Ch 4. Statistics - Yonsei Universitychem.yonsei.ac.kr/~mhmoon/pdf/AnalChem/Ch4.pdf · Ch 4....

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


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


* 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


4.5


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 %


Gaussian curve

4.6


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


4.7


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


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


: 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


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


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


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


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


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


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


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


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


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


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


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


4.21


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


4.22


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


4.23


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


3. Plot the absorbances vs. Concentration then do least squares.


4.25


Uncertainty Propagation in Calibration curve

m : slope

Depends on # of calibration points.Lowest error data from the center of calibration


4.26


Homework

4-F, 13, 14, 16, 20, 33, Additional Problems Set

4.27


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


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


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 ?

Ch 4. Statistics - Yonsei Universitychem.yonsei.ac.kr/~mhmoon/pdf/AnalChem/Ch4.pdf · Ch 4....

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