Post on 23-Jun-2018
4.1
Anal. Chem. by Prof. Myeong Hee Moon
Ch 4. Statistics
Quantitative analysis requires: sound knowledge of chemistry: possibility of interferences
WHY do we need to use STATISTICS in Anal. Chem. ?
uncertainty exists. will we accept uncertainty always ?if not, from how will we disregard the data ?
by statistical treatment
Random Events follows Gaussian Distribution
4.2
Anal. Chem. by Prof. Myeong Hee Moon
4-1 Gaussian Distribution
test of the life times of 4768 light bulbs
1) mean value & standard deviation
x
n
xx i
i
* mean : : or average
4.3
Anal. Chem. by Prof. Myeong Hee Moon
* standard dev. : s : measures how closely the data are clustered
around the mean
1
)( 2
n
xxs i
n-1 : degrees of freedom
for an infinite set of data:
iance
x
var: or
deviation) standard popular (sigma, s
mean) popular (mu, (mean)
2
4-1 Gaussian Distribution (Cont.)
4.4
Anal. Chem. by Prof. Myeong Hee Moon
4.5
Anal. Chem. by Prof. Myeong Hee Moon
2) std.dev. & probability
)2
)(exp(
2
1 y
2
2
x
tells the broadness of Gaussian curve
in a gaussian curvearea under 1 = 68.3 %
2 = 95.5 %3 = 99.7 %
4-1 Gaussian Distribution (Cont.)
Gaussian curve
4.6
Anal. Chem. by Prof. Myeong Hee Moon
3) std.dev. of meanmore measurements more confident on average
(nearly the true value)uncertainty decreases by : n = number of meas.
standard deviation of mean = : s = std.dev.
n
1
n
s
* relative standard deviation = (RSD)
or into percentage = = C.V.
precision of mean =
average deviation of mean = ( )
x
s
100x
s
n
x
n
dn
xxd
4-1 Gaussian Distribution (Cont.)
4.7
Anal. Chem. by Prof. Myeong Hee Moon
4-2 Confidence Intervals
1) confidence interval : an expression stating that true mean, , is likely to lie within a certain distanceour measurements , s (instead of , )
True mean () is likely to lie within a certain range from x
n
stx
Confidence intervals
4.8
Anal. Chem. by Prof. Myeong Hee Moon
Ex. The content of carbohydrate in a glycoprotein (a protein with sugars attached to it) is determined to be 12.6, 11.9, 13.0, 12.7, and 12.5 g per 100 g of protein in replicated analysis. Find the 50% and 90% confidence intervals for the carbohydrate content.
mean = 12.5, std = 0.4
4.9
Anal. Chem. by Prof. Myeong Hee Moon
: tool for expressing confidence interval for comparing results from other experimental tech.
Normally, 95% confidence level: Two results do not differ from each other IF there is 95% chance that our conclusion is correct.
4-3 Comparison of means with Student's t(from different measurements)
4.10
Anal. Chem. by Prof. Myeong Hee Moon
Case 1. t test : measured result with known value
ex) Ni content; known value : 0.0319% (from std. Material)measured value : 0.0329, 0.0322, 0.0330, 0.0323 %The 95% confidence interval ?
0.0006 0.0326 4
0.00043.182 x
this interval doesn't cover 0.0319, thus, measured value are different from known val.
Not within the random error boundary.(it implies there exists systematic errors)
: when we test a new analytical method,we want to see if it agrees to a known value.
4.11
Anal. Chem. by Prof. Myeong Hee Moon
1. <t-test> You are developing a procedure for determining traces of copper in biological materials using a wet digestion followed by measurements by atomic absorption spectrophotometry. In order to test the validity of the method, you obtain a NIST orchard leaves standard reference material and analyze this material. Five replicas are sampled and analyzed, and the mean of the results is found to be 10.08 ppm with a standard deviation of 0.7ppm. The listed value is 11.7ppm. Does your method gives a statistically correct value at the 95% confidence level ?
4.12
Anal. Chem. by Prof. Myeong Hee Moon
Case 2. t test: comparing replicate measurements
(test of two sets of measurements) : test the two techniques are statistically the SAME or NOT
for two sets of data, n1, n2 measurements
21
2121x
nn
nn
S
xt
pooled
2
)1()1(
2
)()(
21
2221
21
21
22
21
nn
nsns
nn
xxxxS ji
pooled
If tcal > ttable (within 95%)
this difference is significant(out of random error range)
there exists systematic error
4.13
Anal. Chem. by Prof. Myeong Hee Moon
Ex) The average mass of nitrogen from air in Table 4-3 is =2.31011 g, with a standard deviation of s1=0.00014, (for n1=7 measurements). The average mass from chemical sources is =2.29947 g, with a standard deviation of s2=0.00138 (for n2=8 measurements)
4.14
Anal. Chem. by Prof. Myeong Hee Moon
2. <t-test> A new gravimetric method is developed for iron (II) in which the iron is precipitated in crystalline form with an organocarbon "cage" compound. The accuracy of the method is checked by analyzing the iron in an ore sample and comparing with the results using the standard precipitation with ammonia and weighing of Fe2O3. The results, reported as % Fe for each analysis, were as follows.
Test method Reference Method20.10% 18.89%20.50 19.2018.65 19.0019.25 19.7019.40 19.4019.99 19.40
=19.65% =19.24%Is there a difference between the two methods ?
