INTERMEDIATE STATISTICS and LEAST SQUARES METHOD

Transcription

1 INTERMEDIATE STATISTICS and LEAST SQUARES METHOD Educational Objective: To introduce several intermediate concepts in statistics for scientific measurements and data analysis, propagation of errors and the least squares method in particular, and to utilize spreadsheet software for the regression analysis. Safety: There are no safety concerns for this lab. Section A. Propagation of Random Errors in Calculations The following rules and formulae should be used when one handles random errors (uncertainties) associated with measured quantities in scientific calculations. I. With Addition & Subtraction: When measured quantities are added or subtracted, the uncertainty in the results of operations is equal to the square root of a sum of squares of each absolute uncertainty in the measured values. Namely, (A ) + (B ) (C ) = (A+B-C) ( ) 1/2 Where,, and are the uncertainties in the measured quantities of A, B and C, (A+B-C) is the result, and ( ) 1/2 is the absolute uncertainty in the result. Example #1: Suppose that a 50.00mL buret was used to dispense 20.00mL of a dilute acid into a flask. The initial volume reading was 1.50 ml and the final volume reading was ml. Uncertainty of the volume measured with the buret is 0.04 ml. What is the volume of the acid delivered and the uncertainty associated with it? V delivered = V final - V initial = ( ) ( ) ml = ( ) ( ) 1/2 ml = /2 ml = ml II. With Multiplication and Division:

2 When measured quantities are multiplied or divided, the absolute uncertainty in the results of operations is equal to the square root of a sum of each relative absolute uncertainty squared times the quantity being calculated. Namely, [(A )(B )] / (C ) = AB/C (AB/C) [( /A) 2 + ( /B) 2 +( /C) 2 ] 1/2 where,, and are the absolute uncertainties in the measured quantities of A, B and C /A, /B, and /C are the relative uncertainties in A, B and C. AB/C is the result, [( /A) 2 + ( /B) 2 +( /C) 2 ] 1/2 is the relative uncertainty in the results, and (AB/C) [( /A) 2 + ( /B) 2 +( /C) 2 ] 1/2 is the absolute uncertainty in the result. Example #2: A metal piece weighs g with a Triple-Beam (Centigram) Balance. As the name suggests error with a centigram balance is 0.01g. Volume of the metal is found to be ml as determined from the Water Displacement Method. What are the density of the metal and its uncertainty? Density = Mass/Volume = ( )g / ( ) ml = (57.32/6.5) (57.32/6.5)[(0.01/57.32) 2 + (0.6/6.5) 2 ] 1/2 g/ml = [ ] 1/2 g/ml = ( ) g/ml = g/ml = g/ml

3 Example #3: A sample of a gas occupies a volume of 3.2 ml at 4.5 o C and 736 mmhg of pressure. Uncertainties in measurements of V, P & T are 0.2 ml, 0.5mmHg & 0.2 o C (or 0.2 K) respectively in the experiment. Find the number of moles of the gas and the uncertainty associated with it. Without considering uncertainty, n = PV/RT = (736/760atm)(3.2mL) / [( L atm/mol K) ( )K] = (736/760atm)(0.0032L) / [( L atm/mol K) (277.65) K] = x 10-4 mol = 1.36 x 10-4 mol = 1.4 x 10-4 mol Considering uncertainty, n = (( )/760atm)( mL) / (( L atm/mol K) ( K)) = 1.36 x 10-4 mol 1.36 x 10-4 [(0.5/736) 2 +(0.2/3.2) 2 +(0.2/277.65) 2 ] 1/2 mol = 1.36 x 10-4 mol 1.36 x 10-4 ( ) mol = 1.36 x 10-4 mol x10-6 mol = (1.36 x x10-4 ) mol = (1.36 x x10-4 ) mol or = (1.4 x x10-4 ) mol Section B. Least Squares Analysis - How do you to draw a best-fitting straight line through a set of scattered data? - Finding a trend line through a set of scattered data points is a routine practice among experimental scientists and engineers. In statistics, this practice is called a regression analysis. The Least Squares Method of Gauss is one of the most popularly employed regression methods. For a linear relationship of y= mx + b, Gauss first derived the following formulae for a slope (m) and an intercept (b) for a straight line that best-fits a set of n data points with coordinates (x 1,y 1 ), (x 2,y 2 ),.. (x i,y i ),..., (x n,y n ).

4 m = (n x i y i - x i y i )/{n x 2 i -( x i ) 2 } where, n is the number of measurements b = ( y i - m x i )/n 2 r = (n x i y i - x i y i )/[{n x i - ( x i ) 2 2 }{n y i - ( y i ) 2 }] ½ r=0, if no correlation. r=1, if a perfect correlation

5 Example #5: From spectrophotometric measurements for a colored compound, the following data are obtained for absorbance and the concentration for a set of five standard solutions: at M, at M, at 0.100M, at M and at M. Plot the data manually on graph paper, then draw a best-fitting straight line (estimate) through the points assuming a linear relationship between x and y (i.e., y = mx+b). Calculate the slope(m) and intercept (b) from any two points, which are on the line for the algebraic equation. A solution of the same compound yielded an absorbance value of What is the concentration? If you have a solution of mm, what do you expect the absorbance value to be? Repeat (1) (4) using spreadsheet software with the least squares method. Namely, find the best-fitting straight line using the least squares method, and give the algebraic formula for it. Compare your result from the graph paper with the result from the spreadsheet. You may try out (1) and (2) on your own at home. Here, however, only the least squares method with spreadsheet software will be illustrated. Solution # x(conc, M) y(abs) x squared y squared xy Sum m = b = r = Slope Intercept Corr. coeff. m = (n x i y i - x i y i )/{n x 2 i -( x i ) 2 } = (5x x2.625) / (5x ) = b = ( y i - m x i )/n = ( x 0.525)/n = r = (n x i y i - x i y i )/[{n x i - ( x i ) 2 2 }{n y i - ( y i ) 2 }] ½ = (5x x2.625)/[{5 x }{5x }] 1/2 =

6 As seen above, obtaining the slope and intercept with the least squares method involves many tedious calculations. In practice, one rarely goes through all of the calculations above. This is because the formulae for the slope, intercept and the correlation coefficient are already programmed into scientific calculators and coded into spreadsheet software (such as Lotus, QuattroPro and MS Excel). All one has to do is provide a set of x vs. y data to the program, and then call for m, b and r accordingly. One should refer to the manuals for calculators and spreadsheet software. A spreadsheet and an output from a Least Squares Program package using the same data are given below. This is performed with MS EXCEL. The constant is the intercept; the X coefficient is the slope. The correlation coefficient squared (R 2 ) is given as a regression output instead of the correlation coefficient (R) to make the value positive. Therefore, y = mx +b = x So y = x

7 The line for this equation is given in the following figure. Beer's Law Plot Cobalt(II) Solution Absorbance Co(II) Concentration (in M) EXPTL LST SQ Namely, (Absorbance) = x (Concentration) And (Concentration) = (Absorbance) / Therefore, the concentration of a solution that exhibits an absorbance of must be: Concentration = (0.350/5.002) M= M = M. A solution of M will exhibit an absorbance (=5.002x0.120). end