1. Statistics Presentation
Ch En 475
Unit Operations
Dr. Randy Lewis
September 2007

2. Statistics of Measured Variables
Lesson 1
Statistics of Measured Variables

3. Quantifying variables (i.e. answering a questions with a number)
Directly measure the variable noted as “measured” variable ex. Temperature measured with thermocouple
Calculate variable from “measured” or “tabulated” variables noted as “calculated” variable ex. Flow rate (Q) = r A v (measured or tabulated)
Each has some error or uncertainty

4. A. Error of Measured Variable
Questions
Some definitions:
x = sample mean
s = sample standard deviation
m = exact mean
s = exact standard deviation
As the sampling becomes larger:
x  m s  s t chart z chart
not valid if bias exists (i.e. calibration is off)
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?

5. How do you determine the error?
Let’s assume “normal” Gaussian distribution
For small sampling: s is known
For large sampling: s is assumed
we’ll pursue
this approach
small
large
(n>30)
Use t tables for this approach
Don’t often have
this much data
Use z tables for this approach

6. Example n
Temp 1
40.1 2
39.2 3
43.2 4
47.2 5
38.6 6
40.4 7
37.7

7. Standard Deviation Summary
Normal
distribution:
40.9 ± (3.27*1) s: 68.3% of data is within this range
40.9 ± (3.27*2) s: 95.4% of data is within this range
40.9 ± (3.27*3) s: 99.7% of data is within this range
If normal distribution is questionable, use Chebyshev's inequality:
At least 50% of the data are within 1.4 s from the mean.
At least 75% of the data are within 2 s from the mean.
At least 89% of the data are within 3 s from the mean.
The above ranges don’t state how accurate the mean is!
Ranges only state % of where the data is located

8. Student t-test (gives confidence where m (not data) is located)
measured mean 5% 5% t
true mean
2a=1- probability
r = n-1 = 6
2-tail
Prob. a t +-
90%
.05
1.943
2.40
95%
.025
2.447
3.02
99%
.005
3.143
3.88
See for expanded table

9. T-test Summary = exact mean 40.9 is sample mean
40.9 ± % confident m is somewhere in this range
40.9 ± % confident m is somewhere in this range
40.9 ± % confident m is somewhere in this range

10. Comparing averages of measured variables
Day 1:
Day 2:
What is your confidence that mx≠my (i.e. they are different)?
Larger t:
More likely
different
1-tail
1-a confident different
a confident same
nx+ny-2

11. Review Review Questions: We covered 3 topics
Explain what the following refers to (and it usefulness):
a) ± 3.27 (1s)
b) ± 2.40 (90% confidence)
Explain another use of the t statistic

12. Class Example Data points Pressure Day 1 Day 2 1 750 730 2 760 3 752
Calculate average and s for both sets of data
Find range in which 95.4% of the data falls (for each set).
Determine range for m for each set at 95% probability
At what confidence are pressures different each day?
Data points
Pressure
Day 1
Day 2 1
750
730 2
760 3
752
762 4
747
749 5
754
737

13. Lesson 2
Statistics of Calculated Variables: Confidence Interval and Propagation of max error

14. Uncertainty of Calculated Variable
Calculate variable from multiple input
(measured, tabulated, …) variables (i.e. m = rAv)
What is the uncertainty of your “calculated” value?
Example: You take measurements of r, A, v
to determine m = rAv. What is the
range of m and its associated uncertainty?
Value used to make decisions by managers- manager needs to know uncertainty of value
Ethics and societal impact is important
How do you determine the uncertainty of the value?
Details provided in Applied Engineering Statistics, Chapters 8 and 14, R.M. Bethea and R.R. Rhinehart, 1991).

15. Example: Uncertainty of Calculated Variable
(g/s) r
(g/cm3) A
(cm2)
V (cm/s) 1
66.55
1.02
20.2
3.23 2
63.97
20.1
3.12 3
71.91
3.49 4
64.75
19.9
3.19 5
68.95
20.3
3.33 6
68.06
19.8
3.37
Avg
67.36
3.29
Stdev
2.92
0.19
0.13

16. Note: If n is large (>10),
Step 1: Student t-test
measured mean
Standard error: stdev of means
if previous experiment was
repeated numerous times.
As n  ∞, x  m (the real mean).
Std error
true mean
2a=1- probability
r = n-1=5
Prob. a t +- t∞
90%
.05
2.015
2.40
1.645
95%
.025
2.571
3.06
1.96
99%
.005
3.365
4.01
2.326
Note: If n is large (>10),
t  2 for 95%. Thus,
confidence is approx
twice std. error

