Download Precision - Chemistry

9/23/2009
Precision: The Population
Standard Deviation
N
2
xi
i 1
N
Precision: The Sample
Standard Deviation
N
xi
x
2
i 1
s
N 1
Side by side
N
xi
i 1
N
N
2
xi
s
x
2
i 1
N 1
1
9/23/2009
N -1 = Degrees of Freedom
By calculating the mean, we use up a
degree of freedom.
Thus, only N -1 remain.
As N
,N-1
N
The z - statistic
z
xi
or z
xi x
s
The deviation from the mean given
as multiples of the standard deviation.
2
9/23/2009
Watch those calculators!
xn-1
xn
sample
population
Watch Excel!
(sample)
=STDEVP(cells) (population)
=STDEV(cells)
Let’s play some more.
Example 6-1, page 126
 Do it with Excel.

3
9/23/2009
The Standard Error, or The
Standard Deviation of the Mean
m
N
Find the mean many, many times using
N data each time, and then find the
standard deviation of the mean. This
equation gives the same value if is
the standard deviation of any subset.
Pooling Data
If we have a number of different people or
a number of different labs perform an
analysis, we can “pool” the data to get a
better estimate of the standard deviation.
Each set of data has its own sample mean
and its own sample standard deviation, so
we lose some degrees of freedom in the
process.
Calculating spooled
N1
spooled
i
Fx x I2
G
Hi 1J
K
1
N2
Nn
j
m 1
Fx x I2 L
Gj 2 J
K
1H
Fxm x nI2
H
K
N1 N 2 L N n n
Where
n = the number of data sets
Nn = the number of data in each data set
4
9/23/2009
Hg in Chesapeake Bay Fish
The mercury in sample of seven fish taken
from the Chesapeake Bay was determined
by a method based on the absorption of
light by mercury atoms. We wish to
estimate the standard deviation of the
method by pooling the data from the seven
fish. Why? Let’s look at the data using
Excel.
The Standard Deviation s and
the Variance s 2
N
i
s
F
IJ2
x
x
G
Hi K
1
N 1
N
s2
i
F
IJ2
x
x
G
Hi K
1
N 1
variance
The Relative Standard
Deviation
RSD
s
x
RSD,%
RSD,ppt
s 100 %
x
s 1000 ppt
x
5
9/23/2009
One of the Data Sets
2.06
1.93
2.12
2.16
1.89
1.95
Mean = 2.018333
= 0.110529
RSD
RSD 0.110529 0.054763
2.018333
=0.055
RSD,% 0.110529 100 %
2.018333
5.476253%
=5.5%
RSD,ppt 0.110529 1000 ppt = 54.762526 ppt
2.018333
=55 ppt
Standard Deviation of
Computed Results: Addition
and Subtraction
+0.50 ( 0.02)
+4.10 ( 0.03)
-1.97 ( 0.05)
2.63 ( ?????)
What is the uncertainty in the answer?
6
9/23/2009
Standard Deviation of
Computed Results: Addition
and Subtraction
Variances are additive.
y a( sa ) b( sb) c( sc)
s2y sa2 sb2 sc2
sy
sa2 sb2 sc2
Standard Deviation of
Computed Results: Addition
and Subtraction
sy
0.02
2
0.0004
0.03
2
2
0.05
0.0009
2
2
0.0025
2
0.0038
0.0616441
0.06
Standard Deviation of
Computed Results: Addition
and Subtraction
+0.50
+4.10
-1.97
2.63
( 0.02)
( 0.03)
( 0.05)
( 0.06)
What is the uncertainty in the answer?
7
9/23/2009
Standard Deviation of Computed
Results: Multiplication and Division
671
. ( 003
. ) 00071
.
( 00004
.
) 00087737
.
( ??)
543
. ( 006
. )
What is the uncertainty in the answer?
Standard Deviation of Computed
Results: Multiplication and Division
y
sy
a b
c
2
2
sa
a
y
sy
sa
a
y
2
sb
b
2
sb
b
2
sc
c
2
sc
c
2
Standard Deviation of Computed
Results: Multiplication and Division
y
sy
2
y
sy
y
6.71 0.0071
0.0087737
5.43
0.03
6.71
2
0.0004
0.0071
0.004471
2
2
0.06
5.43
0.056338
2
2
0.011050
2
0.00001999 0.00317397 0.00012210
0.0031606
0.05758523
8
9/23/2009
Standard Deviation of Computed
Results: Multiplication and Division
sy
y 0.05758523
y 00087737
.
sy 0.05758523y
0.05758523 00087737
.
0.000505
y 00088
.
00005
.
9