1-12 Basic Statistics I[1]

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Basic StatisticsBasic Statistics


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Copyright NN, Inc. 2004 Company Confidential


Serves as a means to analyze data collected in themeasurement phase.

Allows us to numerically describe the data which

characterizes our process KPIVs & KPOVs.

Uses past process and performance data to make

inferences about the future.

Serves as a foundation for advanced statistical

problem solving methodologies.

Provides a language based on numerical facts and

not intuition.

WHY DO WE NEED BASIC STATISTICS?


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The primary objective of statistical analysis is to determine certain

characteristics of a group from a representative sample.

To be valid, this generalization must consider certain important

concepts.

The Random Sample:A random sample is a subgroup of elements gathered so that the results of

statistical analysis can be extended to the population from which it came.

One frequently applied sampling method is to choose the elements

at random. However, this assumes that each element of the population hasan equal chance of being selected.

DATA CHARACTERISTICS


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Variable Data:

Variable data can be expressed in the form of numbers and measured

with a measuring instrument. Since they supply a lot of information

and are sensitive to small variations, variable data are widely used.

Examples of variable data The diameter of the holes in a part. The thickness of an aluminum plate. The time taken by a machine tool to perform an operation.

Room temperature variations. Light intensity variations.



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Attribute Data:Attribute data are based on the number of conforming or nonconforming

characteristics, but not on their measurement. For example, the attribute

data indicate whether a part is good or not, or how many parts conform to

the specifications.

Attribute data are determined with an instrument like a go/no-go gauge(passes the test or not; yes or no; conforming or nonconforming, etc. ).


Examples of attribute dataNumber of parts whose dimensions do not conform to the tolerances. Number of stored parts. Assessment of the quality of a surface coating ( a lot, moderate, or

little orange peel). Number of errors on purchase orders. Number of packing slips with the wrong address.

The type of data, variable or attribute, will determine

the statistical tools to use in processing the data.



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Accurate but Imprecise Process

LS US

Nominal

Inaccurate but Precise Process

LS US

Nominal

PROCESS VARIATION


7/24Copyright NN, Inc. 2004 Company Confidential


Accurate & Precise Process

LS US

Nominal

Inaccurate & Imprecise Process

LS

Nominal

US

PROCESS VARIATION




LS US

LS US

The dispersion is within the tolerance. The distribution is well centered between

the limits. The risk of defect is minimal.

The dispersion is within the tolerance. The distribution is off-center. Production of defects is likely.

PROCESS CENTERING




LS US

LS US

The dispersion is less than the tolerance. The distribution is very off-center and

exceeds the upper tolerance. Production of defects has occurred.

The dispersion is equal to the tolerance. The distribution is centered. The risk of defects is high.

PROCESS VARIATION




LS US

LS US

The dispersion is greater than the tolerance. The distribution is also off-center. Production of defects has occurred.

The dispersion is greater than the tolerance. The distribution is centered. Production of defects has occurred.

PROCESS VARIATION




LS US

LS US

The distribution is bimodal (two peaks). Both dispersions are within the tolerance. Indicates that the data comes from two

distinct populations ( two machines, two

shifts, two suppliers, etc.)

The dispersion is within the tolerance. The distribution is asymmetrical (shifted

towards one limit). There is a possibility that the data are

distorted or inaccurate. However, certain

processes are naturally asymmetrical.

PROCESS VARIATION




Population:- The entire group of objects that

have been made or will be made

containing the characteristic of

interest.- Highly unlikely we can ever obtain

population parameters. all NN-made bearing balls. all Millionaires in the world

Sample:-The group of objects on which one

actually gathers data in a statistical

study.-Usually a sample is a subset of the

population.

NN-made balls produced today. Millionaires at NN.

Population Parameters Sample Parameters

= Population Mean= Population Standard

Deviation

X = Sample Mean

S = Sample StandardDeviation

POPULATION vs. SAMPLE




Mean:-The arithmetic average of a set of values.

uses the quantitative value of each data point. strongly influenced by extreme values.