17. Step 2: Propagation of Maximum Error
Plan: obtain max error (d) for each input variable then obtain max error of calculated variable
Method 1: Propagation of max error- brute force Method 2: Propagation of max error- analytical
Sources of error:
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, …

18. 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, d = 2500 people

19. 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 d = 5 m/s
Judgment for a variable: use judgment for d Example: Read pressure to ± 1 psi, d = 1 psi

20. Method 1: Propagation of max error- brute force
m = r A v
Brute force method:
max
min r A v
r = 1.02 g/cm3 (table)
A = cm2 (avg)
v = 3.29 cm/s (avg)
Additional information:
sA = 0.19 cm2
sv = 0.13 cm/s
All combinations
What is d for each input variable?
mmin < m < mmax
(low 60.0, high 80.4)

21. Method 2: Propagation of max error- analytical
Additional information:
r = 1.02 g/cm3 (table)
A = 20.8 cm2 (avg)
v = 3.29 cm/s (avg)
m = r A v
sA = 0.19 cm2
sv = 0.13 cm/s
y x1 x2 x3
* Remember to take the absolute value!!
Av rv rA
m = mavg ± dm
= rAv ± dm
= ± 10.2 g/s
= (20.8)(3.29)(0.025) = 0.17
(10.2)
ferror,r=

22. 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 (<10%). If large errors, use brute force.
Must use brute force if errors are dependant on each other and/or asymmetric (i.e. Q = 10 ml/min, “+” 2 ml/min and “-” 0.5 ml/min)
Analytical method is easier to assess if lots of inputs. Also gives info on % contribution from each error.
All propagation methods assume there is no bias on the inputs (i.e. calibration is not off, etc.- see next example if calibration is off). We will assume all inputs are not biased (ex. computer noise >> computer bias) in lab unless you have evidence to the contrary.

23. Step 3: Propagation of variance- analytical
Maximum error can be calculated from max errors of input variables as shown previously:
Brute force
Analytical (if assumptions are valid)
Probable error is more realistic
Errors are independent (some may be “+” and some “-”). Not all will be in same direction and at their largest value.
Thus, propagate variance rather than max error to obtain better estimate of error (probable error rather than max error)
You need variance (s2) of each input to propagate variance. If unknown, estimate s2 = (d/2.5)2
Same assumptions apply as with propagation of max error: symmetric error, linear assumption (error <10%), errors are independent, and no bias.
use propagation of max error if not much data, use propagation of variance if lots of data

24. Propagation of variance- analytical
y = yavg ± 1.96 SQRT(s2y) 95%
y = yavg ± 2.57 SQRT(s2y) 99%
m = r A v
r = 1.02 g/cm3 (table)
A = 20.8 cm2 (avg)
v = 3.29 cm/s (avg)
y x1 x2 x3
Additional information:
m = ± 5.7 g/s
sA = 0.19 cm2
sv = 0.13 cm/s

25. Summary Step 1: Student t-test
helps provide confidence level, but not what contributes to error
improves with more experiments (std error)
includes interactions among variables
limitation: can’t be < than accuracy of table, device, calibration, etc. (doesn’t account for error)- if time, include bias for better estimate
m = ± 3.06 g/s (95% confidence)
Step 2: Propagation of maximum error- brute force
gives maximum error
easy to use if equation is difficult to take partials
equation linearity is not required
symmetry of errors is not required
m = (high 80.4, low 60.0)

26. Summary Step 2: Propagation of maximum error- analytical
helps determine what contributes to error
never improves with more experiments
equation given does not include interactions among variables
assumes linear equation (not always true)
m = ± 10.2 g/s (17% error from density)
Step 3: Propagation of variance- analytical
same assumptions as maximum error-analytical
accounts for errors not always at maximum value
s = d/2.5, accounts for error of tables, calibration, etc.
m = ± 5.7 g/s (95% confidence)
t-test < variance < max error (likely between t-test and variance)
The basis for both “analytical” methods is that the error on a measurement must be much less (an order of magnitude or more) than the measurement itself.

27. Data and Statistical Expectations
Summary of raw data (table format)
If measured variable: average and standard deviation
Table showing estimated errors of input variables
If calculated variable: Student t-test to give confidence level (recognizing limitations) and propagation of maximum error (analytical) to show how much error associated with input variables- see instructor for what to propagate
Sample calculations– including one example of ALL statistical calculations
Summary of all calculations- table format is helpful