Statistical Methods For UO Lab — Part 1. Calvin H. Bartholomew Chemical Engineering Brigham Young University. Background. Statistics is the science of problem-solving in the presence of variability (Mason 2003). Statistics enables us to: Assess the variability of measurements
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Calvin H. Bartholomew
Brigham Young University
Characterizes the entire population, which is generally the unknown information we seek
Mean generally designated m
Variance & standard deviation generally designated as s 2, and s, respectively
Characterizes a random, hopefully representative, sample – typically data from which we infer population statistics
Mean generally designated
Variance & standard deviation generally designated as s2 and s, respectivelyPopulation vs. Sample Statistics
Characterizes a single, usually global measurement
Generally simple mathematic and statistical analysis
Procedures are unambiguous
Characterizes a function of dependent variables
Complexity of parameter estimation and statistical analysis depend on model complexity
Parameter estimation and especially statistics are somewhat ambiguousPoint vs. Model Estimation
x = sample mean
s = sample standard deviation
m = exact mean
s = exact standard deviation
As the sampling becomes larger:
x m s st chart z chart
not valid if bias exists (i.e. calibration is off)
Random Error: Single Variable (i.e. T)
How do you determine bounds of m?
we’ll pursue this approach
Use z tables for this approach
a = 1- probability
r = n -1
error = ts/n0.5
e.g. From Example 1: n = 7, s = 3.27
What is your confidence that mx≠my?
99% confident different
1% confident same
Obtain value (i.e. from model) using multiple input variables.
What is the uncertainty of your value?
Each input variable has its own error
Example: How much ice cream do you buy for
the AIChE event? Ice cream = f (time of day, tests, …)
Example: You take measurements of r, A, v
to determine m = rAv. What is the
range of m and its associated uncertainty?
(Methods or rules)
Ψ = y ± 1.96 SQRT(s2y) 95%
Ψ = y ± 2.57 SQRT(s2y) 99%
1.Brute force method: substitute upper and lower limits of all x’s into function to get max and min values of y. Range of y (Ψ ) is between ymin and ymax.
2.Differential method: from a given model
y = f(a,b,c…, x1,x2,x3,…)
Range of y (Ψ) = y ± dy
m = rA v
y x1 x2 x3
x1 = r= 2.0 g/cm3 (table)
x2 = A = 3.4 cm2 (measured avg)
x3 = v = 2 cm/s (calibration)
d1 = 0.257 g/cm3 (Rule 2)
d2 = 0.2 cm2 (Rule 1)
d3 = 0.1 cm/s (Rule 4)
Ψ = 13.6 ± 3.2 g/s
y = (2.0)(3.4)(2) = 13.6 g/s
dy = (6.8)(0.257)+(4.0)(0.2)+(6.8)(0.1) = 3.2 g/s
Which product term contributes the most to uncertainty?
This method works only if errors are symmetrical