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ISE 261 PROBABILISTIC SYSTEMS

Learn how to collect, summarize, and draw conclusions from data types in engineering statistics using graphical and numerical methods. Topics include stem-and-leaf displays, histograms, scatter diagrams, and more.

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ISE 261 PROBABILISTIC SYSTEMS

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  1. ISE 261PROBABILISTIC SYSTEMS

  2. Chapter OneDescriptive Statistics

  3. Engineering StatisticsCollect DataSummarizeDraw Conclusions

  4. Data TypesCategorical (Qualitative) > Attribute Variable (Quantitative)

  5. PopulationDefined collection or group of objects

  6. CensusData is available for all objects in the population

  7. SampleSubset of the population

  8. VariableAny characteristic whose value may change from one object to another in the population

  9. Empirical DataBased on Observation

  10. Data CollectionBasic Principles of Design:ReplicationRandomizationBlocking

  11. Descriptive StatisticsGraphical (Visual) Numerical

  12. Graphical Stem-and-Leaf DisplaysDotplotsHistogramsPareto DiagramScatter Diagrams

  13. NumericalMean MedianTrimmed MeansStandard DeviationVarianceRange

  14. Stem-and-Leaf DisplaysData Format:>Numerical > At Least Two Digits

  15. Information Conveyed:> Identification of a typical value > Extent of spread about typical value > Presence of any gaps in the data > Extent of symmetry in the distribution > Number and location of peaks > Presence of any outlying valuesInformation Not Displayed: > Order of Observations

  16. Construction of Stem-and-Leaf:>Select 1 or more leading digits for stem values. The trailing digits becomes the leaves.>List possible stem values in a vertical column>Record the leaf for every observation beside the corresponding stem>Label or indicate the units for stems and leaves someplace in the display

  17. DOTPLOTSData Format: Numerical Distinct or Discrete ValuesInformation Conveyed:LocationSpreadExtremesGapsConstruction: Each observation is a dot Stack dots above the value on a horizontal scale

  18. Dotplot ExampleData Set: Temperatures F084 49 61 40 83 67 45 66 70 69 80 58 68  60 67 72 73 70 57 63 70 78 52 67 53  67 75 61 70 81 76 79 75 76 58 31

  19. Histograms (Pareto)Data Format: Qualitative (Categorical)Frequency: Number of times that a data value occurs in the data set. Relative Frequency: A proportion of time the value occurs.

  20. Constructing a Pareto Histogram > Above each value (label), draw a rectangle whose height corresponds to the frequency or relative frequency of that value.> Ordering can be natural or arbitrary (eg. Largest to smallest).

  21. Pareto Histogram ExampleDuring a week’s production a total of 2,000 printed circuit boards (PCBs) are manufactured. List of non-conformities:Blowholes = 120Unwetted = 80Insufficient solder = 440Pinholes = 56Shorts = 40Unsoldered = 64 Improvements, Efforts, Time/Money?

  22. Histograms Data Format: >Numerical >Discrete or ContinuousData displayed by magnitude.Observed frequency is a rectangle.Height corresponds to the frequency in each cell.

  23. Histogram ConstructionDiscrete Data:>Find Frequency of each x value>Find Relative Frequency>Mark possible x values on a horizontal scale>Above each value, draw a rectangle whose height corresponds to the frequency or relative frequency of that value

  24. Histogram ConstructionContinuous Data: (Equal Widths)> Count the number of observations (n)> Find the largest & smallest (n)> Find the Range (largest- smallest)> Determine the number and width of the class intervals by the following rules:

  25. Rules> Use from 5 to 20 intervals. Rule of Thumb: # of Intervals = √n> Use class intervals of equal width. Choose values that leave no question of the interval in which a value falls.> Choose the lower limit for the first cell by using a value that is slightly less than the smallest data value.> The class interval (width) can be determined by w = range/number of cells.

  26. Build HistogramContinuous Data:> Tally Data for each Interval> Draw Rectangular Boxes with heights equal to the frequencies of the number of observations.

  27. Histogram ShapesUnimodal (1 single peak)Bimodal (2 different peaks)Multimodal (more than 2 peaks)Symmetric (mirror image) Positively Skewed (R-stretched)Negatively Skewed (L-stretched)Uniform (straight)Truncated (limited)

  28. Scatter DiagramsData Format:ContinuousTwo Random VariablesConstruction:Each Ordered Pair is plottedPatterns:Positive CorrelationNo CorrelationNegative Correlation

  29. MEANSample Mean: _ x =  Data Values nn = Number of Observations in SamplePopulation Mean:u =  Data Values NN = Number of Objects in Population

  30. MedianMiddle value after the observations are ordered from smallest to largest 50% of the values to the right. 50% of the values to the left.Odd number of samples: Middle value of the ordered arrangement.Even number of samples: Average of the two middle values.

  31. MODEThe most frequent value that occurs in the data set.

  32. QuartilesDivides data into four equal parts.Interquartile Range = Q3 – Q1

  33. Trimmed MeansMean obtained from trimming off % of the observations from “each” side of a data set.

  34. Range Difference between the largest & smallest values.

  35. Standard DeviationThe square root of the averagesquared deviation from the mean. _ s = [(xi– x)2 / (n-1)]1/2Short Cut Method:s = [( xi2– ( xi)2 / n) / (n-1)]1/2

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