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