Statistical Analysis

1
Statistical Analysis
2
Definition of Statistics

• Descriptive Statistics numerical facts, figures
  and information
• Goal
• Describe a set of numbers
• Make accurate inferences about process/behavior
  based upon incomplete information

3
Statistical Steps

• Steps
• Gather data
• Organize data
• Analyze data

4
Population and Sample

• Population collection of all elements of
  interest
• Sample subset of the population

Sample
Population
5
Measurement Scale

• Nominal measures by labels/names
• Ordinal measures by rank order
• Interval numerical difference between values
• Ratio ratio of data values are meaningful, zero
  in the scale

6
Numerical Measures

• Measurement of central tendency
• Mean
• Median
• mode
• Measurement of variations
• Range
• Variance
• Standard deviation

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Mean

• The mean is the average value

Sample mean
Population mean
8
Median

• The middle value when the numbers are arranged in
  ascending or descending order.

1 2 3 4 5
Ex
median
9
Mode

• The data value that occurs with greatest frequency

1 1 1 3 5
mode
Ex
10
Central Tendency Example

• Test scores

Median
Mode 83
11
Range

• Difference between the largest and smallest value
  in the dataset

1 2 3 4 5
Range 5-1 4
Ex
12
Variance

• Measurement of the dispersion of values from the
  mean

Sample variance
Population variance
Statistics for Business and Economics,5th ed.,
pg68.
13
Standard Deviation

• The positive square root of the variance

Sample standard deviation
Population standard deviation
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Percentile

• The value such that p of the total items lie
  below the value
• Lower quartile, middle quartile, upper quartile
• Steps
• Arrange data in ascending order
• Compute an index i as follows
• If i is not an integer, round up.
• If i is an integer, the p is the average
  between the values in position i and i1.

15
Probability
P(A) number of favorable outcomes
total number of possible outcomes

• Areas of Probability
• simple events
• combinations of events

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Probability Distribution

• Continuous Distribution
• continuous scale
• Discrete Distribution
• discrete values

17
Discrete Vs. Continuous Data

• Discrete Examples
• Yes/No
• Throws of a die
• Go / No go
• Heads/Tails

• Continuous Examples
• segment time
• degrees of temperature
• measurements like inches, miles, feet, etc
• weight

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Discrete Distribution

• Binomial Distribution
• Hypergeometric Distribution
• Poisson Distribution

19
Binomial Probability Distribution

• Only two discrete outcome
• Consists of a sequence of n identical trials
• Trials are independent

Statistics for Business and Economics,5th ed.,
pg159.
20
Poisson Probability Distribution

• The probability is the same for any two intervals
  of equal length.
• The occurrence in any interval is independent of
  the occurrence in any other interval.

Statistics for Business and Economics,5th ed.,
pg165.
21
Hypergeometric Distribution

• Similar to binomial
• Trials are not independent

for xlt r
Statistics for Business and Economics,5th ed.,
pg170.
22
Continuous Distribution

• Uniform Distribution
• Exponential Distribution
• Normal Distribution

23
Exponential Distribution

• Used to describe time to complete a task
• Used in calculations of reliability

for x gt 0, m gt 0
Statistics for Business and Economics,5th ed.,
pg199.
24
Uniform Distribution

• Value of function is constant

for a lt x lt b elsewhere
Statistics for Business and Economics,5th ed.,
pg179.
25
Normal Distribution

• Symmetric distribution
• Highest point occurs at mean
• Mean, median and mode are at center point

Statistics for Business and Economics,5th ed.,
pg184.
26
Normal Distribution
34.13
34.13
13.06
13.06
2.14
2.14
0.13
0.13
-3s -2s -1s m 1s 2s
3s
68.26
95.46
99.73
68.26 of the population is within /- 1
of the
??
?
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Z-scores

• A Z score is a data point's position between the
  mean and another location as measured by the
  number of standard deviations.
• Z is a universal measurement because it can be
  applied to any unit of measure.

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Standard Normal Probability Table
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Standard Normal Probability Area
0.64
0.7393
0.2607
30
Capability Studies

• Purpose
• Determine whether or not a process or machine is
  in a state of statistical control.
• Define the distribution which describes the
  output of the process or machine.
• Relate the mean and variability o f the process
  or machine to the permissible range of dimensions
  allowed by the specification or tolerance.

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Capability Index
Capability Ratio of the specification width to
6 times the process or standard
deviation.
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Capability Index, Cp
USL - 20 LSL - 10 ToleranceUSL - LSL10 s 1
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Actual Capability Index, Cpk
Actual Capability Ratio of the difference
between the process mean and the nearest
specification limit to 3 times the standard
deviation.
USL
LSL
X
3s
tolerance
CPK Min CPL, CPU
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Actual Capability Index, Cpk
35
Interpreting Process Capability Indexes
CP
CPK
Interpretation
1.80 1.80 1.80 0.50 0.30 0.30 0.40 0.1
5
LOW VARIABILITIY, CENTERED LOW VARIABILITY, NOT
CENTERED HIGH VARIABILITY, CENTERED HIGH
VARIABILITY, NOT CENTERED
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Summary

• Statistical Analyses
• Process Characterization
• Discrete/Continuous Data
• Distribution
• Characterize variations and behavior
• Capability
• Stability

3s
37
Credits

• This module is intended as a supplement to design
  classes in mechanical engineering. It was
  developed at The Ohio State University under the
  NSF sponsored Gateway Coalition (grant
  EEC-9109794). Contributing members include
• Gary Kinzel. Project supervisors
• Phuong Pham.. ... Primary authors
• L. Pham ... Audio voice

Reference Statistics for Business and
Economics, 5th ed., Anderson, Sweeney,
Williams, West Publishing Co., New York1993.
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