chegr 3745 statistical treatment of experimental data

1

CHEGR 3745Statistical Treatment of

Experimental Data

2

Normal Distribution

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

60 65 70 75 80 85 90

Measured Value

Pro

bab

ilit

y

3

Mean, Variance, and Standard Deviation

1 , =AVERAGE(.....) in Excel

N

ii

xx

N

2

2 1 , =VAR(...) in Excel1

N

ii

x xs

N

2 , =STDEV(...) in Excels s

4

Normal Probability Function

5

Population A

Population B

Population C

Population D

• Is Population D different from– Population A?– Population B?– Population C?– -2?

6

How Willing are You to be Wrong?

• Never!

• 1% of the time?

• 5% of the time?

• 50% of the time?

7

Confidence Intervals

; ( , )

1

tsMean x

Nt student's t - statistic f df

alpha risk

df N

8

d.f 0.01 0.05 0.10

1 63.66 12.71 6.31

2 9.92 4.30 2.92

3 5.84 3.18 2.35

4 4.60 2.78 2.13

5 4.03 2.57 2.02

6 3.71 2.45 1.94

7 3.50 2.36 1.89

8 3.36 2.31 1.86

9 3.25 2.26 1.83

10 3.17 2.23 1.81

11 3.11 2.20 1.80

12 3.05 2.18 1.78

13 3.01 2.16 1.77

14 2.98 2.14 1.76

15 2.95 2.13 1.75

16 2.92 2.12 1.75

17 2.90 2.11 1.74

18 2.88 2.10 1.73

19 2.86 2.09 1.73

20 2.85 2.09 1.72

t-distribution

9

ExampleSample Corrosion Rate

MPY

1 3.1

2 2.2

3 3.6

213..709.0503.0

97.22

fdssx

10

d.f 0.01 0.05 0.10

1 63.66 12.71 6.31

2 9.92 4.30 2.92

3 5.84 3.18 2.35

4 4.60 2.78 2.13

5 4.03 2.57 2.02

6 3.71 2.45 1.94

7 3.50 2.36 1.89

8 3.36 2.31 1.86

9 3.25 2.26 1.83

10 3.17 2.23 1.81

11 3.11 2.20 1.80

12 3.05 2.18 1.78

13 3.01 2.16 1.77

14 2.98 2.14 1.76

15 2.95 2.13 1.75

16 2.92 2.12 1.75

17 2.90 2.11 1.74

18 2.88 2.10 1.73

19 2.86 2.09 1.73

20 2.85 2.09 1.72

t-distributiont = 4.30

11

Confidence Intervals

4.3 0.7092.96

32.96 1.76

tsMean x

nMean

12

Is the Mean Equal to 5?Comparing a Mean with a Constant

2

2.96 54.96

0.5033

(5 for this example)

( , ) 4.30

;

;

Mean is not equal to 5

obs

critical

obs critical

obs critical

x Mt

sN

x Measured mean

M Target mean

t t table df

If t t x M

If t t x M

13

Are the Means of Two Populations Equal?Comparing Two Means

1 2

1 2

2 21 1 2 2

1 2

22 21 2

1 2

2 22 21 2

1 2

1 2

1 1

1 1

2

. .

1 1

obs

pool

pool

x xt

sN N

N s N ss

N N

s sN N

d fs sN N

N N

14

1 2

1 2

( , . .)

;

;

critical

obs critical

obs critical

t f d f

If t t x x

If t t x x

Example

15

Sample Corrosion Rate, MPY

1.0 M NaOH

Sample Corrosion Rate, MPY

0.5 M NaOH

1 4.0 1 3.5

2 4.6 2 3.2

3 3.7 3 3.7

4 3.9 4 3.8

5 3.1

Mean 4.05 3.46

Variance 0.15 0.093

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Are the Means of Two Populations Different? Comparing Two Means

2 21 1 2 2

1 2

1 2

1 2

2

2 2

1 1 4 1 0.15 5 1 0.0930.34

2 4 5 2

4.03 3.462.47

1 1 1 10.34

4 5

0.15 0.0934 5

. . 5.7 6.00.15 0.093

4 54 1 5 1

( , . .) 2.4

pool

obs

pool

critical

N s N ss

N N

x xt

sN N

d f

t f d f

1 2

1 2

5

;

;obs critical

obs critical

If t t x x

If t t x x

17

Does a Treatment Change Anything?Paired Testing

2

2

0

1

( , )

;

;

diff

obs

diff

diff

diff

critical

obs critical

obs critical

xt

s

Nx Average of the differences

s Variance of the differences

df N

t t table df

If t t Treatment is Significant

If t t No Change