Class 05. Using the Normal
Intro to Descriptive Statistics
What we learned last class
• To love the NORMAL distribution?
– IT IS FLEXIBLE
• Parameter μ lets you center it anywhere.
• Parameter σ lets you specify the width (variance,
uncertainty)
– If X is highly variable, pick a large σ
– NORMDIST makes probability calculations easy.
• No need for converting x to z in order to use the table in the
book. No need for NORMSINV[(x-μ)/σ].
– It approximates the binomial when n is big
• Just set μ=n*p and σ = [n*p*(1-p)]^.5
EMBS Fig 6.4, p 249
You can’t make it
skewed.
You can’t change
the 68/95/99
breakdown.
It is a pdf (prob
density function).
The height is
probability density.
Areas under the
curve are
probabilities
If X~N(μ,σ)
And Y=a+b*X,
Then
Y~N(a+b*μ,b*σ)
Temp(C)~N(10,10)
Temp(F)= 32 + (9/5) Temp(C)
Temp(F)~N(50,18)
-20
-10
0
10
20
30
40
-4
14
32
50
68
86
104
Case: Lorex Pharmaceuticals
• Lorex produces Linatol (liquid medicine for
high blood pressure)
– Sold in 10-ounce bottles.
– 600*12 = 7,200 fill attempts per shift
– Linatol costs them $0.40 per ounce to make.
– Bottles filled to more than 10 ounces get sold for
$186 per case of 12.
– Bottles under-filled get sold as seconds at a 20%
discount ($148.8 per case)
Case: Lorex Pharmaceuticals
• The Filling Machine has an adjustable target.
Test have shown that the fill amounts will be
independent (across fill attempts) and
– Normally distributed
– μ = target
– σ = 0.16 fluid ounces
• In other words (symbols?)
– X ~ N(target,0.16)
What target should
they use?
Worksheets….
Lorex Worksheet
Target
P(underfilled bottle)
Minimum required
10
$186
$148.80
Expected Good
Cases
Expected Bad
Cases
Total Cases
10.0
600
10.1
600
10.2
600
10.3
600
Expected
Revenue
Fill Test Data
Normal(10.2,0.16)?
EXHIBIT 2
LOREX PHARMACEUTICALS
Filling Line Test Results with Target = 10.2
9.89
10.17
10.29
10.00
10.04
10.35
10.05
10.19
10.21
9.79
10.53
10.47
10.24
10.17
10.15
10.06
10.13
10.04
10.20
10.37
10.24
10.03
10.01
10.20
10.41
10.17
10.35
10.06
10.19
10.17
10.07
10.27
10.13
10.24
10.14
9.84
10.36
10.11
10.42
10.16
10.21
10.01
10.25
10.44
10.44
10.44
10.13
10.29
10.53
10.32
10.16
10.21
10.09
10.02
10.32
10.14
10.11
10.20
10.35
9.96
10.30
10.33
10.36
10.17
10.15
10.14
10.07
10.37
10.40
10.25
10.24
10.03
10.20
10.04
10.16
10.22
10.12
10.36
10.24
10.07
10.40
10.29
10.21
10.10
10.23
10.19
10.19
10.29
10.25
10.18
10.42
9.85
10.45
10.37
10.22
10.19
10.23
10.48
10.17
9.76
10.06
10.17
10.04
10.41
10.27
10.00
10.23
10.11
10.19
9.97
10.05
10.12
10.33
10.18
10.54
9.91
10.28
10.23
9.98
9.99
10.15
10.11
10.19
10.22
10.10
9.99
10.40
10.76
10.20
10.31
10.16
10.23
10.17
10.00
10.11
10.30
10.64
10.10
10.23
10.45
10.17
10.19
9.98
10.13
Fill Test Data
Descriptive Statistics
Summary Statistics
Amount
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
10.198
0.014
10.190
#N/A
0.163
0.026
0.771
0.245
0.997
9.758
10.756
1468.542
144
Fill Test Data
Histogram
Frequency
1
2
3
10
17
33
36
14
16
7
3
1
1
Histogram
40
35
30
Frequency
Bin
9.758
9.841
9.925
10.008
10.091
10.174
10.257
10.340
10.423
10.506
10.590
10.673
More
25
20
15
Frequency
10
5
0
Bin
Assignment for class 06
• Find some numerical data
• Get it into excel in one column with a label at the top
• Data Analysis (File, Options, Addins, Go, check analysis
ToolPak, okay)
– Descriptive Statistics
• Check “summary statistics”
• One sheet
– A brief description of the data and where you got it.
– A printout (or hand copy) of the summary statistics
– A brief answer to the question…could these data have
come from a normal distribution…why or why not?
Completed and extended worksheet
Lorex Worksheet
Minimum
$186
10
$148.80
Target
P(underfilled bottle)
Expected Good
Cases
Expected Bad
Expected Expected Expected
Cases
Total Cases Revenue Cost
Contribution
10.0
0.5
300.00
300.00
600
$100,440
$28,800
$71,640
10.1
0.266
440.41
159.59
600
$105,663
$29,088
$76,575
10.2
0.106
536.61
63.39
600
$109,242
$29,376
$79,866
10.3
0.030
581.76
18.24
600
$110,922
$29,664
$81,258
10.4
0.006
596.27
3.73
600
$111,461
$29,952
$81,509
10.5
0.001
599.47
0.53
600
$111,580
$30,240
$81,340
10.6
0.000
599.95
0.05
600
$111,598
$30,528
$81,070