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					  Probabilistic and
Statistical Techniques
            Lecture 6

   Eng. Ismail Zakaria El Daour

              2011
                                  1
Probabilistic and Statistical Techniques




             Chapter 2 (part 4)
   Summarizing and Graphing Data




                                           2
Probabilistic and Statistical Techniques


             Measures of position
                Definition
   z Score (or standardized value)
       the number of standard deviations
       that a given value x is above or below
       the mean



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Probabilistic and Statistical Techniques




     Measures of Position z score

       Sample                  Population
          xx                          x
       z                        z
           s                               
             Round z to 2 decimal places


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Probabilistic and Statistical Techniques


       Interpreting Z Scores




Whenever a value is less than the mean, its
corresponding z score is negative
      Ordinary values:   z score between –2 and 2
      Unusual Values:    z score < -2 or z score > 2

                                                       5
  Probabilistic and Statistical Techniques


                  Percentiles
 A percentile provides information about how the
  data are spread over the interval from the smallest
  value to the largest value.

 Admission test scores for colleges and universities
  are frequently reported in terms of percentiles.



                                                        6
Probabilistic and Statistical Techniques



                Percentiles


    The pth percentile of a data set is a
     value such that at least p percent of the
     items take on this value or less and at
     least (100 - p) percent of the items
     take on this value or more.


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Probabilistic and Statistical Techniques


   Arrange the data in ascending order.

   Compute index i, the position of the pth percentile.

                       i = (p/100)n

   If i is not an integer, round up. The p th percentile
   is the value in the i th position.

   If i is an integer, the p th percentile is the average
   of the values in positions i and i +1.



                                                            8
Probabilistic and Statistical Techniques

              90th Percentile
          i = (p/100)n = (90/100)70 = 63
    Averaging the 63rd and 64th data values:
    90th Percentile = (580 + 590)/2 = 585
  425   430   430   435   435   435   435   435   440   440
  440   440   440   445   445   445   445   445   450   450
  450   450   450   450   450   460   460   460   465   465
  465   470   470   472   475   475   475   480   480   480
  480   485   490   490   490   500   500   500   500   510
  510   515   525   525   525   535   549   550   570   570
  575   575   580   590   600   600   600   600   615   615
                                                              9
Probabilistic and Statistical Techniques




                    Quartiles

   Quartiles are specific percentiles.
   First Quartile = 25th Percentile
   Second Quartile = 50th Percentile = Median
   Third Quartile = 75th Percentile



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Probabilistic and Statistical Techniques



                  Definition
 Q1 (First Quartile) separates the bottom
     25% of sorted values from the top 75%.
 Q2 (Second Quartile) same as the median;
      separates the bottom 50% of sorted
      values from the top 50%.
 Q3 (Third Quartile) separates the bottom
      75% of sorted values from the top 25%.

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Probabilistic and Statistical Techniques

                    Third Quartile
                Third quartile = 75th percentile
             i = (p/100)n = (75/100)70 = 52.5 = 53
                     Third quartile = 525
 425   430    430   435   435   435   435   435      440   440
 440   440    440   445   445   445   445   445      450   450
 450   450    450   450   450   460   460   460      465   465
 465   470    470   472   475   475   475   480      480   480
 480   485    490   490   490   500   500   500      500   510
 510   515    525   525   525   535   549   550      570   570
 575   575    580   590   600   600   600   600      615   615
                                                                 12
Probabilistic and Statistical Techniques



                    Quartiles

                    Q1, Q2, Q3
   divide ranked scores into four equal parts
               25%    25%   25% 25%

        (minimum)
                    Q1 Q2 Q3      (maximum)

                       (median)


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Probabilistic and Statistical Techniques


         Some Other Statistics
 Interquartile Range (or IQR): Q3 - Q1
                                    Q3 - Q1
 Semi-interquartile Range:
                                       2
 Midquartile:        Q3 + Q1
                          2
 10 - 90 Percentile Range: P90 - P10
                                              14
Probabilistic and Statistical Techniques



                      Definitions

     A boxplot is a graph of a data set that consists of
     a line extending from the minimum value to the
     maximum value, and a box with lines drawn at
     the first quartile, Q1; the median; and the third
     quartile, Q3.



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Probabilistic and Statistical Techniques




                 Boxplots




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Probabilistic and Statistical Techniques




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