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ISE 261 PROBABILISTIC SYSTEMS

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Chapter One Descriptive Statistics

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Engineering Statistics Collect Data Summarize Draw Conclusions

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Data Types Categorical (Qualitative) > Attribute Variable (Quantitative)

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Population Defined collection or group of objects

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Census Data is available for all objects in the population

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Sample Subset of the population

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Variable Any characteristic whose value may change from one object to another in the population

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Empirical Data Based on Observation

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Data Collection Basic Principles of Design: Replication Randomization Blocking

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Descriptive Statistics Graphical (Visual) Numerical

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Graphical Stem-and-Leaf Displays Dotplots Histograms Pareto Diagram Scatter Diagrams

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Numerical Mean Median Trimmed Means Standard Deviation Variance Range

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Stem-and-Leaf Displays Data Format: > Numerical > At Least Two Digits Stem-and-Leaf Displays Data Format: > Numerical > At Least Two Digits

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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 values Information Not Displayed: > Order of Observations 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 values Information Not Displayed: > Order of Observations

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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

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DOTPLOTS Data Format: Numerical Distinct or Discrete Values Information Conveyed: Location Spread Extremes Gaps Construction: Each observation is a dot Stack dots above the value on a horizontal scale

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Dotplot Example Data Set: Temperatures F 0 84 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 Dotplot Example Data Set: Temperatures F 0 84 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

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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.

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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).

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

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

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Histogram Construction Discrete 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

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Histogram Construction Continuous 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:

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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.

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

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Histogram Shapes Unimodal (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)

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Scatter Diagrams Data Format: Continuous Two Random Variables Construction: Each Ordered Pair is plotted Patterns: Positive Correlation No Correlation Negative Correlation

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MEAN Sample Mean: _ x = Data Values n n = Number of Observations in Sample Population Mean: u = Data Values N N = Number of Objects in Population

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Median Middle 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.

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MODE The most frequent value that occurs in the data set.

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Quartiles Divides data into four equal parts. Interquartile Range = Q 3 – Q 1

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Trimmed Means Mean obtained from trimming off % of the observations from “each” side of a data set.

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Range Difference between the largest & smallest values.

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Standard Deviation The square root of the average squared deviation from the mean. _ s = [ (x i – x) 2 / (n-1)] 1/2 Short Cut Method: s = [( x i 2 – ( x i ) 2 / n) / (n-1)] 1/2

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Variance Square of the Standard Deviation.

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Boxplots Information Conveyed: > Center > Spread > Nature of Symmetry > Identification of Outliers

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Build Boxplots On 1. Smallest Value 2. Lower Fourth 3. Median 4. Upper Fourth 5. Largest Value Fourth Spread = Upper Fourth – Lower Fourth

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Construction Of Boxplot 1. Order data from smallest to largest. 2. Separate smallest half from the largest half. (If n is odd include the median in both halves). 3. Lower fourth is the median of the smallest half. 4. Upper fourth is the median of the largest half. 5. Fourth spread = Upper fourth – Lower fourth. 6. On a horizontal measurement scale, the left edge of a rectangle is the lower fourth & the right edge is the upper fourth. 7. Place a vertical line inside the rectangle at the location of the median. 8. Draw whiskers out from ends of the rectangle to the smallest and largest data values.

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