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

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

- 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

- 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

Using Spreadsheets for Plotting Calibration Curves

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

• 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

• 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

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

• 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

Slope, Intercept, and Coefficient Of Determination

• 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.

• 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