April 13

Friday the 13th Statistics

Everything could go wrong

Murphy's Laws:

• If anything can possibly go wrong, it will
• Nature always sides with the undiscovered fault
• Jelly bread always falls jelly side down.

There are some unappreciated features of those problems.

First, an overview

• Regression-- finding the best fit of data to a curve
  • for discussion, limit to best straight line
  • we can, of course fit to polynomials or other more complex functions
  • best straight line y = a + b x
  • we want to go beyond simply knowing the best equation
  • we want to know how reliable, useful the results will be.
• Residuals are defined for each point
  • r_i = [ y_i - (a + b x_i) ]²
  • "best fit" is the one that minimizes the sum of the residuals
    • you should be aware of a bias and limitation here
    • this assumes that the error is associated with determining y
    • it assumes that x is known with much higher certainty
      • for many analytical determinations this is true
      • x is the concentration of carefully prepared standards of pure materials
      • good balances, volumetric glassware, good technique and pure reagents can keep the uncertainty of x small
      • y is the instrumental measurement, subject to a variety of S/N problems
    • you can however, encounter cases were x is subject to errors, perhaps errors exceeding those of y.
    • you might swap the role of x and y and proceed
    • if x and y have comparable errors you need another set of statistical tools
    • we will not consider that complication further... x is assumed known accurately.
• Standard Statistical Tools give us additional information
  • Appendix A in the text has formulas you can use
  • Excel has a Tool (Data Analysis/ Regression) that you can use
  • Other math packages have more statistics
  • Assume a package gives us the first four parameters
    • the best slope, m (eq. a1-32)
    • the standard deviation of the slope, s_m (eq. a1-35)
    • the best intercept, b (eq. a1-33)
      • (this is the y-intercept, the value of y when x=0)
    • the standard deviation of the intercept, s_b (eq. a1-36)
      • usually we use slope and intercept to covert an experimental y value to determine x : ____ x = (y - intercept) / slope
      • standard deviations give us some numbers to use in estimating the error of any point evaluated using the formula
  • Another useful value is the standard deviation of the residuals s_r (eq. a1-34)
    • really a measure of how much difference there is between the y-values and the best curve.

• This is a standard additions problem--
  • There is only one set of data being analyzed-- fixed sample + known standard additions
  • the calibration curve is essentially being determined simultaneously
• The set of data is extrapolated to determine the x intercept (where the curve has y=0)
  • x-intercept = -(y-intercept ) / (slope)
  • the uncertainties in y- intercept and slope fix the uncertainties in the x- intercept
  • when you multiply or divide, the variance of the result is the sum of the variance of the two values
    • variance is the square of the standard deviation (noted here as S)
  • S(result)² = S(intercept)² + S(slope)²
  • S(result) = square root (S_a2 + S_b2)

One set of data is used to determine a calibration curve

The additional measurements are made on the samples being analyzed

• the results involve two sources of error
• those of the calibration data
• those of the sample measurements

in practice, the sample measurements will be less accurate

• we can reduce the sample measurement errors by making replicate measurements
• we need to calculate the error for each sample we analyze

Appendix , page A19, equation (eq. a1-37)

• sc = (s_r/m) sqrt { 1/M + 1/N + (Y^av_c + Y^av)2/(m²Sxx) }
  • m is slope
  • M is number of analyses (replicates)
  • N is number of points used for the calibration curve
  • y^av_c is the average of the replicate measurements
  • y^av is the average of the values from the calibration curve
  • S_xx is computed from the calibration data
  • S_xx = (Sum) X_i2 - {(Sum) x_i }²/ N