Statistics Review ChE 477 Winter 2018 Dr. Harding

Statistical Methods Repeated Data Points Comparing Averages of Measured Variables Linear Regression Propagation of Errors Propagation of max error - brute force Propagation of max error – analytical Propagation of variance – analytical Propagation of variance - brute force – Monte Carlo simulation

1. Repeated Data Points measured mean true mean Use t-test based on measured st dev (s) measured mean true mean In Excel, =T.INV(,r) for one-tailed test ( =0.025 for 95% confidence interval) =T.INV.2T(,r) for two-tailed test ( =0.05 for 95% confidence interval) r = n-1

Gaussian Distribution 68.27% of distribution lies within one  95.45% of distribution lies within two  99.73% of distribution lies within three  t-test is used when we do not have enough data points (<30) 68.27% 95.45% 99.73%

2. Comparing averages of measured variables Experiments were completed on two separate days. When comparing means at a given confidence level (e.g. 95%), is there a difference between the means? Day 1: Day 2:

2. Comparing averages of measured variables New formula: Larger |T|: More likely different Step 1 (compute T) For this example, Step 2 Compute net r r = nx1+nx2-2 In Excel, =T.INV(,r) for one-tailed test ( =0.025 for 95% confidence interval) =T.INV.2T(,r) for two-tailed test ( =0.05 for 95% confidence interval)

2. Comparing averages of measured variables Step 3 Compute net t from net r Step 4 Compare |T| with t 2-tail At a given confidence level (e.g. 95% or a=0.05), there is a difference if: 𝑇 >𝑡 𝛼 2 ,𝑟 T t 2.5 > 2.145 95% confident there is a difference! (but not 98% confident)

3. Linear Regression y = mx + b Fit m and b, get r2 Find confidence intervals for m and b m = 3.56  0.02, etc. Use standard error and t-statistic Excel add-on (or Igor) Find confidence intervals for line Use standard error around mean Narrow waisted curves around line Depends on n Meaning: How many ways can I draw a line through data Find prediction band for line Meaning: Where are the bounds of where the data lie

3. Linear Regression (Confidence Interval) Prediction Band

Example

Confidence Intervals and Prediction Bands What good is the confidence interval for a line? Shows how many ways the line can fit the points Let’s you state the confidence region for any predicted point r2 still helps determine how good the fit is What good is the prediction band? Shows where the data should lie Helps identify outlying data points to consider discarding

4. Propagation of Errors Quantifying variables (i.e. answering a question with a number) Directly measure the variable. - referred to as “measured” variable ex. Temperature measured with thermocouple Calculate variable from “measured” or “tabulated” variables - referred to as “calculated” variable ex. Flow rate m = r A v (measured or tabulated) Each has some error or uncertainty

4. Propagation of Errors To obtain uncertainty of “calculated” variable DO NOT just calculate variable for each set of data and then average and take standard deviation DO calculate uncertainty using error from input variables: use uncertainty for “calculated” variables and error for input variables Plan: Obtain max error (d) for each input variable then obtain uncertainty of calculated variable Method 1: Propagation of max error - brute force Method 2: Propagation of max error - analytical Method 3: Propagation of variance - analytical Method 4: Propagation of variance - brute force – Monte Carlo simulation

Sources of Uncertainty Estimation - we guess! Discrimination - device accuracy (single data point) Calibration - may not be exact (error of curve fit) Technique - i.e. measure ID rather than OD Constants and data - not always exact! Noise - which reading do we take? Model and equations - i.e. ideal gas law vs. real gas Humans - transposing, …

Estimates of Error (d) for input variables (d’s are propagated to find uncertainty) Measured: measure multiple times; obtain s; d ≈ 2.5s Reason: 99% of data is within ± 2.5s Example: s = 2.3 ºC for thermocouple, d = 5.8 ºC Tabulated : d ≈ 2.5 times last reported significant digit (with 1) Reason: Assumes last digit is ± 2.5 (± 0 assumes perfect, ± 5 assumes next left digit is fuzzy) Example: r = 1.3 g/ml at 0º C, d = 0.25 g/ml Example: People = 127,000 d = 2500 people

Estimates of Error (d) for input variables Manufacturer spec or calibration accuracy: use given spec or accuracy data Example: Pump spec is ± 1 ml/min, d = 1 ml/min Variable from regression (i.e. calibration curve): d ≈ 2.5*standard error (std error is stdev of residual) Example: Velocity is slope with std error = 2 m/s Judgment for a variable: use judgment for d Example: Read pressure to ± 1 psi, d = 1 psi

If none of the above rules apply, give your best guess! Estimates of Error (d) for input variables If none of the above rules apply, give your best guess!

