Statistical Data Analysis Prof. Terry A. Ring, Ph. D. Dept. Chemical & Fuels Engineering University of Utah http://www.che.utah.edu/~geoff/writing/index.html

Making Measurements Choice of Measurement Equipment Accuracy – systematic error associated with measurement. Precision – random error associated with measurement.

Definitions Error – the difference between the measured quantity and the ”true value.” The “true value” is not known!!! So how do you calculate the error?? Random errors - the disagreement between the measurements when the experiment is repeated Is repeating the measurement on the same sample a new experiment? Systematic errors - constant errors which are the same for all measurements. Bogus Data – mistake reading the instrument

Random Error Sources Judgement errors, estimate errors, parallax Fluctuating Conditions Digitization Disturbances such as mechanical vibrations or static electricty caused by solar activity Systematic Error Sources Calibration of instrument Environmental conditions different from calibration Technique – not at equilibrium or at steady state.

Statistics Mean xM Deviation xi-xM Standard Deviation  Confidence level or uncertainty, 95% confidence = 1.96  99% confidence = 2.58  Please note that Gaussian distributions do not rigorously apply to particles- log-normal is better. Mean and standard deviation have different definitions for non-Gaussian Distributions

Comparison of means – Student’s t-test v= n1+n2-2 use the t-value to calculate the probability, P, that the two means are the same.

Estimating Uncertainties or Estimating Errors in Calculated Quantities –with Partial Derivatives G=f(y1,y2,y3,…)

Way around the Partial Derivatives This approach applies no matter how large the uncertainties (Lyons, 1991).   (i) Set all xi equal to their measured values and calculate f. Call this fo. (ii) Find the n values of f defined by fi = f(x1,x2,...,xi+si,...,xn) (11) (iii) Obtain sf from (12) If the uncertainties are small this should give the same result as (10). If the uncertainties are large, this numerical approach will provide a more realistic estimate of the uncertainty in f. The numerical approach may also be used to estimate the upper and lower values for the uncertainty in f because the fi in (11) can also be calculated with xi+ replaced by xi-.

Table 4. Uncertainties in Gas Velocity Calculated from (15) and (17) Now try the same calculation using the spread sheet method. The dimensionless form of (12) is (after taking the square root)   (17) The propagated fractional uncertainties using (15) and (17) are compared in Table 4. A further advantage of the numerical approach is that it can be used with simulations. In other words, the function f in (12) could be a complex mathematical model of a distillation column and f might be the mole fraction or flow rate of the light component in the distillate. Table 4. Uncertainties in Gas Velocity Calculated from (15) and (17) Equation Used f/f0 (9) 0.011968 (12) with + 0.011827 (12) with - 0.012113 See web page with sample calculation done with Excel  

Fitting Data Linear Equation – linear regression Non-linear Equation Linearize the equation- linear regression Non-linear least squares

Rejection of Data Points Maximum Acceptable Deviations (Chauvenet’s Criterion)

Example xi-xM/=8.9-8.2/0.3=2.33 xi-xM/=7.9-8.2/0.3=1.0

Regression Linear Regression Linearize non-linear equation first good for linear equations only Linearize non-linear equation first linearization leads to errors Non-linear Regression most accurate for non-linear equations See Mathcad example

Residence Time Measurements Time(min) Time(min) T=(To-Tin)exp(-t/tau)+Tin

Non-Linear Fit Linearized Eq. Fit Flow Calc.s Temperature(C) Time (min)

Results

Fitting Linear Data Non-linear Data Linear Regression Non-linear Fit – Most Accurate Linearization of Equation then Linear Fit – Least Accurate