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					                                   Chapter 3
                                   Section 4
                                 Measures of
                                  Position



Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 1 of 23
                     Chapter 3 – Section 4

● Mean / median describe the “center” of the data
● Variance / standard deviation describe the
  “spread” of the data
● This section discusses more precise ways to
  describe the relative position of a data value
  within the entire set of data




   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 2 of 23
                     Chapter 3 – Section 4

● The standard deviation is a measure of
  dispersion that uses the same dimensions as the
  data (remember the empirical rule)
● The distance of a data value from the mean,
  calculated as the number of standard deviations,
  would be a useful measurement
● This distance is called the z-score




   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 3 of 23
                     Chapter 3 – Section 4

● If the mean was 20 and the standard deviation
  was 6
   The value 26 would have a z-score of 1.0 (1.0
    standard deviation higher than the mean)
   The value 14 would have a z-score of –1.0 (1.0
    standard deviation lower than the mean)
   The value 17 would have a z-score of –0.5 (0.5
    standard deviations lower than the mean)
   The value 20 would have a z-score of 0.0




   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 4 of 23
                     Chapter 3 – Section 4

● The population z-score is calculated using the
  population mean and population standard
  deviation
                                                    x
                                           z
                                                       

● The sample z-score is calculated using the
  sample mean and sample standard deviation
                                                     xx
                                            z
                                                      s


   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 5 of 23
                     Chapter 3 – Section 4

● z-scores can be used to compare the relative
  positions of data values in different samples
   Pat received a grade of 82 on her statistics exam
    where the mean grade was 74 and the standard
    deviation was 12
   Pat received a grade of 72 on her biology exam
    where the mean grade was 65 and the standard
    deviation was 10
   Pat received a grade of 91 on her kayaking exam
    where the mean grade was 88 and the standard
    deviation was 6


   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 6 of 23
                     Chapter 3 – Section 4

● Statistics
    Grade of 82
    z-score of (82 – 74) / 12 = .67
● Biology
    Grade of 72
    z-score of (72 – 65) / 10 = .70
● Kayaking
    Grade of 81
    z-score of (91 – 88) / 6 = .50
● Biology was the highest relative grade

   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 7 of 23
                     Chapter 3 – Section 4

● The quartiles are the 25th, 50th, and 75th
  percentiles
    Q1 = 25th percentile
    Q2 = 50th percentile = median
    Q3 = 75th percentile
● Quartiles are the most commonly used
  percentiles
● The 50th percentile and the second quartile Q2
  are both other ways of defining the median


   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 8 of 23
                     Chapter 3 – Section 4

● Quartiles divide the data set into four equal parts




● The top quarter are the values between Q3 and
  the maximum
● The bottom quarter are the values between the
  minimum and Q1



   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 9 of 23
                      Chapter 3 – Section 4

● Quartiles divide the data set into four equal parts




● The interquartile range (IQR) is the difference
  between the third and first quartiles
                   IQR = Q3 – Q1
● The IQR is a resistant measurement of
  dispersion

   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 10 of 23
                      Chapter 3 – Section 4

● Extreme observations in the data are referred to
  as outliers
● Outliers should be investigated
● Outliers could be
     Chance occurrences
     Measurement errors
     Data entry errors
     Sampling errors
● Outliers are not necessarily invalid data


   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 11 of 23
                      Chapter 3 – Section 4

● One way to check for outliers uses the quartiles
● Outliers can be detected as values that are
  significantly too high or too low, based on the
  known spread
● The fences used to identify outliers are
   Lower fence = LF = Q1 – 1.5  IQR
   Upper fence = UF = Q3 + 1.5  IQR
● Values less than the lower fence or more than
  the upper fence could be considered outliers


   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 12 of 23
                      Chapter 3 – Section 4

● Is the value 54 an outlier?
             1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54
● Calculations
      Q1 = (4 + 7) / 2 = 5.5
      Q3 = (27 + 31) / 2 = 29
      IQR = 29 – 5.5 = 23.5
      UF = Q3 + 1.5  IQR = 29 + 1.5  23.5 = 64
● Using the fence rule, the value 54 is not an
  outlier


   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 13 of 23
     Summary: Chapter 3 – Section 4

● z-scores
   Measures the distance from the mean in units of
    standard deviations
   Can compare relative positions in different samples
● Percentiles and quartiles
   Divides the data so that a certain percent is lower and
    a certain percent is higher
● Outliers
   Extreme values of the variable
   Can be identified using the upper and lower fences


   Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 14 of 23

				
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