Method 1: Propagation of max error- brute force Brute force method: obtain upper and lower limits of all input variables (from maximum errors); plug into equation to get uncertainty of calculated variable (y). Uncertainty of y is between ymin and ymax. This method works for both symmetry and asymmetry in errors (i.e. 10 psi + 3 psi or - 2 psi)

Example: Propagation of max error- brute force m = r A v Brute force method: min max r A v r = 2.0 g/cm3 (table) A = 3.4 cm2 (measured avg) v = 2 cm/s (slope of graph) Additional information: sA = 0.03 cm2 std. error (v) = 0.05 cm/s All combinations What is d for each input variable? mmin < m < mmax

Example: Propagation of max error- brute force m = r A v Brute force method: min max r 1.75 2.25 A 3.325 3.475 v 1.875 2.125 r = 2.0 g/cm3 (table) A = 3.4 cm2 (measured avg) v = 2 cm/s (slope of graph) Additional information: sA = 0.03 cm2 std. error (v) = 0.05 cm/s All combinations What is d for each input variable? mmin < m < mmax 10.9 13.6 16.6 2.69 3.01

Method 2: Propagation of max error- analytical Propagation of error: Utilizes maximum error of input variable (d) to estimate uncertainty range of calculated variable (y) Uncertainty of y: y = yavg ± dy Assumptions: input errors are symmetric input errors are independent of each other equation is linear (works o.k. for non-linear equations if input errors are relatively small) * Remember to take the absolute value!!

Example: Propagation of max error- analytical m = r A v y x1 x2 x3 Av rv rA 3.42 2 2 2 3.4 r = 2.0 g/cm3 (table) A = 3.4 cm2 (measured avg) v = 2 cm/s (slope of graph) m = mavg ± dm = rAv ± dm = 13.6 ± 2.85 g/s Additional information: For r, s = 0.1 g/cm3 d = 0.25 sA = 0.03 cm2, d = 0.075 std. error (v) = 0.05 cm/s d = 0.125 = (3.4)(2)(0.25) = 0.60 (2.85) ferror,r= (fractional error)

Propagation of max error If linear equation, symmetric errors, and input errors are independent  brute force and analytical are same If non-linear equation, symmetric errors, and input errors are independent  brute force and analytical are close if errors are small. If large errors (i.e. >10% or more than order of magnitude), brute force is more accurate. Must use brute force if errors are dependent on each other and/or asymmetric. Analytical method is easier to assess if lots of inputs. Also gives info on % contribution from each error.

Method 3: Propagation of variance- analytical Maximum error can be calculated from max errors of input variables as shown previously: Brute force Analytical Probable error is more realistic Errors are independent (some may be “+” and some “-”). Not all will be in same direction. Errors are not always at their largest value. Thus, propagate variance rather than max error You need variance (s2) of each input to propagate variance. If s (stdev) is unknown, estimate s = d/2.5

Method 3: Propagation of variance- analytical y = yavg ± 1.96 SQRT(s2y) 95% y = yavg ± 2.57 SQRT(s2y) 99% gives propagated variance of y or (stdev)2 gives probable error of y and associated confidence error should be <10% (linear approximation) use propagation of max error if not much data, use propagation of variance if lots of data

Example: Propagation of variance Calculate r and its 95% probable error All independent variables were measured multiple times (Rule 1); averages and s are given M = 5.0 kg s = 0.05 kg L = 0.75 m s = 0.01 m D = 0.14 m s = 0.005 m

Propagation of Errors

Method 4: Monte Carlo Simulation (propagation of variance – brute force) Choose N (N is very large, e.g. 100,000) random ±δi from a normal distribution of standard deviation σi for each variable and add to the mean to obtain N values with errors: rnorm(N,μ,σ) in Mathcad generates N random numbers from a normal distribution with mean μ and std dev σ Find N values of the calculated variable using the generated x’i values. Determine mean and standard deviation of the N calculated variables.

Propagation of Errors Overall Summary measured variables: use average, std dev (data range), and student t-test (mean range and mean comparison) calculated variable: determine uncertainty -- Max error: propagating error with brute force -- Max error: propagating error analytically -- Probable error: propagating variance analytically -- Probable error: propagating variance with brute force (Monte Carlo)