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Correlation Coefficient
-used as a measure of correlation between 2 variables-the closer observed values are to the most probable values, the more definite the relationship between x and y
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Pearson Correlation Coefficient
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- Maximum value of r is 1- If value r is 1, there is exact correlation between the 2 variables- Variables completely independence if r is 0- Minimum value of r is -1- Negative r value indicates that the assumed dependence opposite to what exist- And therefore a positive coefficient for the reverse relation - A correlation coefficient near 1 means there is a direct relationship between
two variables- Eg. Absorbance & concentration.
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Data for Example 3.19
Sample Your method(mg/dL) Standard method(mg/dL)
A 10.2 10.5
B 12.7 11.9 C 8.6 8.7 D 17.5 16.9 E 11.2 10.9 F 11.5 11.1
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- a correlation coefficient can be calculated for a calibration curve to ascertain the degree of correlation between the measured instrumental variable and the sample concentration- As a general rule : # 0.90 < r < 0.95, indicates fair curve # 0.95 < r < 0.99, indicates good curve # r > 0.99, indicates excellent linearity- correlation coefficient is usually not true in scientific measurements- the square of the correlation coefficient, r², is a more conservative measure of closeness of fit
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- r value of 0.90 corresponds to an r² value of only 0.81 - r value 0f 0.95 equivalent to an r² value of 0.90 - goodness of fit is judged by the number of 9’s, so three 9’s (0.999) better represents an excellent fit - this is called the coefficient of determination
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Using Spreadsheets for Plotting Calibration Curves
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The availability of spreadsheets makes it unnecessary to plot data on graph paper and do hand calculations for the
least-squares regression analysis statistics.
Fluorescence
Riboflavin,Intensity arbitary
µg/mL (xi) Units (yi) Xi2 xiyi
0.000 0.0 0.0000 0.00
0.100 5.8 0.0100 0.58
0.200 12.2 0.0400 2.44
0.400 22.3 0.1600 8.92
0.800 43.3 0.6400 34.64
∑xi = 1.500 ∑yi = 83.6 ∑xi2 = 0.8500 ∑xiyi = 46.58
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• Open a new spreadsheet and enter• Cell A1: Riboflavin, ppm (adjust the column width to
incorporate the text)• Cell B1: Fluorescence intensity • Cell A3: 0.000• Cell A4: 0.100• Cell A5: 0.200• Cell A6: 0.400• Cell A7: 0.800• Cell B3: 0.0• Cell B4: 5.8• Cell B5: 12.2• Cell B6: 22.3• Cell B7: 43.3
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• Format the cell numbers to have three decimal places for column A and one for column B
• Click on the chart wizard icon on the toolbar (the one with the vertical bars).
• Step 1- Chart Type- of the Wizard will appear
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Follow the following sequence
• Select XY (scatter), and Scatter (no line) for Chart subtype• Next• Data Range: enter A3:B7 (click on series, and note the X values and
Y values addresses• Check columns (after going back to data range)• Next• Chart title: enter Calibration Curve• Value (X) axis: enter Riboflavin• Value (Y) axis: enter Fluorescence intensity• Gridlines: uncheck Major gridlines• Legend: Delete Show legend• Data labels: None (Try Show Value, and note the data entered on
each point on the line)• Next• Click on As New Sheet: chart 1• Finish
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• The calibration graph is plotted on new Excel Sheet.• Enter the least – squares equation line and the r2
value. Click on the figure, and chart will appear in the toolbar. Click on it and continue:
• Add Trendline• Linear• Options• Display Equation on chart• Display R-squared value on chart• OK
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Slope, Intercept, and Coefficient Of Determination
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• We can use the Excel statistical functions to calculate the slope and intercept for a series data, and the R2 value, without a plot.
• Open a new spreadsheet and enter the calibration data from Example 3.21
• As in figure 3.9, in cells A3:B7. in cell A9 type intercept, in cell A10, slope, and in cell A11, R2 Highlight cell B9, click on fx: Statistical, and scroll down to INTERCEPT under function name, and click ok
• For known_x’s, enter the array A3:A7, and known_y’s, enter B3:B7.
• Click OK.
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• The intercept is displayed in cell B9.• Repeat highlighting cell B10, scrolling to slope, and
entering the same arrays. The slope appears in cell B10. • Repeat again, highlighting cell B11, and scrolling to RSQ. R2
appears in cell B11.• Compare with the values in figure 3.9