Transcript standard deviation

ERT 207-ANALYTICAL CHEMISTRY
SIGNIFICANT FIGURES AND STANDARD
DEVIATION
DR. SALEHA SHAMSUDIN
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What is a "significant figure"?
The number of significant figures in a result is simply the
number of figures that are known with some degree of
reliability. The number 13.2 is said to have 3 significant
figures. The number 13.20 is said to have 4 significant
figures.
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Rules for Counting Significant Figures Overview
1. Nonzero integers
2. Zeros
 leading zeros
 captive zeros
 trailing zeros
3. Exact numbers
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Rules for Counting Significant
Figures - Details
Nonzero integers always count as
significant figures.
3456 has
4 sig figs.
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Rules for Counting Significant
Figures - Details
Zeros

Leading zeros do not count as
significant figures.
0.0486 has
3 sig figs.
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Rules for Counting Significant
Figures - Details
Zeros
Captive zeros always count as
significant figures.
16.07 has
4 sig figs.
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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.
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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
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100. has 3 sig. fig. = 1.00 x 102
100 has 1 sig. fig. = 1 x 102
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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
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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
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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)
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Example : Give the correct answer for the
following operation to the maximum number
of significant figures.
1.0923 x 2.07
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Solution:
1.0923 x 2.07 = 2.261061
2.26
The correct answer is therefore 2.26 based on
the key number (2.07).
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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)
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Example : Give the answer for the following
operation to the maximum number of
significant figures: 43.7+ 4.941 + 13.13
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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
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Rules for Counting Significant Figures.
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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
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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.
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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
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The standard deviation of the mean:
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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
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Relative Standard Deviation (rsd)
Dimensionless, but
expressed in the
same units as the
value
x
rsd 
x
or
s
rsd 
x
Coefficient of
s
CV   100%
variation (or
x
variance), CV, is the
percent rsd:
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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
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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.
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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
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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
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