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