Transcript analytical chem lecture 4

ERT 207
ANALYTICAL CHEMISTRY
SIGNIFICANT FIGURES AND
STANDARD DEVIATION
LECTURE 4
13 JAN 2011
MISS NOORULNAJWA DIYANA YAACOB
1
2
Significant Figures
Significant figures – all known digits, plus the first uncertain
digit.
In significant figure convention:
We first determine the number of significant figures in
our data (measurements).
We use that knowledge to report an appropriate number of
digits in our answer.
Significant figure convention is not a scientific law!
Significant figure convention is a set of guidelines to ensure
that we don’t over- or underreport the precision of results –
at least not too badly…
Rules for Counting Significant Figures Overview
1. Nonzero integers
2. Zeros
 leading zeros
 captive zeros
 trailing zeros
3. Exact numbers
3
Rules for Counting Significant
Figures - Details
Nonzero integers always count as
significant figures.
3456 has
4 sig figs.
4
Rules for Counting Significant
Figures - Details
Zeros

Leading zeros do not count as
significant figures.
0.0486 has
3 sig figs.
5
Rules for Counting Significant
Figures - Details
Zeros

Captive zeros always count as
significant figures.
16.07 has
4 sig figs.
6
Rules for Counting Significant
Figures - Details
Zeros

Trailing zeros are significant only
if the number contains a decimal
point.
9.300 has
4 sig figs.
7
Rules for Counting Significant
Figures - Details
Exact numbers have an infinite number of significant
figures.
Independent of measuring device:
1 apple, 10 students, 5 cars….
2πr The 2 is exact and 4/3 π r2 the 4 and 3 are exact
From Definition: 1 inch = 2.54 cm exactly
The 1 and 2.54 do not limit the significant figures
8
100. has 3 sig. fig. = 1.00 x 102
100 has 1 sig. fig. = 1 x 102
9
Rules For Rounding
1. In a series of calculations, carry the extra digits
through to the final result, then round.
2. If the digit to be removed:
A. Is less than 5, then no change e.g. 1.33 rounded to 2
sig. fig = 1.3
B. Is equal or greater than 5, the preceding digit increase
by 1 e.g. 1.36 rounded to 2 sig. fig = 1.4
10
3. If the last digit is 5 and the second last digit is an
even number, thus the second last digit does not
change.
Example, 73.285
73.28
4. If the last digit is 5 and the second last digit is an
odd number, thus add one to the last digit.
Example, 63.275
63.28
11
Rules for Significant Figures in
Mathematical Operations
Multiplication and Division: # sig figs in
the result equals the number in the least
precise measurement used in the calculation.
6.38  2.0 =
12.76  13 (2 sig figs)
12
Example : Give the correct answer for the following
operation to the maximum number of significant
figures.
1.0923 x 2.07
13
Solution:
1.0923 x 2.07 = 2.261061
2.26
The correct answer is therefore 2.26 based on the
key number (2.07).
14
Rules for Significant Figures in
Mathematical Operations
Addition and Subtraction: # decimal places
in the result equals the number of decimal
places in the least precise measurement.
6.8 + 11.934 =
18.734  18.7 (3 sig figs)
15
Example : Give the answer for the following
operation to the maximum number of significant
figures: 43.7+ 4.941 + 13.13
16
43.7
4.941
+ 13.13
61.771
61.8
answer is therefore 61.8 based on the key number
(43.7).
Rules Exponential
The exponential can be written as follows. Example,
0.000250
2.50 x 10-4
Rules for logarithms and
antilogarithms
Log (3.1201)
mantissa
characteristics
The number of significant figures on the right of the
decimal point of the log result is the sum of the
significant figures in mantissa and characteristic
19
Rules for Counting Significant Figures.
20
Example 1
List the proper number of significant figures in the
following numbers and indicate which zeros are
significant
0.216 ; 90.7 ; 800.0; 0.0670
Solution:
0.216
3 sig fig
90.7
3 sig fig; zero is significant
800.0 4 sig fig; all zeroes are significant
0.0670 3 sig fig; 0nly the last zero is significant
21
Standard Deviation
The standard deviation is calculated to indicate the
level of precision within a set of data.
Abbreviations include sdev, stdev, s and s.
s is called the population standard deviation – used
for “large” data sets.
s is the sample standard deviation – used for “small”
data sets.
22
STANDARD DEVIATION
The most important statictics.
Recall back: Sample standard deviation.
For N (number of measurement) < 30
For N > 30,
The Mean Value
The “average” ( x)
Generally the most
appropriate value to
report for replicate
measurements when
the errors are random
and small.
x1  x2  ...  xn
x
n
25
The standard deviation of the mean:
26
The standard deviation of the mean is sometimes
referred to as the standard error
The standard deviation is sometimes expressed as the
relative standard deviation (rsd) which is just
the standard deviation expressed as a fraction of
the mean; usually it is given as the percentage of
the mean (% rsd), which is often called the
coefficient of variation
27
Relative Standard Deviation (rsd)
Dimensionless, but
expressed in the same
units as the value
x
rsd 
x
or
s
rsd 
x
Coefficient of variation
(or variance), CV, is the
percent rsd:
s
CV   100%
x
28
Example 1
Calculate the mean and the standard deviation of the
following set of the analytical results: 15.67g,
15.69g, and 16.03g
Answer: 0.20g
29
30
Example 2
The following replicate weighing were obtained:
29.8, 30.2, 28.6, and 29.7mg. Calculate the
standard deviation of the individual values and the
standard deviation of the mean. Express these as
absolute (units of the measurement) and relative
(% of the measurement) values.
31
32
Median and Range
Median is the middle value of a
data set
If there is an even number
of data, then it is the
average of the two
central values
Useful if there is a large
scatter in the data set –
reduces the effect of
outliers
Use if one or more points
differ greatly from the
central value
Range is the difference between
the highest and lowest point
in the data set.
Mean and Median
Range
Median
Mean
Range
33
Median and Range
For example:
6.021, 5.969, 6.062, 6.034, 6.028, 5.992
Rearrange: 5.969, 5.992, 6.021, 6.028, 6.034, 6.062
Median = (6.021+6.028)/2 = 6.0245 = 6.024
Range = 6.062-5.969 = 0.093
34