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Chapter6
Random Error in
Chemical Analyses
6A THE NATURE OF RANDOM ERRORS
1.Error occur whenever a measurement is made.
2.Random errors are caused by the many
uncontrollable variables that are an inevitable
part of every analysis.
3.If we can identify sources of uncertainty, it is
usually impossible to measure them because
most are so small that they cannot be detected
individually.
4.The accumulated effect of the individual random
uncertain.
5.we cannot positively identify or measure them
Fig 6-1, p.106
6A-1 What Are the Source of
Random Errors?

It is four small random errors combine to give
an overall error. Assume that each error has
an equal probability of occurring and that
each can cause the final result to be high or
low by fixed amount  U

When the same procedure is applied to a very
large number of individual errors, a bellshaped curve like that shown in Figure 6-1c
result. Such a plot is called a Gaussian curve
or a normal error curve.

Figure 6-2(a) Four random
uncertainties ,

Figure 6-2(b) shows the
theoretical distribution for ten
equal-sized uncertain.

Figure 6-1( C) is called a
Gaussian curve or a normal
error curve.

It is a curve that shows the
symmetrical distribution of a
data around the mean of an
infinite set of data.
6A-2 Describing the Distribution Of
Experimental Data

Table 6-2 are typical of those obtained by
an worker weighing to the nearest milligram
(which correspond to 0.001mL) on a top
loading balance and making every effort to
avoid systematic error.
Table 6-2, p.108
Table 6-3, p.108



Number of measurement increases, the
histogram approaches the shape of the
continuous curve shown as plot B in figure
6-2.
Plot A in figure 6-2 is a bar graph.
This curve is a Gaussian curve, or normal
error curve, derived for an infinite set of
data.
6B TREATING RANDOM ERRORS
WITH STATISTICS?
1.Statistical methods to evaluate the random errors
discussed in the preceding section, we base
statistic analyses on assumption that random
errors in analytical results follow a Gaussian,
2.Statistics only reveal information that is already
present in a data set. No new information is
created by statistical treatments. Statistical
analysis can look at our data in different ways and
make objective and intelligent decisions regarding
their quality and use.
6B-1 Sample and Populations
1.We infer information about a population or universe
from observations made on a subset or sample.
2.The population is the collection all measurements of
interest and must be carefully defined by the
experimenter. It is finite and real. （conceptual）
3. Statistical laws have been derived for populations ,
often they must be modified substantially when applied
to a small sample because a few data points may not
represent the entire population.
4.Do not confuse the statistical sample with the analytical
sample. Four analytical samples analyzed in the
laboratory represent a single statistical sample.
6B-2 Properties Gaussian Curves The


Population Mean  and The Sample mean 
Sample mean is the mean of a limited sample
drawn from a population of data.
In the absence of any systematic error, the
population mean is also the true value for the
measured quantity


N↓
i
i
i 1
i 1

x


x

By the time N reaches 20 to 30 ,this difference is negligible. Note that the
sample mean is a statistic that estimates the population parameter.

The Population Standard
Deviation(
)




The Population standard deviation , which is a
measure of the precision of a population of data,
The quantity （Xi － ）is the deviation
Figure 6-4b shows another type of normal error
curve in which the axis is a new variable Z
Z represents the deviation of a result from the
population mean relative to the standard
deviation .

 
 x
i 1
i
 

2

x  


Fig6-4a shows two Gaussian curves
which we plot the relative frequency Y of various deviations from
the mean versus the deviation from the mean. The standard
deviation for the data set yielding the broader but lower curve B
is twice that for the measurements yielding curve A. The
precision of the data set leading to curve A is twice as good as
that of the data set represented by curve B
Normal error curve has several
general properties
1、The mean occurs at the central point of maximum
frequency.
2、There is a symmetrical distribution of positive
and negative about the maximum,
3、There is an exponential decrease in frequency as
the magnitude of the deviations increases.
Thus, small uncertainties are observed much more
often than very large ones.
Gaussian Curves
The equation for the Gaussian error curve is （6-3）
Areas under a Gaussian curve （P113）
± σ＝68.3％； ± 2σ＝95.4％； ±3σ＝99.7％
( x   )
y 
2
2
e

2
2
The area beneath a Gaussian curve for a population
lies within one standard deviation (±σ)
p.114
The area beneath a Gaussian curve for a population
lies within one standard deviation (±2σ)
p.114
The area beneath a Gaussian curve for a
population lies within one standard deviation (±3σ)
p.115
6B-3 Finding the Sample Standard
Deviation
When it is applied to a small sample of data. Equation 6-4
differs from equation 6-1 in two ways.
First, the sample mean appears in the numerator in place of the
population mean
Second , N in 6-1 is replaced by the number of degrees of
freedom（N-1）
 x  x 

s
i 1

2
i
 1

d
i 1
2
i
 1
An Alternative expression for
sample standard Deviation


X



i
N
2
i 1


Xi 



N
i 1




s
N 1
N
2
6-5
Example 6-1 The following results were obtained in the
replicate determination of the lead content of a blood
sample: 0.752,0.756,0.752,0.751,and 0.760 ppm pb.
Calculate the mean and the standard deviation of this set
of data （p116）



