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STATISTICS IN DATA EVALUATION Defining confidence limits
Estimating the different of two means
(t test)
Estimating the precision of data from twoexperiments (F test)
Deciding to accept or reject outliers (Q test)
Calibration graphs
Methods of validation
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CONFIDENCE LIMITS ANDCONFIDENCE INTERVAL
Confidence- assert a certain probability thatthe confidence interval does include the true
value. The greater the certainty, the greater the
interval required.
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CONFIDENCE LIMITS (CL) OF MEAN Since the exact value of population mean,
cannot be determined, one must usestatistical theory to set limits around themeasured mean, , that probably contain .
CL only have meaning with the measuredstandard deviation, s, is a good
approximation of the population standarddeviation, , and there is no bias in themeasurement.
x
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0 2 4-2-4 -3 -1 1 3
dN/N
80%
+1.29-1.29
CONFIDENCE LIMITS (CL)
In the absence of any systematic errors, the limits withinwhich the population mean () is expected to lie with a
given degree of probability.
0 2 4-2-4 -3 -1 1 3
dN/N
50%
+0.67-0.67
0 2 4-2-4 -3 -1 1 3
dN/N
95%
-1.96 +1.96
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CONFIDENCE INTERVAL (CI)
CI when is known (Population)
N = Number of measurements
N
zx =
forCI
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VALUES FOR z AT VARIOUSCONFIDENCE LEVELS
Confidence Level, % z
50 0.67
68 1.080 1.2990 1.6495 1.9696 2.00
99 2.5899.7 3.0099.9 3.29
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CI For Small Data Set (N < 20)Not Known
Values of t depend on degree of freedom,
(N - 1) and confidence level (from Table t).
t also known as students t and will be used in
hypothesis test.
Example 2
N
tsx =forCI
http://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/Example2.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/Example2.pdf8/13/2019 Ssck 1203 Data Analysis 090214 Students 02
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VALUES OF t AT VARIOUSCONFIDENCE LEVEL
Degrees of Freedom 80% 90% 95% 99%(N-1)
1 3.08 6.31 12.7 63.7
2 1.89 2.92 4.30 9.923 1.64 2.35 3.18 5.844 1.53 2.13 2.78 4.605 1.48 2.02 2.57 4.036 1.44 1.94 2.45 3.717 1.42 1.90 2.36 3.508 1.40 1.86 2.31 3.369 1.38 1.83 2.26 3.25
19 1.33 1.73 2.10 2.8859 1.30 1.67 2.00 2.66 1.29 1.64 1.96 2.58
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OTHER USAGE OF CONFIDENCEINTERVAL
To determine number of replicates neededfor the mean to be within the confidenceinterval.
To determine systematic error.
http://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/replicate.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/systematic_error.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/systematic_error.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/replicate.pdf8/13/2019 Ssck 1203 Data Analysis 090214 Students 02
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SIGNIFICANT TESTS
Approach tests whether the differencebetween the two resultsis significant (due to
systematic error) or notsignificant(merely
due to random error).
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NULL HYPOTHESIS, Ho The values of two measured quantities do not differ
(significantly)UNLESS we can prove it that the two
values are significantly different.
Innocent until proven guilty
The calculated valueof a parameter from theequation is compared to the parameter value from
the table.
If the calculated value is smallerthan the table value,the hypothesis is acceptedand vice-versa.
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NULL HYPOTHESIS, Ho
Can be used to compare:
and and s and s1and s2
2x
x
1x
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APPLICATION OF t-TEST
A t-test is used to compareone set of
measurement with another to decide
whether or not they are significantlydifferent.
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t TEST
1. Comparison between experimental mean
and true mean (and )
To check the presence of systematic error.
Stepsfor t test.
Example 4.
x
http://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/steps_t.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/Example4.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/Example4.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/steps_t.pdf8/13/2019 Ssck 1203 Data Analysis 090214 Students 02
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t TEST
2. Compare and from two sets of data
Normally used to determine whether the twosamples are identical or not.
The difference in the mean of two sets of thesame analysis will provide information onthe similarity of the sample or the existence
of random error.
Steps Example 5
1x 2x
http://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/steps_t2.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/Example5.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/Example5.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/steps_t2.pdf8/13/2019 Ssck 1203 Data Analysis 090214 Students 02
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Comparing the precisionof twomeasurements
Is Method A more precise than Method B?
Is there any significant difference betweenboth methods?
F TEST
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DIXONS TEST OR Q TEST
A way of detectingoutlier, a data which isstatistically does not belong to the set.
Data:10.05, 10.10, 10.15, 10.05, 10.45, 10.10
By inspection, 10.45 seems to be out of thedata normal range.
Should this data be eliminated? Example 7 Table Q
http://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/Example7.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/TableQ.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/TableQ.pdfhttp://sscc%202243%20space/SSC%202243%20201112%20I/CHAPTER_2/Example7.pdf8/13/2019 Ssck 1203 Data Analysis 090214 Students 02
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CALIBRATION GRAPHS Commonly used in analytical chemistry to find the
quantitative relationbetween two variables (e.g.
response and concentration).
The calibration curves are normally linear, howevernot all the points are located on the drawn straightline (random error).
Regression analysiscan be done on the data to see
how good the linearityof the data is.(Method of least squares)
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METHOD OF LEAST SQUARES
Linear relationship between analytical signal (y)andconcentration (x).
Calculate best straight line through data points, eachof which is subject to experimental errors.
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CALIBRATION CURVES
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Concentration (X)
Respons
e(Y) y = mx c
m = slope
c = intercept
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CALIBRATION METHODS Standard Calibration Method Standard Addition Method
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STANDARD CALIBRATION METHOD
1 ppm 2 ppm 3 ppm 4 ppm 5 ppm
SampleBlank
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Calibration Plot for Absorbance versus Concentration
0.000.05
0.10
0.15
0.20
0.25
0.30
0.35
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Concentration
Absorbance
STANDARD CALIBRATION METHOD
y = 0.06x 0.0067
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STANDARD ADDITION METHOD
(x + 0) ppm (x + 10) ppm (x + 20) ppm ( x + 50) ppm
(x + 100) ppm Blank
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STANDARD ADDITION METHOD
Concentration (ppm) Signal
(x + 0.00) 5.0
(x + 10.00) 11.0
(x + 20.00) 17.0
(x + 50.00) 28.0
(x + 100.00) 55.0
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STANDARD ADDITION METHOD
0
10
20
30
40
50
60
-20 -10 0 10 20 30 40 50 60 70 80 90 100 110 120
Concentration
Abs
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The closer the R value to 1 (or 1), the betterthe correlation between y and x.
R = +1: perfect positive correlation with all
points lying on a straight line withpositive slope.
R = 1: perfect negative correlation.
Correlation coefficient, R2of > 0.999:evidence of acceptable fitof the data to theregression line.
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METHOD VALIDATIONDEFINITION
Method validation is the process to confirmthat the analytical procedureemployed for aspecific testis suitable for its intended use.
The process of verifyingthat a procedure or
methodyields acceptable results.
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PURPOSE OF VALIDATION
To defend validity of the resultand
demonstrate method is fit for the intended
purpose.
Responsibility of the laboratories.
Based on evaluation of the method
performanceand the estimated uncertainty
on the result.
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VALIDATION OF ANALYTICAL METHOD
(METHOD VALIDATION) Analysis of Standard Samples (SRM) Analysis by Other Methods
Standard Addition to the Sample
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