Statistical Methods For UO Lab Part 1

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Statistical Methods For UO Lab Part 1

• Calvin H. Bartholomew
• Chemical Engineering
• Brigham Young University

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Background

• Statistics is the science of problem-solving in
  the presence of variability (Mason 2003).
• Statistics enables us to
• Assess the variability of measurements
• Avoid bias from unconsidered causes variation
• Determine probability of factors, risks
• Build good models
• Obtain best estimates of model parameters
• Improve chances of making correct decisions
• Make most efficient and effective use of
  resources

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Some U.S. Cultural Statistics

• 58.4 have called into work sick when we weren't.
• 3 out of 4 of us store our dollar bills in rigid
  order with singles leading up to higher
  denominations.
• 50 admit they regularly sneak food into movie
  theaters to avoid the high prices of snack foods.
• 39 of us peek in our host's bathroom cabinet.
• 17 have been caught by the host.
• 81.3 would tell an acquaintance to zip his
  pants.
• 29 of us ignore RSVP.
• 35 give to charity at least once a month.
• 71.6 of us eavesdrop.

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Population vs. Sample Statistics

• Population statistics
• 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

• Sample statistics
• 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, respectively

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Point vs. Model Estimation

• Model development
• Characterizes a function of dependent variables
• Complexity of parameter estimation and
  statistical analysis depend on model complexity
• Parameter estimation and especially statistics
  are somewhat ambiguous

• Point estimation
• Characterizes a single, usually global
  measurement
• Generally simple mathematic and statistical
  analysis
• Procedures are unambiguous

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Overall Approach

• Use sample statistics to estimate population
  statistics
• Use statistical theory to indicate the accuracy
  with which the population statistics have been
  estimated
• Use linear or nonlinear regression
  methods/statistics to fit data to a model and to
  determine goodness of fit
• Use trends indicated by theory to optimize
  experimental design

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Sample Statistics

• Estimate properties of probability distribution
  function (PDF), i.e., mean and standard deviation
  using Gaussian statistics
• Use student t-test to determine variance and
  confidence interval
• Estimate random errors in the measurement of data
• For variables that are geometric functions of
  several basic variables, use the propagation of
  errors approach estimate (a) probable error (PE)
  and (b) maximum possible error (MPE)
• PE and MPE can be estimated by differential
  method MPE can also be estimated by brute force
  method
• Determine systematic errors (bias)
• Compare estimated errors from measurements with
  calculated errors from statisticswill reveal
  whether methods of measurement or quantity of
  data is limiting

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Random Error Single Variable (i.e. T)
Questions

• Several measurements
• are obtained for a
• single variable (i.e. T).
• What is the true value?
• How confident are you?
• Is the value different on
• different days?

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How do you determine bounds of m?

• Lets assume a normal Gaussian distribution
• For small sample s is known
• For large sample s is assumed

well pursue this approach
Use z tables for this approach
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Example 1
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Properties of a Normal PDF

• About 68.26, 95.44, and 99.74 of data lie
  within 1, 2, and 3 standard deviations of the
  mean, respectively.
• When mean is zero and standard deviation is 1, it
  is referred to as a standard normal distribution.
• Plays fundamental role in statistical analysis
  because of the Central Limit Theorem.

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Central Limit Theorem

• Distribution of means calculated from a large
  data set is approximately normal
• Becomes more accurate with larger number of
  samples
• Sample mean approaches true mean as n ? ?
• Assumes distributions are not peaked close to a
  boundary and variances are finite

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Student t-Distribution

• Widely used in hypothesis testing and determining
  confidence intervals
• Equivalent to normal distribution for large
  sample size
• Student is a pseudonym, not an adjective actual
  name was W. S. Gosset who published in early
  1900s.

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Student t-Distribution

• Used to compute confidence intervals according to
  
• Assumes mean and variance are estimated by sample
  values
• Value of t decreases with DOF or number of data
  points n increases with increasing confidence

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Student t-test (determine error from s)
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t
a 1- probability r n -1 error t s /n 0.5
e.g. From Example 1 n 7, s 3.27
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Values of Student t Distribution

• Depend on both confidence level desired and
  amount of data.
• Degrees of freedom are n-1, where n number of
  data points (assumes mean and variance are
  estimated from data).
• This table assumes two-tailed distribution of
  area.

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Example 2

• Five data points with sample mean and standard
  deviation of 713.6 and 107.8, respectively.
• The estimated population mean and 95 confidence
  interval is (from previous table ta 2.77645)

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Example 3 Comparing Averages
Day 1 Day 2
What is your confidence that mx?my?
99 confident different 1 confident same
nxny-2
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Error Propagation Multiple Variables
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?
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Value and Uncertainty

• Values are used to make decisions by managers
  uncertainty of a value must be specified
• Ethics and societal impact of values are
  important
• How do you determine the uncertainty of a value?

• 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,

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Estimates of Error (d ) for Input
Variable (Methods or rules)

• Measured variable (as we just did) measure
  multiple times obtain s
• d 2.57 s (t chart shows gt 2.57 s for 99
  confidence
• e.g. s 2.3 ºC for thermocouple, d 5.8
  ºC2. Tabulated variable d 2.57 times last
  reported significant digit (e.g. r 1.0 g/ml
  at 0º C, d 0.257 g/ml)

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Estimates of Error (d) for Variable

• Manufacturer specs use given accuracy data
  (ex. Pump is 1 ml/min, d 1 ml/min)
• Variable from regression (i.e. calibration
  curve) d standard error (e.g. Velocity from
  equation with std error 2 m/s )
• Judgment for a variable use judgment for d
  (e.g. graph gives pressure to 1 psi, d 1 psi)

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Calculating Maximum or Probable Error

• Maximum error can be calculated as shown
  previously
• Brute force method
• Differential method
• Probable error is more realistic positive and
  negative errors can lower the error. You need
  standard deviations (s or s) to calculate
  probable error (PE) (i.e. see previous
  example). PE d 2.57 s

? y 1.96 SQRT(s2y) 95
? y 2.57 SQRT(s2y) 99
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Calculating Maximum (Worst) Error
1. Brute force method substitute upper and
lower limits of all xs into function to get
max and min values of y. Range of y (? ) is
between ymin and ymax. 2. Differential method
from a given model
y f(a,b,c, x1,x2,x3,)
Exact constants
Independent variables
Range of y (?) y dy
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Example 4 Differential method
m r A 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)
? 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