Statistical Analysis

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Systems Engineering Program
Department of Engineering Management, Information
and Systems
EMIS 7370/5370 STAT 5340 PROBABILITY AND
STATISTICS FOR SCIENTISTS AND ENGINEERS
Statistical Analysis Descriptive Statistics
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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• Basic Concepts
• Analysis of Location, or Central Tendency
• Analysis of Variability
• Analysis of Shape

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Population vs. Sample
Population the total of all possible values
(measurement, counts, etc.) of a particular
characteristic for a specific group of
objects. Sample a part of a population selected
according to some rule or plan. Why sample? -
Population does not exist - Sampling and testing
is destructive
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Sampling

• Characteristics that distinguish one type of
  sample
• from another
• the manner in which the sample was obtained
• the purpose for which the sample was obtained

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Types of Samples

• Simple Random Sample
• The sample X1, X2, ... ,Xn is a random sample if
  
• X1, X2, ... , Xn are independent identically
• distributed random variables.
• Remark Each value in the population has an
• equal and independent chance of being included
• in the sample.
• Stratified Random Sample
• The population is first subdivided into
• sub-populations for strata, and a simple random
• sample is drawn from each strata

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Types of Samples - Continued

• Censored Samples
• Type I Censoring - Sample is terminated at a
• fixed time, t0. The sample consists of K times to
  
• failure plus the information that n-k items
• survived the fixed time of truncation.
• Type II Censoring - Sampling is terminated
• upon the Kth failure. The sample consists of K
• times to failure, plus information that n-k items
  
• survived the random time of truncation, tk.
• Progressive Censoring - Sampling is reduced in
• stage.

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Types of Samples - Continued

• Systematic Random Sample
• The N items in the population are arranged in
• some order.
• Select an item at random from the first K N/n
• items, where n is the sample size.
• Select every Kth item thereafter.

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Statistical Analysis Objective

• Data represents the entire population
• Statistical analysis is primarily descriptive.
• Data represents sample from population
• Statistical analysis
• - describes the sample
• - provides information about the population

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Analysis of Location or Central Tendency

• Sample (Arithmetic) Mean
• Sample Midrange
• Sample Mode
• Sample Median
• Sample Percentiles

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Sample Mean
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Sample Mode

• Definition
• Most frequently occurring value in the sample
• Remarks
• A sample may have more than one mode
• The mode may not be a central value
• Not well understood, nor frequently used

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Sample Median
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Analysis of Variability

• Sample Range
• Sample Variance
• Sample Standard Deviation
• Sample Coefficient of Variation

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Sample Range

• Formula R Xmax - Xmin
• where Xmax is the largest value in the sample
• and Xmin is the smallest sample value
• Remarks
• Easy to determine
• Easily understood
• Determined by extreme values
• Does not use all sample data

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Sample Variance Standard Deviation

• Sample Variance
• Sample Standard Deviation
• s (sample variance)1/2
• Remarks
• Most frequently used measure of variability
• Not well understood

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Sample Coefficient of Variation
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Analysis of Shape

• Skewness
• Kurtosis

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Estimate of Skewness
For a unimodal distribution, xr is an
indicator of distribution shape lt 1 ,
indicates skewed to the left xr 1 ,
indicates symmetric gt 1 , indicates
skewed to the right
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Measure of Skewness

• The third moment about the mean is related to
• the asymmetry or skewness of a distribution
• For a unimodal (i.e., a single peaked)
  distribution
• ?3 lt 0 , distribution is skewed to the left
• ?3 0 , distribution is symmetric
• ?3 gt 0 , distribution is skewed to the right
• Measure of skewness relative to degree of spread

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Comparison of Distribution Skewness

• Normal
• Exponential

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Estimation of Skewness

• Estimate of skewness of a distribution from a
• random sample
• where
• and

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Measurement of Kurtosis

• The fourth moment about the mean is related to
• the peakedness, called kurtosis, of a
  distribution
• Relative measure of Kurtosis
• where

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Estimation of Kurtosis

• Estimate of kurtosis of a distribution (?2) from
  a
• random sample
• where
• and

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Comparison of Kurtosis
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Presentation of Data
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40 Specimens
40 specimens are cut from a plate for tensile
tests. The tensile tests were made, resulting in
Tensile Strength, x, as follows
Perform a statistical analysis of the tensile
strength data.
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40 Specimens
The following descriptive statistics were
calculated from the data