4.15
Anal. Chem. by Prof. Myeong Hee Moon
Case 3; Comparing individual differences
Two different methods on several different samples (no duplication)Cholesterol content (g/L)
d
Plasma sample
Method A Method B Difference (di)
123456
1.462.222.841.971.132.35
1.422.382.671.801.092.25
0.04-0.160.170.170.040.10
=+0.06
nS
dt
d
cal 1
)( 2
n
dds i
d
4.16
Anal. Chem. by Prof. Myeong Hee Moon
Is my red blood cell count high today ?
Red cell counts on five “normal” days: 5.1, 5.3, 4.8, 5.4, and 5.2x106 cells/L =5.16 s=0.23
Today’s value = 5.6x106 cells/L
2845230
65165.
.
..'
n
S
xcountstodayt
d
cal
x
What is the probability of finding t=4.28 for 4 degrees of freedom ?
See table 4.2: at 4 degrees of freedom, 4.28 lies between 98 & 99% There is less than a 2% probability of observing a count of 5.6x106 cells/L on normal days.
reasonable to conclude that today’s count is elevated.
4.17
Anal. Chem. by Prof. Myeong Hee Moon
4-4 Comparison of st.dev. with the F test
F test ---- check two std.devs are significantly different each other.
22
21
S
SFcalc If Fcalc > Ftable then significant
4.18
Anal. Chem. by Prof. Myeong Hee Moon
4-6. Grubbs test for an outlier
during measurements of mass lost of zinc, we need to discard some questionable data10.2, 10.8, 11.6, 9.9, 9.4, 7.8, 10.0, 9.2, 11.3, 9.5, 10.6, 11.6
If Gcalc > Gtab, then rejected.
4.19
Anal. Chem. by Prof. Myeong Hee Moon
1. Finding the BEST STRAIGHT LINE
; correlation between data points
1) Method of Least Squares
y = m x + bm: slope, b: y-intercept
each data --- ( xi, yi )vertical deviation = di = yi - y
= yi - (mxi + b)
4-7. Method of Least Squares
4.20
Anal. Chem. by Prof. Myeong Hee Moon
we want to MINIMIZE di (whether positive or neg.)
method of maximum likelihood: Assume a gaussian distribution with std.dev. i. for the observations about the actual value y(xi) at x=xi
2
21
exp2
1
i
i
ii
yyPthe probability Pi
maximize the probability ? minimize the sum in the exponential…
-- direct summation of each di ? no good
4-7. Method of Least Squares
4.21
Anal. Chem. by Prof. Myeong Hee Moon
2
2
i
iddi
2 = (yi - y)2 = (yi - mxi -b)2
minimizing (assume )2
2m
2
b
METHOD OF LEAST SQUARES
22
2
22
)x()x(n
x)yx(y)x(b
)x()x(n
yx)yx(nm
ii
iiiii
ii
iiii
4-7. Method of Least Squares
4.22
Anal. Chem. by Prof. Myeong Hee Moon
2) How reliable are least-squares parameters ?
estimate UNCERTAINTY in slope & intercept
std. dev. of y
2)-n( freedom of reesdeg
)d(s i
yy 2
22
222
22
22
)x()x(n
)x(
)x()x(n
n
ii
iyb
ii
ym
4-7. Method of Least Squares
4.23
Anal. Chem. by Prof. Myeong Hee Moon
4-8. Calibration Curves
Std. Solution : solutions with known concentrationsHow to build calibration ?
1. prepare a series of std. Solutions (varying conc.)measure absorbance.
2. subtract the absorbance of blank solution
4.24
Anal. Chem. by Prof. Myeong Hee Moon
3. Plot the absorbances vs. Concentration then do least squares.
4-8. Calibration Curves
4.25
Anal. Chem. by Prof. Myeong Hee Moon
Uncertainty Propagation in Calibration curve
m : slope
Depends on # of calibration points.Lowest error data from the center of calibration
4-8. Calibration Curves
4.26
Anal. Chem. by Prof. Myeong Hee Moon
Homework
4-F, 13, 14, 16, 20, 33, Additional Problems Set
4.27
Anal. Chem. by Prof. Myeong Hee Moon
Additional Problems Set
1. The following replicate calcium determinations on a blood sample using AAS and a new colorimetric method were reported. Is there a significant difference in the precision of the two methods ?AAS (mg/dL) 10.9, 10.1, 10.6, 11.2, 9.7, 10.0Colorimetric (mg/dL) 9.2, 10.5,9.7, 11.5,11.6, 9.3, 10.1, 11.2
4.28
Anal. Chem. by Prof. Myeong Hee Moon
2. Students measured the concentration of HCl in a solution by titrations using different indicators to find the end point. Is the difference between indicators 1 and 2 significant at the 95% confidence level ? Answer the same question for indicator 2 and 3.
Indicator Mean HCl concentration (M) (+std.dev.)
Number of Measurements
1. Bromothymol blue2. Methyl red
3. Bromocresol green
0.09565 + 0.002250.08686 + 0.000980.08641 + 0.00113
281829
4.29
Anal. Chem. by Prof. Myeong Hee Moon
3. A Standard Reference Material is certified to contain 94.6 ppm of an organic contaminant in soil. Your analysis gives values of 98.6, 98.4, 97.2, 94.6, and 96.2 ppm. Do your results differ from the expected results at the 95% confidence level ? If you made one more measurement and found 94.5, would your conclusion change ?