1. ANALYTICAL CHEMISTRY ERT 207
Lecture 5 Coefficient correlation Coefficient determination Calibration curve (slope & intercept)

2. Mid-term preparation Wednesday 6 November, 9am BASIC STATISTICS
Define accuracy and precision, remember ways of describing accuracy and precision, types of errors, understand the concept of significant figures, standard deviation.
UTILIZATION OF STATISTICS IN DATA ANALYSIS
Identify the significant testing. Calculate the T test and Q test.

3. Scatter Plots and Correlation
A scatter plot (or scatter diagram) is used to show the relationship between two variables
Correlation analysis is used to measure strength of the association (linear relationship) between two variables
Only concerned with strength of the relationship
No causal effect is implied

4. Scatter Plot Examples Linear relationships Curvilinear relationships y

5. Scatter Plot Examples Strong relationships Weak relationships y y x x

6. Scatter Plot Examples
No relationship y x y x

7. Correlation Coefficient
The population correlation coefficient ρ (rho) measures the strength of the association between the variables
The sample correlation coefficient r is an estimate of ρ and is used to measure the strength of the linear relationship in the sample observations

8. Features of ρ and r Unit free Range between -1 and 1
The closer to -1, the stronger the negative linear relationship
The closer to 1, the stronger the positive linear relationship
The closer to 0, the weaker the linear relationship

9. Examples of Approximate r Valuesy y y x x x
r = -1
r = -.6
r = 0 y y x x
r = +.3
r = +1

10. Calculating the Correlation Coefficient
Sample correlation coefficient:
or the algebraic equivalent:
where:
r = Sample correlation coefficient
n = Sample size
x = Value of the independent variable
y = Value of the dependent variable

11. Example:
You are developing a new analytical method for the determination of blood urea nitrogen (BUN). You want to determine whether your method differs significantly from a standard one for analyzing a range sample concentrations expected to be found in the routine laboratory. It has been ascertained that the two methods have comparable precisions. Following are two sets of the results for a number of individual samples:

12. Sample
Your Method (mg/dL) ,x
Standard Method (mg/dL) ,y A
10.2
10.5 B
12.7
11.9 C
8.6
8.7 D
7.5
16.9 E
11.2
10.9 F
11.5
11.1

14. Coefficient of Determination, R2
The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable
The coefficient of determination is also called R-squared and is denoted as R2
where

15. Coefficient of Determination, R2
Note: In the single independent variable case, the coefficient of determination is
where:
R2 = Coefficient of determination
r = Simple correlation coefficient

16. Total variation is made up of two parts:
Total sum of Squares
Sum of Squares Error
Sum of Squares Regression
where:
= Average value of the dependent variable
y = Observed values of the dependent variable
= Estimated value of y for the given x value

17. SST = total sum of squares
Measures the variation of the yi values around their mean y
SSE = error sum of squares
Variation attributable to factors other than the relationship between x and y
SSR = regression sum of squares
Explained variation attributable to the relationship between x and y

18. Explained and Unexplained Variationy yi   y
SSE = (yi - yi )2 _
SST = (yi - y)2  _ y 
SSR = (yi - y)2 _ _ y y x Xi

19. Examples of Approximate R2 Valuesy
R2 = 1
Perfect linear relationship between x and y:
100% of the variation in y is explained by variation in x x
R2 = 1 y x
R2 = +1

20. Examples of Approximate R2 Valuesy
0 < R2 < 1
Weaker linear relationship between x and y:
Some but not all of the variation in y is explained by variation in x x y x

21. Examples of Approximate R2 Valuesy
No linear relationship between x and y:
The value of Y does not depend on x. (None of the variation in y is explained by variation in x) x
R2 = 0

22. Introduction to Regression Analysis
Regression analysis is used to:
Predict the value of a dependent variable based on the value of at least one independent variable
Explain the impact of changes in an independent variable on the dependent variable
Dependent variable: the variable we wish to explain
Independent variable: the variable used to explain the dependent variable

23. Simple Linear Regression Model
Only one independent variable, x
Relationship between x and y is described by a linear function
Changes in y are assumed to be caused by changes in x

24. Types of Regression Models
Positive Linear Relationship
Relationship NOT Linear
Negative Linear Relationship
No Relationship

25. Method of Least Squares
Find “best” line by minimizing vertical deviation between the points and the line.
Chemistry 215 Copyright D Sharma

26. Calculating the Residual

27. Linear Regression Fitting a straight line to observations
Small residual errors Large residual error
Error = (Actual value) – (Predicted value)

28. Least Squares Parameters
SLOPE
INTERCEPT

29. Calibration Curves
A calibration curve shows the response of an analytical method to known quantities of analyte.
For example, a spectroscopic analysis of a protein sample…
Necessary solutions:
Standard solutions
Blank solution
Sample solution(s)
Protein from the cancer-causing oncogene called ras (Credit: Sung-Hou Kim/UC Berkeley)

30. Constructing a Calibration Curve
Spectroscopic analysis of a protein sample…

31. Constructing a Calibration Curve
Spectroscopic analysis of a protein sample…cont.
Determination of an unknown value (x) based on its response (y)
Equation of linear response
y = m  (x) + b
Abs = m  (µg protein) + b
y = (x)
…where y is the corrected abs.
Determine the unknown concentration based on its absorbance

32. Tips for Calibrating Instruments
Know the limitations of your instrument
Limits of detection (or LOD)
Range of linearity
Watch-out for interferences
Overlapping spectral responses (e.g., from impurities)
Unwanted sample precipitation
Use serial dilutions where possible
Less error than preparing individual samples

33. Serial Dilution (A Review)

35. Using spreadsheet for plotting calibration curves
Some useful statistical syntaxes:
AVERAGE = mean of series of number
MEDIAN = median of series of number
STDEV = standard deviation
VAR = variance
RSQ = R- squared

36. Fluoresence intensity
Riboflavin (ppm)
Fluoresence intensity
0.000
0.00
0.100
5.80
0.200
12.20
0.400
22.30
0.800
43.30

38. Slope, intercept and coefficient determination
We can use the Excel statistical functions to calculate the slope and intercept for a series of data , and the R2 value without a plot
Open a new spreadsheet and enter the calibration data from the previous example.
In cell A9 type INTERCEPT, cell A10, SLOPE AND cell A11, R-Squared.
Highlight cell B9
Click on fx: Statistical
And scroll down to INTERCEPT and click OK

39. For known_x’s, enter the array A3:A7 and for known_y’s, enter B3:B7, then click OK.
The INTERCEPT is displayed in cell B9.
Now repeat , highlighting cell B10, scrolling to SLOPE , and entering the same arrays. The Slope appears in cell B10.
Followed the same way for R-squared.

40. Exercise
The following data were obtained to get a calibration curve for the determination of Zn in the wastewater by using atomic absorption spectrometry (AAS). Using calculator or computer, plot the data and find the best straight line equation and correlation determination.

41. Zn concentration (ppm)
Absorbance 2
0.095 4
0.194 6
0.290 8
0.390 10
0.466

42. Solution:
The equation of the straight line is:
Y =0.047X
Correlation determination (R2) = 0.998