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 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
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.
Population vs. Sample Statistics
Population statistics Characterizes the entire
population, which is generally the unknown information we seek
Mean generally designated
Variance & standard deviation generally designated as , and , 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
x
Point vs. Model Estimation
Point estimation Characterizes a single,
usually global measurement
Generally simple mathematic and statistical analysis
Procedures are unambiguous
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
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
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 statistics—will reveal whether methods of measurement or quantity of data is limiting
Some definitions:
x = sample means = sample standard deviation
= exact mean = exact standard deviation
As the sampling becomes larger:
x s
t chart z chartnot valid if bias exists (i.e. calibration is off)
Random Error: Single Variable (i.e. T)
Several measurementsare obtained for a single variable (i.e. T). • What is the true value?• How confident are you?• Is the value different on different days?
Questions
Let’s assume a “normal” Gaussian distribution For small sample: s is known For large sample: is assumed
How do you determine bounds of
i
i
nx
x
small
large(n>30)
i
i xxn
s 22
11
i
i xxn
22
11
we’ll pursue this approach
Use z tables for this approach
Example 1
n Temp
1 40.1
2 39.2
3 43.2
4 47.2
5 38.6
6 40.4
7 37.7
9.407
)7.374.406.382.472.432.391.40( x
7.10
9.407.37
9.404.409.406.38
9.402.479.402.43
9.402.399.401.40
17
1
2
22
22
22
2
s
27.3s
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.
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
xZx
n
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.
0.4
0.3
0.2
0.1
0.0
Pro
babi
lity
Den
sity
-4 -2 0 2 4
Value of Random Variable
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
60
50
40
30
20
10
Qua
ntile
Val
ue o
f t
Dis
trib
utio
n
20151050
Degrees of Freedom
99 % confidence interval 95 % confidence interval 90 % confidence interval
s tx
n
Student t-test (determine error from s)
where t f , 12
sx t n
n
= 1- probabilityr = n -1error = t s /n
0.5
Prob. /2 t t s/n 0.5
90% 0.05 1.943 2.40
5% 5%
t
e.g. From Example 1: n = 7, s = 3.27
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.
Two-tailed confidence leveldf 90% 95% 99%
1 6.31375 12.7062 63.65672 2.91999 4.30265 9.924843 2.35336 3.18245 5.840914 2.13185 2.77645 4.604095 2.01505 2.57058 4.032146 1.94318 2.44691 3.707437 1.89457 2.36458 3.498928 1.85954 2.30598 3.35519 1.83311 2.26214 3.24968
10 1.81246 2.22813 3.1691811 1.79588 2.20098 3.1057512 1.78229 2.17881 3.054513 1.77093 2.16037 3.0122514 1.76131 2.14479 2.9768315 1.75305 2.13145 2.946716 1.74588 2.1199 2.9207717 1.73961 2.10982 2.8982218 1.73406 2.10092 2.8784419 1.72913 2.09302 2.8609320 1.72472 2.08596 2.8453421 1.72074 2.07961 2.8313622 1.71714 2.07387 2.8187523 1.71387 2.06866 2.8073324 1.71088 2.0639 2.7969425 1.70814 2.05954 2.78743
inf 1.64486 1.95997 2.57583
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 t = 2.77645):
107.8*2.77645713.6
5
713.6 133.9
713.6(133.9)
s tx
n
Example 3: Comparing Averages
Day 1:
Day 2: 9n 2.67 s 2.37
7n 3.27s 9.40
yy
xx
y
x
What is your confidence that x≠y
5.211
2
)1()1( 22
yxyx
yyxx
nnnn
snsn
yxt
nx+ny-2
99% confident different1% confident same
Error Propagation: Multiple Variables
Example: How much ice cream do you buy for the AIChE event? Ice cream = f (time of day, tests, …)
Example: You take measurements of , A, v to determine m = Av. What is the range of m and its associated uncertainty?
Obtain value (i.e. from model) using multiple input variables. What is the uncertainty of your value?Each input variable has its own error
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:1. Estimation- we guess!2. Discrimination- device accuracy (single data point)3. Calibration- may not be exact (error of curve fit)4. Technique- i.e. measure ID rather than OD5. Constants and data- not always exact!6. Noise- which reading do we take?7. Model and equations- i.e. ideal gas law vs real gas8. Humans- transposing, …
Estimates of Error () for Input Variable(Methods or rules)
1. Measured variable (as we just did): measure multiple times; obtain s;
≈ 2.57 s (t chart shows > 2.57 s for 99% confidence
e.g. s = 2.3 ºC for thermocouple, = 5.8 ºC
2. Tabulated variable: ≈ 2.57 times last reported significant digit (e.g. = 1.0 g/ml at 0º C, = 0.257 g/ml)
Estimates of Error (d) for Variable
3. Manufacturer specs: use given accuracy data (ex. Pump is ± 1 ml/min, d = 1 ml/min)
4. Variable from regression (i.e. calibration curve): ≈ standard error (e.g. Velocity from equation with std error = 2 m/s )
5. Judgment for a variable: use judgment for (e.g. graph gives pressure to ± 1 psi, = 1 psi)
Calculating Maximum or Probable Error
1. Maximum error can be calculated as shown previously:a) Brute force methodb) Differential method
2. Probable error is more realistic – positive and negative errors can lower the error. You need standard deviations ( or s) to calculate probable error (PE)
(i.e. see previous example). PE = = 2.57
ixi
iy x
y 2
2
2
Ψ = y ± 1.96 SQRT(y) 95%
Ψ = y ± 2.57 SQRT(y) 99%
Calculating Maximum (Worst) Error
y = f(a,b,c…, x1,x2,x3,…)
Exact constants Independent variables
Range of y (Ψ) = y ± y
ii i
y xy
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
32 3
1
21 3
2
1 23
v 6.8 cm /
v 4.0 g/cm /
6.8 g/cm
yx x A s
x
yx x s
x
yx x A
x
Example 4: Differential method
m = A vy x1 x2 x3
x1 = = 2.0 g/cm3 (table)x2 = A = 3.4 cm2 (measured avg)x3 = v = 2 cm/s (calibration)
1 = 0.257 g/cm3 (Rule 2)2 = 0.2 cm2 (Rule 1)3 = 0.1 cm/s (Rule 4)
y = (2.0)(3.4)(2) = 13.6 g/sy = (6.8)(0.257)+(4.0)(0.2)+(6.8)(0.1) = 3.2 g/s Which product term contributes the most to uncertainty?
Ψ = 13.6 ± 3.2 g/s
ii i
y xy
This method works only if errors are symmetrical
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