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					   Chapter 13

Normal Distributions




     Chapter 13        1
        Thought Question 1
Birth weights of babies born in the United
States follow, at least approximately, a bell-
shaped curve. What does that mean?

         Thought Question 2
What does it mean if a person’s SAT score
falls at the 20th percentile for all people
who took the test?


                  Chapter 13                 2
          Thought Question 3
A study in 1976-80 found that males (ages 18-24)
have a mean height of 70 inches and a standard
deviation of 2.8 in., while females (ages 18-24)
have a mean height of 65 in. and a standard
deviation of 2.5 in. A “standardized score” is the
number of standard deviations an individual falls
above or below the mean for the whole group (is
positive for values above the mean, and negative
for those below the mean). Thus, a man who is
72.8 inches tall has a standardized score of 1.
What is the standardized score for your height?

                    Chapter 13                  3
         Thought Question 4

Many measurements in nature tend to follow a
similar pattern. The pattern is that most of the
individual measurements take on values that
are near the average, with fewer and fewer
measurements taking on values that are
farther from the average in either direction.
Describe what shape the distribution of such
measurements would have.



                  Chapter 13                   4
  Bell-Shaped Curve:
The Normal Distribution
 of Population Values




        Chapter 13        5
Asymmetric Distributions
of the Population Values




        Chapter 13         6
The Normal Distribution



           standard deviation

      mean


        Chapter 13              7
   With the Mean and Standard
Deviation of the Normal Distribution
        We Can Determine:
 What    proportion of individuals fall into
  any range of values
 At what percentile a given individual
  falls, if you know their value
 What value corresponds to a given
  percentile


                  Chapter 13                    8
         Empirical Rule for
         Any Normal Curve
 68%  of the values fall within one
  standard deviation of the mean
 95% of the values fall within two
  standard deviations of the mean
 99.7% of the values fall within three
  standard deviations of the mean

            “68-95-99.7 Rule”


                 Chapter 13               9
       Empirical Rule for
       Any Normal Curve

   68%                         95%
-1sd   +1sd           -2 sd          +2 sd


               99.7%
       -3 sd                +3 sd

               Chapter 13              10
 Health and Nutrition Examination
        Study of 1976-1980
            (HANES)
 Heights   of adults, ages 18-24
  – women
     mean: 65.0 inches
     standard deviation: 2.5 inches

  – men
     mean: 70.0 inches
     standard deviation: 2.8 inches




                   Chapter 13          11
    Health and Nutrition Examination
           Study of 1976-1980
               (HANES)
   Empirical Rule
    – women
        68% are between 62.5 and 67.5 inches
            [mean  1 std dev = 65.0  2.5]
        95% are between 60.0 and 70.0 inches
        99.7% are between 57.5 and 72.5 inches

    – men
        68% are between 67.2 and 72.8 inches
        95% are between 64.4 and 75.6 inches
        99.7% are between 61.6 and 78.4 inches




                        Chapter 13                12
      Health and Nutrition Examination
             Study of 1976-1980
                 (HANES)
    What    proportion of men are less than
       72.8 inches tall? 68%
                                                         (by Empirical Rule)


           32% / 2 = 16%
                                     ?
                             -1                     +1

 standard                             70       72.8         (height values)
 deviations: -3        ?
                      -2     = 84%
                               -1      0        1           2      3
height values: 61.6   64.4    67.2     70      72.8        75.6   78.4


                                  Chapter 13                             13
      Health and Nutrition Examination
             Study of 1976-1980
                 (HANES)
    What    proportion of men are less than
       68 inches tall?



                             ?
 standard                        68 70   (height values)
 deviations: -3       -2     -1     0       1      2       3
height values: 61.6   64.4   67.2   70     72.8   75.6   78.4


                              Chapter 13                        14
         Standardized Scores
 How  many standard deviations is 68
  from 70?
 standardized score =
     (observed value minus mean) / (std dev)
     [ = (68 - 70) / 2.8 = -0.71 ]
 The value 68 is 0.71 standard
  deviations below the mean 70.


                   Chapter 13                  15
          Standardized Scores
 standardized      score =
       (observed value minus mean) / (std dev)
     z is the standardized score
     x is the observed value
     m is the population mean
     s is the population standard deviation
                     x-m
              z
                        s
                    Chapter 13                 16
 Health and Nutrition Examination
        Study of 1976-1980
            (HANES)
 What proportion of men are less than
 68 inches tall?