Median:- Number reflecting the 50% rank of a set of values.

can be easily identified as the center after all of

the values are sorted from high to low. hardly affected by extreme values.

Mode:- Most frequently occurring value in a data set.

n

n

ii

=

= 1

MEASUREMENTS OF CENTRAL TENDENCY

12

14

17

18

19

2227

27

28




Range:- Numerical distance between the highest and thelowest values in a data set.

very sensitive to extreme values in the data.maxmin=Range

Variance :- Sum of the distances between individual data points and

population or sample mean. Distances are squared toremove negative.

very sensitive to extreme values in the data.

);(22

s

1

)(

1

2

2

=

=

n

s

n

i

i

Standard Deviation :- The square root of the variance.

most commonly used measurement to

quantify variability.

);( s

1

)(1

2

=

=

n

s

n

i

i

Adding Variances :)(2

2

2

1 +

2

1

2

2 = Variance of sample 2

= Variance of sample 1

2

2

2

1

2+=Then

)(2

2

2

1 +=

MEASUREMENTS OF VARIABILITY




Rules:= 68.25% of the data will fall within +/- 1 s from the mean

= 95.46% of the data will fall within +/- 2 s from the mean



= 99.999943% of the data will fall within +/- 5 s from the mean= 99.9999998% of the data will fall within +/- 6 s from the mean

-4 -3 -2 -1 0 1 2 3 4

68.3%

95.5%

99.73%

NORMAL CURVE PROPERTIES




LSL USL

Long-Term

Capability

Short-Term

Capability

Over time, a process tends to shift by approximately 1.5.

THE DYNAMIC PROCESS




If a manufacturing process is represented by:

The variation of Y is driven by the variation of the Xs.

The nature of the variation of each x can differ from the others. Some Xs vary over short cycles, others over long cycles.

Thus, a process generally exhibits different variation patterns

over the long term than it does over the short term.

Y = f(X X X X1 2 3 k , , ,..., )

LONG-TERM vs. SHORT-TERM VARIATION




Long Term Capability Covers a relatively long period of time (e.g. weeks, months) Will include effect of long term noise variables . Generally consists of 100-200 points.

Short Term Capability Covers a relatively short period of time (e.g. days, weeks).

Will include effect of short term noise variables.

Generally consists of 30 to 50 data points.

Instantaneous Capability Covers a very short period of time (e.g. one shift) Minimal effects due to noise variables.

LONG-TERM vs. SHORT-TERM VARIATION



PROCESS CAPABILITY RATIO - Cp


LSL USLTolerance

Process Width

-3 +3

(Max Allowable Range of Characteristic)=

(Normal Variation of the Process)

USL - LSL=

6OR

99.7% of values.

pCpC




This index accounts for the static mean shift in the

process (the amount that the process is off target).

Both the Cp andCpk indexes should be considered if upper and

lower specifications exist. Why?

USL - X X - LSLCpk= Min or

3 3

PROCESS CAPABILITY RATIO - Cpk




Lower Spec Upper Spec

3.5 4.5 5.5 6.5

No Mean Shift

3.5 6.5 7.55.54.5


Mean Shift of 1.5 Sigma

Cp = 1.02

Cpk = 1.02

Cp = 1.00

Cpk = 0.52

3 SIGMA PROCESS





4.0 5.0 6.0 7.0

No Mean Shift

3.02.0 8.0


4.0 5.0 6.0 7.0

Mean Shift of 1.5 Sigma

3.02.0 8.0

Cp = 2.04

Cpk = 2.04

Cp = 2.01

Cpk = 1.52

6 SIGMA PROCESS




Relation Cp, Cpk and PPM (Parts Per Million)


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Predict the percent of time that the process will fail

to operate as required.- defects, downgrades or rework

Set the performance baseline from which to

measure any improvements made.

Establish a benchmark against which we cancompare other equipment, other plants, etc.

KNOWING OUR PROCESS CAPABILITY CAN.

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1-12 Basic Statistics I[1]