Note in Example 6-1 that the difference between
is very small. If we had rounded these numbers
before subtracting them, a serious error would
have appeared in the computed value of S.
To avoid this source of error, never round a
standard deviation calculation until the very end.
進行標準偏差計算時，應該到最後才進行四捨五
入
Standard Error of the Mean Sm

The standard deviation of each mean is known
as the standard error of the mean and is given
the symbol Sm
S
Sm 
N
(6  6)
6B-4 Reliability of S as a Measure of
Precision
1、The probability that these statistical tests provide
correct results increases as the reliability of S becomes
greater.
2、The rapid improvement in the reliability of S with
increases in N makes it feasible to obtain a good
approximation of σ when the method of measurement is
not excessively time consuming and when an adequate
supply of sample is available.
3.The pooled estimate ofσ , which we call Spooled is a
weighted average of the individual estimates. P124
6B-5 variance and other measures of
precision
1.Variance（S2）：is the square of the standard deviation .
Note that the standard deviation has the same units as the
data, while the variance has the units of the data squared.
2、Relative standard Deviation （RSD）
We calculate the relative standard deviation by dividing the
standard deviation by the mean value of the data set.
S
 1000 ppt
Sr＝
X
3、Coefficient of variation （Cv）
S
＝ X ×100％
4、Spread or Range （w）：
It is the difference between the largest value in the set and
the smallest.
Ex 6-3 For the set of data in EX6-1 Calculate (a) the
variance, (b) the relative standard deviation in parts
per thousand ,( C) the coefficient of variation , and
( d ) the spread.
(sample: 0.752,0.756,0.752,0.751,0.760 ppm pb.

Sol:
X  0.754 ppm and
s  0.0038 ppm Pb
(a ) S  (0.0038)  1.4  10
2
2
5
0.0038
(b) RSD 
 1000 ppt  5.0 ppt
0.754
0.0038
(c)CV 
 100%  0.50%
0.754
(d ) w  0.760  0.751  0.009 ppmPb
6C THE STANDARD DEVIATION OF
COMPUTED RESULTS
As the Table 6- 4,the way such estimates are made depends on the
type of arithmetic that involved.
6C-1 The Standard Deviation of
Sums or Differences
y  a  b  c (1)
y  y  (a  a )  (b  b)  (c  c) (2)
(2)  (1)
y  a  b  c
y  (a  b  c)
2
2
y  (a  b  c)
sy  s  s  s
2
a
2
b
2
c
2
6C-2 The Standard Deviation of a
Products and Quotients
ab
y
c
2
(6-9)
2
 sa   sb   sc 
      
y
a b c
sy
2
(6-10)
y  a  b(1)
y  y  ( a  a )(b  b)
y  y  ab  ab  ba  ab ( 2)
( 2)  (1) y  ab  ba  ab (3)
(3)
y
ab ba
ab



(1)
y
ab
ab
ab
y
b a
ab



y
b
a
ab
ab
b a


ab
b
a
y 
y


y
b 2
a 2
(
) (
) y
b
a
b a 2
(

)
b
a
6D REPORTING COMPUTED DATA

The significant figure in a number are all
the certain digits plus the first uncertain
digit.
6D-1 The Significant Figure
Convention


Express data in scientific notation to avoid
confusion in determining whether terminal
zeros are significant.
Rules for determining the number of
significant figures:
1. Disregard all initial zeros.
2. Disregard all final zeros unless they
follow a decimal point.
3. All remaining digits including zeros
between nonzero digits are significant
6D-2 Significant Figures in
Numerical Computations
Sum and Differences
For addition and subtraction, the weak link is the
number of decimal place in the number with the
smallest number of decimal places.
Products and Quotients
When adding and subtracting numbers in
scientific notation, express the numbers to the
same power of ten
Ex:
2.432  106  2.432  106


 1.227  105  0.1227  105
6D-2 Significant Figures in
Numerical Computations

Logarithms and Antilogarithms
The number of significant figures in the mantissa,
or the digits to the right of the decimal point of a
logarithm, is the same as the number of
significant figures in the original number. Thus,
log( 9.57  10 4 )=4.981
6D-3 Rounding Data


Note that 0.635 rounds to 0.64 and 0.625
rounds to 0.62
In rounding a number ending in 5, always
round so that the result ends with an even
number.
6D-4 Rounding the Results from
Chemical Computations


It is especially important to postpone
rounding until the calculation is completed.
At least one extra digit beyond the
significant digits should be carried through
all the computations to avoid a rounding
error.
Example 6-6 A 3.4842g sample of a solid mixture containing
benzoic acid, C6H5COOH(122.123g/mol) , was dissolved and
titrated with base to a phenolphthalein end point .The acid
consumed 41.36mL of 0.2328 M NaOH. Calculate the percent
benzoic acid in the sample.
Sol:
mmolNaOH 1mmolHBz
122.123g
41.36mL  0.2328


mLNaOH
mmolNaOH 1000mmol  100
HBz 
3.4842 gsample
 33.749