             ?
                 68 70    (height values)
              -0.71   0   (standardized values)


                 Chapter 13                   17
     Table B: Percentiles of the
  Standardized Normal Distribution
 See  Table B (the “Standard Normal Table”) in
  back of the text (or back of the supplement).
 Look up the closest standardized score in
  the table.
 Find the percentile corresponding to the
  standardized score (this is the percent of
  values below the corresponding
  standardized score or z-value).

                  Chapter 13               18
                         Table B
Standard                 Standard                Standard
 Score     Percentile     Score     Percentile    Score     Percentile
  –3.4       0.03          –1.1      13.57          1.2      88.49
  –3.3       0.05          –1.0      15.87          1.3      90.32
  –3.2       0.07          –0.9      18.41          1.4      91.92
  –3.1       0.10          –0.8      21.19          1.5      93.32
  –3.0       0.13          –0.7      24.20          1.6      94.52
  –2.9       0.19          –0.6      27.42          1.7      95.54
  –2.8       0.26          –0.5      30.85          1.8      96.41
  –2.7       0.35          –0.4      34.46          1.9      97.13
  –2.6       0.47          –0.3      38.21          2.0      97.73
  –2.5       0.62          –0.2      42.07          2.1      98.21
  –2.4       0.82          –0.1      46.02          2.2      98.61
  –2.3       1.07           0.0      50.00          2.3      98.93
  –2.2       1.39           0.1      53.98          2.4      99.18
  –2.1       1.79           0.2      57.93          2.5      99.38
  –2.0       2.27           0.3      61.79          2.6      99.53
  –1.9       2.87           0.4      65.54          2.7      99.65
  –1.8       3.59           0.5      69.15          2.8      99.74
  –1.7       4.46           0.6      72.58          2.9      99.81
  –1.6       5.48           0.7      75.80          3.0      99.87
  –1.5       6.68           0.8      78.81          3.1      99.90
  –1.4       8.08           0.9      81.59          3.2      99.93
  –1.3       9.68           1.0      84.13          3.3      99.95
  –1.2      11.51           1.1      86.43          3.4      99.97




                        Chapter 13                                       19
   Table B: Percentiles of the
Standardized Normal Distribution
  Standard Score      Percentile
       (Z)

       -0.8                21.19

       -0.7                24.20

       -0.6                27.42


              Chapter 13           20
 Health and Nutrition Examination
        Study of 1976-1980
            (HANES)
 What proportion of men are less than
 68 inches tall?



          24.20%
               68 70      (height values)
              -0.71   0   (standardized values)


                Chapter 13                    21
 Health and Nutrition Examination
        Study of 1976-1980
            (HANES)
 What height value is the 10th percentile
 for men ages 18 to 24?




            10%
                ? 70    (height values)



                Chapter 13                22
    Table B: Percentiles of the
 Standardized Normal Distribution
 SeeTable B (the “Standard Normal Table”) in
 back of the text (or back of the supplement).
 Look   up the closest percentile in the table.
 Find   the corresponding standardized score.
 Thevalue you seek is that many standard
 deviations from the mean.

                    Chapter 13               23
   Table B: Percentiles of the
Standardized Normal Distribution
  Standard Score      Percentile
       (Z)

       -1.4                8.08

       -1.3                9.68

       -1.2                11.51


              Chapter 13           24
 Health and Nutrition Examination
        Study of 1976-1980
            (HANES)
 What height value is the 10th percentile
 for men ages 18 to 24?




            10%
                ? 70      (height values)
               -1.3   0   (standardized values)

                Chapter 13                    25
         Observed Value for a
          Standardized Score
 What  height value is the 10th percentile
  for men ages 18 to 24?
 observed value =
     mean plus [(standardized score)  (std dev)]
     = 70 + [(-1.3 )  (2.8)]
     = 70 + (-3.64) = 66.36
 Thevalue 66.36 is approximately the
 10th percentile of the population.

