ARI 04 Basic Data Analysis

8/12/2019 ARI 04 Basic Data Analysis

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Overview and Theory of Distributions Histogram

Normal Probability Plots Identifying a Distribution Data Examples


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Properties Of A Normal DistributionThe normal distribution is the concept that is the basis formost statistical tests.

Completely describedby its mean and

standard deviation Tails extend to

Area under curve

represents 100% of possible observations Curve is symmetrical ;

50% each side of mean

60

0

100

200

300

F r e q u e n c y

70 80 90Days

100 110 120


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Normal DistributionThe mean and standarddeviation are required to

fully describe thedistribution.

Compare the means of

each distribution.

The means are the same but the standard deviations differ.

3 rd Distribution

Mean

1 st Distribution

2nd Distribution


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95.44%

68.26%

99.73%

43210-1-2-3-4

.60-75%

.90-98%

.99-100%

The Standard Normal CurveThe standard normal curve is a special case of the normaldistribution where the mean = 0 and the standard deviation= 1.

Theoretical Empirical

99.7% of the population is within approximately 3 standard deviations of the mean.


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Histogram Purpose: To show the

shape of the data

Applications: Show Variation or

range of data Performance of data

around a nominal target To understand the

amount of data at agiven point

To find outliers in theprocess

5

10

15

20.5 23.5 26.5 29.5 32.5 35.5 38.5 41.5


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Anatomy Of A Histogram A. Vertical axis Either Frequency or the Percentage of data points ineach Class

B. Modal Class Class with the highest frequency

C. Frequency Number of data points found in each ClassD. Class Each bar is one Class, or interval, or binE. Horizontal axis Scale of measure for the element being plotted

A

900800700600500400300200100

60

50

40

30

20

10

0

F r e q u e n c y

E

C

D

B


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Open MINITAB data file: 12a_Basic_Stats.mpj Run: Stat Basic Statistics Display

Descriptive Statistics Highlight all columns from Normal to PS 2 andclick Select

Note that the mean and StDev ofNormal, Pos Skew and NegSkew are identical.

Variable N Mean Median StDev SE Mean Normal 500 70.000 69.977 10.000 0.447

Pos Skew 500 70.000 65.695 10.000 0.447 Neg Skew 500 70.000 73.783 10.000 0.447 Mystery 500 100.00 104.20 32.38 1.45

PS 2 500 70.010 66.000 9.981 0.446

12a_ Basic_Stats.MPJ

Follow-me Histogram Example


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Remembering that the means and standard deviationsof the first three data sets were the same Lets graphthem using histograms

Run: Graph Histogram (Select Simple, Click OK) Enter Normal, Pos Skew and Neg Skew into

Graph Variables field

Click OK

Histogram in Minitab


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What The Graphs Show:3 Different Distributions

Neg Skew

F r e q u e n c y

7260483624120

250

200

150

100

50

0

Histogram of Neg Skew

Pos Skew

F r e q u e n c y

130120110100908070

140

120

100

80

60

40

20

0

Histogram of Pos Skew

Normal

F r e q u e n c y

10090807060504030

70

60

50

40

30

20

10

0

Histogram of Normal


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Using the data file: 12a_Basic_Stats.mpjGraph Histogram With fit and Groups

Select Normal Mystery as graph variables

Histogram with Groups


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Mean & Std Devfor 2 data sets

Histogram with Groups


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Data Sets Come inVarious

Shapes

Left Skewed (Negatively Skewed)

Uniform Distribution

Bimodal Distribution

Bell Shape The Normal Distribution

Right Skewed (Positively Skewed)Ex: Tool Wear during machining process

Ex: Random Variation on a stable process

Ex: Torque, capacity of a container

Ex: Pre Sorting, Measurement System not sensitive enough

Ex: Pre Sorting, Measurement System not sensitive enough


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Normal Probability Plots Normal probability plots are a graphical technique to

determine if a distribution is normally distributed

Using the previous data file: 12a_Basic_Stats.mpj Stat Basic Stats Normality Test

Select Normal as Variable

12a_ Basic_Stats.MPJ


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Interpreting The Normal Probability Plot Normally distributed data will appear on the plot as a straight line

Generate plots for Pos Skew, Neg Skew, and Mystery Are they normal?

N o r m a l

P e r c e n t

1 1 01 0 09 08 07 06 05 04 03 0

99.9

99

9590

80

706050403020

10

5

1

0.1

M ean

0.328

70.00S tD e v 10 .00N 500

A D 0.418P -V alu e

P r o b a b i l i ty P l o t o f N o r m a lNor m a l


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Examples:

13012011010090807060

300

200

100

0

C2

F r e q u e n c y

Normal Probability Plots

1101009080706050403020

100

50

0

C1

F r e q u e n c y

Normal Probability Plots

1069686766656463626

.999

.99

.95

.80

.50

.20

.05.01

.001

P r o b a b i l i t y

Normal

p-value: 0.328 A-Squared: 0.418

Anderson-Darling Normality Tes t

N of data: 500Std Dev: 10

Average: 70

Normal D istribution

13012011010090807060

.999.99.95

.80

.50

.20

.05.01

.001

P r o b a b i l i t y

Pos Skew

p-value: 0.000 A-Squared: 46.447

Anderson-Darling Normality Tes t

N of data: 500Std Dev: 10

Average: 70

Positive Skewed D istribution

Normal bell shaped

Not Normal Positive Skewed

P-value:0.328

P-value:0.000


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Anderson-Darling Test If the p-value from the Anderson-Darling test < alpha of .05,

the data is not normal per that test, however: The Anderson-Darling test is not robust to small sample

sizes so for samples less than 50 it is best to rely on the

Fat Pencil test. If a fat pencil can cover all of the pointson the normal probability plot, the data may safely betreated as normal

For large samples the Anderson-Darling can measureslight departures from normality that will have little or noeffect on the level of analysis that we will be performing.

Again use the Fat Pencil test to determine reasonablenormality.


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Identifying an appropriate distribution

This function allows you to fit your data with 14distributions.

Use individual distribution identification to find a useful

distribution to fit the data if a normal distribution does notfit the data well. Why? In capability analysis , finding anappropriate distribution to fit the data is extremely

important.Roughness.MPJ

Open the data file:

Roughness.mpj

Choose Stat Quality ToolsIndividual Distribution

IdentificationComplete the dialog box asshown


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Interpreting your results


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ARI 04 Basic Data Analysis