore, the uncertainty in the answer should be known to one
decimal point. The answer can be written as 8.6 + 0.3. The
relative uncertainty in answer can now be calculated:
Srel = (+0.342/8.6) = +0.04
The relative uncertainty in answer is the same as that of the key
number, which should be the case.
23

24. 2.33 x 6.085 x 2.1= ? How many significant figures should be in the answer?
Answer - two.
Which is the key number?
Answer - the 2.1
Why?
It has the least number of significant figures in the problem. It is, therefore, the
least precise measurement.
Answer = 30.
24

25. Relative uncertainty in key number = ( +1/21) = +0.048
Now find the absolute uncertainty in answer:
(29.77/21) = 1.4
Therefore, the absolute uncertainty in the answer should be
known to integers.
The answer can be written as 30. + 1.4 The relative
uncertainty in answer can now be calculated:
Srel = (+1.4/30.) = +0.033
The relative uncertainty in answer is close to that of the key
number, which should be the case. 25

26. How many significant figures will the answer to 3.10 x 4.520 = (Calculator
gives 14.012) have?
3.10 is the key number which has three significant figures.
Three is supposed to be the correct answer. 14.0 has three significant
figures. Note that the zero in the tenth's place is considered significant.
All trailing zeros in the decimal portion are considered significant.
26

27. Another common error is for the student to think that 14 and
14.0 are the same thing. THEY ARE NOT. 14.0 is ten times
more precise than 14. The two numbers have the same
value, but they convey different meanings about how
trustworthy they are.
However, the correct answer should be reported as 14.01.
Note that an additional significant figure is included in the
answer. This is because the answer is less than the key
number.
27

28. Relative uncertainty in key number = ( +1/310) = +3.2*10-3
Now find the absolute uncertainty in answer:
(14.012/310) = 0.0452
Therefore, the absolute uncertainty in the answer should be known to
one hundredth. The answer can be written as 14.01 + 0.05. The
relative uncertainty in answer can now be calculated:
Srel = (+0.05/14.01) = +3.5*10-3
The relative uncertainty in answer is very close to that of the key
number, which should be the case.
28

29. Why do we add an additional significant figure in the answer
when the answer is less than the key number?
The answer to this question simply is to reduce the
uncertainty associated with the answer. When the answer is
less than the key number, the uncertainty associated with
the answer is unjustifiably large.
29

30. (4.52 x 10-4) ÷ (3.980 x 10-6).
How many significant figures in the answer?
Answer - three.
Which is the key number?
Answer - the 4.52 x 10-4.
Why?
It has the least number of significant figures in the problem. It is, therefore, the
least precise measurement. Notice it is the 4.52 portion that plays the role of
determining significant figures; the exponential portion plays no role. However,
since the answer is less than the key number, an additional significant figure is
used and written as a subscript.
Answer = 113.6
30

31. Relative uncertainty in key number = ( +1/452) =
+2.2*10-3
Now find the absolute uncertainty in answer:
(113.568/452) = 0.251
Therefore, the uncertainty in the answer should be known to
one decimal point. The absolute uncertainty in the answer
is therefore +0.3. The answer can be written as 113.6 +
0.3. The relative uncertainty in answer can now be
calculated:
Srel = (+0.3/113.6) = +2.6*10-3
The relative uncertainty in answer is very close to that of the
key number, which should be the case. 31

32. 4.20 x 3.52 =
Which is the key number?
Both have 3 significant figures. In this case, the number with smaller value,
regardless of the decimal point, is the key number (3.52). The correct answer
should be reported as 14.78
Once again, we have added an additional significant figure as a subscript since
the answer is less than the key number
32

33. Uncertainty in key number = ( +1/352) = +2.84*10-3
Now find the absolute uncertainty in answer :
(14.784/352) = 0.042
Therefore, the absolute uncertainty in the answer should be
known to one hundredth. The absolute uncertainty in the
answer is therefore +0.042. The answer can be written
as 14.78 + 0.04. The relative uncertainty in answer can
now be calculated:
Srel = (+0.042/14.78) = +2.84*10-3
The relative uncertainty in answer is the same as that of the
key number, which should be the case.
33

34. %
5470578
.
88
1689
.
1
05300
.
0
*
5481
.
0
*
63
.
35 
Look at the following multiplication problem:
The key number is 35.63 which has 4 significant figures. Therefore, the answer
should be 88.55%
The answer is larger than the key number, therefore no additional figure is
added.
34

35. Why not keep additional significant figures as we did in the
previous example?
Uncertainty in key number = ( +1/3563) = +2.8*10-4
Now find the absolute uncertainty in answer :
(88.547/3563) = 0.025
Therefore, the uncertainty in the answer should be known to one
hundredth. The absolute uncertainty in the answer is therefore
+0.02 and can be written as 88.55 + 0.02. The relative
uncertainty in answer can now be calculated:
Srel = (+0.025/88.55) = +2.8*10-4
The relative uncertainty in answer is the same as that of the key
number, which should be the case.
35

36.  
687
04
.
36
24
.
32
100.0
x
97.7

Find the answer to the following calculation:
113.67 = 113. 67
When multiple operations are involved, do it in a step-by-step procedure.
The parenthesis above has 97.7 as the key number.
36

37. For the initial calculation :
( 97.7 x 100.0 )
32.42
Uncertainty in key number = ( +1/977) = +1.0*10-3
Now find the absolute uncertainty in answer :
(301.36/977) = 0.308
Therefore, the uncertainty in the answer should be known to one
decimal point. The uncertainty in the answer is therefore +0.31. The
answer can be written as 301.36 + 0.3. The relative uncertainty in
answer can now be calculated:
Srel = (+ 0.31/301.36) = +1.0*10-3
The relative uncertainty in answer is the same as that of the key
number, which should be the case.
37