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2DS00
Statistics 1 for Chemical
Engineering
Lecture 2
Week schedule
Week 1: Measurement and statistics
Week 2: Introduction to regression analysis
Week 3: Simple linear regression analysis
Week 4: Multiple linear regression analysis
Week 5: Nonlinear regression analysis
Detailed contents of week 2
• error propagation
• significant numbers
• Least Squares Method
• basic notions of regression analysis
Error propagation
• measurements usually consist of
submeasurements
• examples:
–titration (begin reading and end reading)
–concentration is function of mass and volume
–...
• how to compute precision and accuracy of
composite measurements?
Expectation and variance
X is random variable with
density f
( ) ( )E X xf x dx
2
( ) ( ) ( )Var X x E x f x dx
Linear combinations
random variables Xi with mean i and variance
linear combination
Rule 1
Rule 2 (independent random variables):
1 1 n na X a X 2i
1 1 1 1n n n nE a X a X a a
2 2 2 21 1 1 1n n n nVar a X a X a a
Example: titration
Begin reading burette: 3.51 ml with =0.02 ml
End reading burette: 15.67 ml with =0.03 ml
Questions:
1. What is of used titrate?
2. The of used titrate should not exceed 0.03.
What should both readings have, assuming
that they have equal ’s?
Error propagation law
Z=f(X1,X2) with E(Xi )= µi and Var(Xi )= i 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
2 2
1 2 1 22 21 2( , ) ( , ) ( , ) ( , )
2 2
1 21 2( , ) ( , ) ( , ) ( , )
1 1( ) ( , ) ( ) ( )
2 2
( ) ( ) ( )
x x x x
x x x x
f fE Z f Var X Var X
x x
f fVar Z Var X + Var X
x x
Roots
2 2
2 3/ 2
2
1( ) ( )
2 8
( ) ( )4
x
2
x
fE Z Var X
x
fVar Z Var X =
x
( )Z = f X X
Logarithms
( ) lnZ = g X X
2 2
2 2
2
2
1( ) ln ( ) ln
2 2
( ) ( )
x
2
x
gE Z Var X
x
gVar Z Var X =
x
Products
1 2 1 2( , )Z = f X X kX X
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
2 2
1 2 1 2 1 22 21 2( , ) ( , ) ( , ) ( , )
2 2
2 2 2 2 2 22 2 1 1 2
1 2( , ) ( , ) ( , ) ( , )
1 1( ) ( , ) ( ) ( )
2 2
( ) ( ) ( )
x x x x
x x x x
f fE Z f Var X Var X k
x x
f fVar Z Var X + Var X k k
x x
2 2 2
1 2
1 2
z
z
σ = +
Quotients
11 2
2
( , )X
Z = f X X kX
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
22 21 2
1 2 1 22 2 31 2 2 2( , ) ( , ) ( , ) ( , )
2 2 22 21 1
2 2 21 2 2 2( , ) ( , ) ( , ) ( , )
1 1( ) ( , ) ( ) ( )
2 2
( ) ( ) ( )
x x x x
x x x x
f fE Z f Var X Var X k k
x x
f fVar Z Var X + Var X k k
x x
2
22
2 2 2
1 2
1 2
z
z
σ = +
Example: pH
z = pH = 3.0 [H+] = 1,0∙10-3 M.; z = 0.1
Calculate the coefficient of variation in [H+]
Significance
Basic rules:
1. Addition and subtraction: as many digits behind the decimal point
as the measurement with the least digits behind the decimal point
0.03 + 0.12 + 0.4576 = 0.61
2. Multiplication and division: as many significant digits as the
measurement with the least significant digits
0.12 *9.678234 = 1.2
Note: 12.40 has 4 significant digits