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Measure Phase Six Sigma Statistics

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Measure PhaseSix Sigma Statistics

Welcome to Measure

Process Discovery

Six Sigma Statistics

Basic Statistics

Descriptive Statistics

Normal Distribution

Assessing Normality

Special Cause / Common Cause

Graphing Techniques

Measurement System Analysis

Process Capability

Wrap Up & Action Items

The purpose of Basic Statistics is to:

Provide a numerical summary of the data being analyzed.

Data (n)

Factual information organized for analysis.

Numerical or other information represented in a form suitable for processing by computer

Values from scientific experiments.

Provide the basis for making inferences about the future.

Provide the foundation for assessing process capability.

Provide a common language to be used throughout an organization to describe processes.

Relax….it won’t be that bad!

Summation

An individual value, an observation

A particular (1st) individual value

The Standard Deviation of sample data

For each, all, individual values

The Standard Deviation of population data

The variance of sample data

The Mean, average of sample data

The variance of population data

The grand Mean, grand average

The range of data

The Mean of population data

The average range of data

A proportion of sample data

Multi-purpose notation, i.e. # of subgroups, # of classes

A proportion of population data

The absolute value of some term

Sample size

Greater than, less than

Population size

Greater than or equal to, less than or equal to

Population Parameters:

Arithmetic descriptions of a population

µ, , P, 2, N

Population

Sample

Sample

Sample

Population: All the items that have the “property of interest” under study.

Frame: An identifiable subset of the population.

Sample: A significantly smaller subset of the population used to make an inference.

- Sample Statistics:
- Arithmetic descriptions of asample
- X-bar , s, p, s2, n

Attribute Data (Qualitative)

Is always binary, there are only two possible values (0, 1)

Yes, No

Go, No go

Pass/Fail

Variable Data (Quantitative)

Discrete (Count) Data

Can be categorized in a classification and is based on counts.

Number of defects

Number of defective units

Number of customer returns

Continuous Data

Can be measured on a continuum, it has decimal subdivisions that are meaningful

Time, Pressure, Conveyor Speed, Material feed rate

Money

Pressure

Conveyor Speed

Material feed rate

Understanding the nature of data and how to represent it can affect the types of statistical tests possible.

Nominal Scale – data consists of names, labels, or categories. Cannot be arranged in an ordering scheme. No arithmetic operations are performed for nominal data.

Ordinal Scale – data is arranged in some order, but differences between data values either cannot be determined or are meaningless.

Interval Scale – data can be arranged in some order and for which differences in data values are meaningful. The data can be arranged in an ordering scheme and differences can be interpreted.

Ratio Scale – data that can be ranked and for which all arithmetic operations including division can be performed. (division by zero is of course excluded) Ratio level data has an absolute zero and a value of zero indicates a complete absence of the characteristic of interest.

Time to weigh in!

Continuous Data is always more desirable

In many cases Attribute Data can be converted to Continuous

Which is more useful?

15 scratches or Total scratch length of 9.25”

22 foreign materials or 2.5 fm/square inch

200 defects or 25 defects/hour

Measures of Location (central tendency)

Mean

Median

Mode

Measures of Variation (dispersion)

Range

Interquartile Range

Standard deviation

Variance

Open the MINITAB™ Project “Measure Data Sets.mpj” and select the worksheet “basicstatistics.mtw”

Mean is:

Commonly referred to as the average.

The arithmetic balance point of a distribution of data.

Stat>Basic Statistics>Display Descriptive Statistics…>Graphs…>Histogram of data, with normal curve

Population

Sample

Descriptive Statistics: Data

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3

Data 200 0 4.9999 0.000712 0.0101 4.9700 4.9900 5.0000 5.0100

Variable Maximum

Data 5.0200

Median is:

The mid-point, or 50th percentile, of a distribution of data.

Arrange the data from low to high, or high to low.

It is the single middle value in the ordered list if there is an odd number of observations

It is the average of the two middle values in the ordered list if there are an even number of observations

Descriptive Statistics: Data

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3

Data 200 0 4.9999 0.000712 0.0101 4.9700 4.9900 5.0000 5.0100

Variable Maximum

Data 5.0200

Trimmed Mean is a:

Compromise between the Mean and Median.

The Trimmed Mean is calculated by eliminating a specified percentage of the smallest and largest observations from the data set and then calculating the average of the remaining observations

Useful for data with potential extreme values.

Stat>Basic Statistics>Display Descriptive Statistics…>Statistics…> Trimmed Mean

Descriptive Statistics: Data

Variable N N* Mean SE Mean TrMean StDev Minimum Q1 Median

Data 200 0 4.9999 0.000712 4.9999 0.0101 4.9700 4.9900 5.0000

Variable Q3 Maximum

Data 5.0100 5.0200

Mode is:

The most frequently occurring value in a distribution of data.

Mode = 5

- Range is the:
- Difference between the largest observation and the smallest observation in the data set.
- A small range would indicate a small amount of variability and a large range a large amount of variability.

- Difference between the largest observation and the smallest observation in the data set.
- Interquartile Range is the:
- Difference between the 75th percentile and the 25th percentile.

Descriptive Statistics: Data

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3

Data 200 0 4.9999 0.000712 0.0101 4.9700 4.9900 5.0000 5.0100

Variable Maximum

Data 5.0200

Use Range or Interquartile Range when the data distribution is Skewed.

- Standard Deviation is:
- Equivalent of the average deviation of values from the Mean for a distribution of data.
- A “unit of measure” for distances from the Mean.

- Use when data are symmetrical.

Population

Sample

Descriptive Statistics: Data

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3

Data 200 0 4.9999 0.000712 0.0101 4.9700 4.9900 5.0000 5.0100

Variable Maximum

Data 5.0200

Cannot calculate population Standard Deviation because this is sample data.

- Variance is the:
- Average squared deviation of each individual data point from the Mean.

Sample

Population

- The Normal Distribution is the most recognized distribution in statistics.
- What are the characteristics of a Normal Distribution?
- Only random error is present
- Process free of assignable cause
- Process free of drifts and shifts

- So what is present when the data is Non-normal?

- The normal curve is a smooth, symmetrical, bell-shaped curve, generated by the density function.
- It is the most useful continuous probability model as many naturally occurring measurements such as heights, weights, etc. are approximately Normally Distributed.

- Each combination of Mean and Standard Deviation generates a unique Normal curve:
- “Standard” Normal Distribution
- Has a μ = 0, and σ = 1
- Data from any Normal Distribution can be made to fit the standard Normal by converting raw scores to standard scores.
- Z-scores measure how many Standard Deviations from the mean a particular data-value lies.

- The area under the curve between any 2 points represents the proportion of the distribution between those points.
- Convert any raw score to a Z-score using the formula:
- Refer to a set of Standard Normal Tables to find the proportion between μ and x.

The area between the Mean and any other point depends upon the Standard Deviation.

m

x

The Empirical Rule…

-6

+6

-5

-4

-3

-2

+1

-1

+2

+3

+4

+5

- 68.27 % of the data will fall within +/- 1 Standard Deviation
- 95.45 % of the data will fall within +/- 2 Standard Deviations
- 99.73 % of the data will fall within +/- 3 Standard Deviations
- 99.9937 % of the data will fall within +/- 4 Standard Deviations
- 99.999943 % of the data will fall within +/- 5 Standard Deviations
- 99.9999998 % of the data will fall within +/- 6 Standard Deviations