                   Chapter 13                   26
          Observed Value for a
           Standardized Score
 observed      value =
       mean plus [(standardized score)  (std dev)]
     x is the observed value
     m is the population mean
     z is the standardized score
     s is the population standard deviation

               x  m  zs
                     Chapter 13                   27
 Health and Nutrition Examination
        Study of 1976-1980
            (HANES)
 RECALL:   What proportion of men are
 less than 68 inches tall?
                               NOW: what proportion
                               of men are greater
                               than 68 inches tall?
                       ?
                  100%-24.2%
            24.20% = 75.8%
                68 70      (height values)
               -0.71   0   (standardized values)


                 Chapter 13                    28
  Health and Nutrition Examination
         Study of 1976-1980
             (HANES)
 The average height of males ages 18–
  24 years old was 70.0 inches with a
  standard deviation of 2.8 inches.
 Itis also known that this distribution of
  heights follows a normal or bell-shaped
  curve.
 What  proportion of men are between
  68 inches tall and 74 inches tall?

                 Chapter 13               29
      Health and Nutrition Examination
             Study of 1976-1980
                 (HANES)
    First,     draw and label a normal curve.

                                                         m = 70 in.
                                                         s = 2.8 in.

 standard
 deviations: -3       -2     -1     0       1      2       3
height values: 61.6   64.4   67.2   70     72.8   75.6    78.4




                              Chapter 13                         30
     Health and Nutrition Examination
            Study of 1976-1980
                (HANES)
   Shade   on the graph the range of
     heights between 68 and 74 inches.


                                ?

(height values)         68 70        74
(standardized values)   ?   0        ?


                        Chapter 13        31
         Standardized Scores
 How  many standard deviations is 68
  from 70?
 standardized score =
     (observed value minus mean) / (std dev)
     [ = (68 - 70) / 2.8 = –0.71 ]
 The value 68 is 0.71 standard
  deviations below the mean 70.


                   Chapter 13                  32
     Health and Nutrition Examination
            Study of 1976-1980
                (HANES)
   What  proportion of men are less than 68
     inches tall?




                    ?%
(height values)        68 70          74
(standardized values) -0.71   0       ?


                         Chapter 13        33
   Table B: Percentiles of the
Standardized Normal Distribution
  Standard Score      Percentile
       (Z)

       -0.8                21.19

       -0.7                24.20

       -0.6                27.42


              Chapter 13           34
     Health and Nutrition Examination
            Study of 1976-1980
                (HANES)
   What  proportion of men are less than 68
     inches tall?




                  24.20%
(height values)        68 70        74
(standardized values) -0.71   0     ?


                       Chapter 13         35
     Health and Nutrition Examination
            Study of 1976-1980
                (HANES)
   What  proportion of men are less than 74
     inches tall?

                  ?%


                  24.20%
(height values)        68 70        74
(standardized values) -0.71   0     ?


                       Chapter 13         36
         Standardized Scores
 How  many standard deviations is 74
  from 70?
 standardized score =
     (observed value minus mean) / (std dev)
     [ = (74 - 70) / 2.8 = 1.43 ]
 The value 74 is 1.43 standard
  deviations above the mean 70.


                   Chapter 13                  37
     Health and Nutrition Examination
            Study of 1976-1980
                (HANES)
   What  proportion of men are less than 74
     inches tall?

                  ?%


                  24.20%
(height values)        68 70        74
(standardized values) -0.71   0     1.43


                       Chapter 13          38
   Table B: Percentiles of the
Standardized Normal Distribution
  Standard Score     Percentile
       (Z)

       1.3                90.32

       1.4                91.92

       1.5                93.32


             Chapter 13            39
     Health and Nutrition Examination
            Study of 1976-1980
                (HANES)
   What  proportion of men are less than 74
     inches tall?

                  91.92%


                   24.20%
(height values)            68 70        74
(standardized values) -0.71     0       1.43


                           Chapter 13          40
     Health and Nutrition Examination
            Study of 1976-1980
                (HANES)
   What   proportion of men are between
     68 inches tall and 74 inches tall? 67.72%

                  91.92%

                             91.92% – 24.20%
                   24.20%             = 67.72%

(height values)            68 70            74
(standardized values) -0.71     0           1.43


                           Chapter 13              41
            Key Concepts
 Population  values are distributed with
  differing shapes, some normal, some
  non-normal.
 Empirical Rule (“68-95-99.7 Rule”)
 Standardized Score
 Percentile
 Standard Normal Table


                 Chapter 13                 42

				
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