17
1 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
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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
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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
