Data606_Lab3_JagdishChhabria
Jagdish Chhabria
In this lab we’ll investigate the probability distribution that is most central to statistics: the normal distribution. If we are confident that our data are nearly normal, that opens the door to many powerful statistical methods. Here we’ll use the graphical tools of R to assess the normality of our data and also learn how to generate random numbers from a normal distribution.
The Data
This week we’ll be working with measurements of body dimensions.This data set contains measurements from 247 men and 260 women, most of whom were considered healthy young adults.
load("more/bdims.RData")Let’s take a quick peek at the first few rows of the data.
head(bdims)## bia.di bii.di bit.di che.de che.di elb.di wri.di kne.di ank.di sho.gi
## 1 42.9 26.0 31.5 17.7 28.0 13.1 10.4 18.8 14.1 106.2
## 2 43.7 28.5 33.5 16.9 30.8 14.0 11.8 20.6 15.1 110.5
## 3 40.1 28.2 33.3 20.9 31.7 13.9 10.9 19.7 14.1 115.1
## 4 44.3 29.9 34.0 18.4 28.2 13.9 11.2 20.9 15.0 104.5
## 5 42.5 29.9 34.0 21.5 29.4 15.2 11.6 20.7 14.9 107.5
## 6 43.3 27.0 31.5 19.6 31.3 14.0 11.5 18.8 13.9 119.8
## che.gi wai.gi nav.gi hip.gi thi.gi bic.gi for.gi kne.gi cal.gi ank.gi
## 1 89.5 71.5 74.5 93.5 51.5 32.5 26.0 34.5 36.5 23.5
## 2 97.0 79.0 86.5 94.8 51.5 34.4 28.0 36.5 37.5 24.5
## 3 97.5 83.2 82.9 95.0 57.3 33.4 28.8 37.0 37.3 21.9
## 4 97.0 77.8 78.8 94.0 53.0 31.0 26.2 37.0 34.8 23.0
## 5 97.5 80.0 82.5 98.5 55.4 32.0 28.4 37.7 38.6 24.4
## 6 99.9 82.5 80.1 95.3 57.5 33.0 28.0 36.6 36.1 23.5
## wri.gi age wgt hgt sex
## 1 16.5 21 65.6 174.0 1
## 2 17.0 23 71.8 175.3 1
## 3 16.9 28 80.7 193.5 1
## 4 16.6 23 72.6 186.5 1
## 5 18.0 22 78.8 187.2 1
## 6 16.9 21 74.8 181.5 1You’ll see that for every observation we have 25 measurements, many of which are either diameters or girths. A key to the variable names can be found at http://www.openintro.org/stat/data/bdims.php, but we’ll be focusing on just three columns to get started: weight in kg (wgt), height in cm (hgt), and sex (1 indicates male, 0 indicates female).
Since males and females tend to have different body dimensions, it will be useful to create two additional data sets: one with only men and another with only women.
mdims <- subset(bdims, sex == 1)
fdims <- subset(bdims, sex == 0)
mdims## bia.di bii.di bit.di che.de che.di elb.di wri.di kne.di ank.di sho.gi
## 1 42.9 26.0 31.5 17.7 28.0 13.1 10.4 18.8 14.1 106.2
## 2 43.7 28.5 33.5 16.9 30.8 14.0 11.8 20.6 15.1 110.5
## 3 40.1 28.2 33.3 20.9 31.7 13.9 10.9 19.7 14.1 115.1
## 4 44.3 29.9 34.0 18.4 28.2 13.9 11.2 20.9 15.0 104.5
## 5 42.5 29.9 34.0 21.5 29.4 15.2 11.6 20.7 14.9 107.5
## 6 43.3 27.0 31.5 19.6 31.3 14.0 11.5 18.8 13.9 119.8
## 7 43.5 30.0 34.0 21.9 31.7 16.1 12.5 20.8 15.6 123.5
## 8 44.4 29.8 33.2 21.8 28.8 15.1 11.9 21.0 14.6 120.4
## 9 43.5 26.5 32.1 15.5 27.5 14.1 11.2 18.9 13.2 111.0
## 10 42.0 28.0 34.0 22.5 28.0 15.6 12.0 21.1 15.0 119.5
## 11 40.3 29.0 33.0 20.1 30.3 13.4 10.4 19.4 14.5 117.1
## 12 43.7 29.0 31.3 20.5 29.7 15.0 11.7 20.9 16.0 123.5
## 13 47.4 29.6 35.7 20.8 31.4 16.1 11.3 21.5 15.4 116.5
## 14 40.3 27.5 31.4 21.7 28.0 13.3 10.3 18.8 13.2 113.0
## 15 41.0 26.8 32.2 21.9 28.6 14.9 10.6 17.8 14.0 107.5
## 16 45.0 27.0 33.2 21.7 30.6 13.7 11.1 20.7 14.0 112.0
## 17 39.9 30.0 34.5 21.0 29.4 15.6 11.9 21.2 16.0 112.2
## 18 43.0 26.5 30.3 19.3 30.0 14.8 11.2 19.7 14.7 120.0
## 19 43.1 28.6 33.4 22.2 29.5 14.9 12.2 20.8 14.8 109.0
## 20 43.6 29.3 34.4 20.2 32.6 15.4 10.9 20.7 15.5 118.5
## 21 42.0 27.5 30.7 21.3 32.0 13.1 11.1 19.2 13.9 116.0
## 22 43.8 28.0 33.3 20.0 32.0 15.0 11.5 20.4 14.4 111.0
## 23 42.3 26.4 31.2 18.0 30.9 14.6 10.8 18.6 13.8 117.7
## 24 42.7 29.9 35.0 21.8 32.8 14.3 11.2 19.8 14.1 123.9
## 25 44.8 27.8 32.2 18.3 31.5 15.2 11.6 19.4 14.7 120.6
## 26 46.0 30.1 34.5 20.2 31.1 16.4 13.3 22.2 14.9 129.5
## 27 45.4 31.8 35.2 20.2 32.3 14.6 10.5 20.2 15.3 115.0
## 28 40.5 28.3 33.4 19.2 28.8 14.6 11.1 20.8 14.5 116.0
## 29 39.4 25.5 30.2 17.6 27.7 13.0 10.2 18.9 13.2 107.8
## 30 40.2 27.2 31.7 18.1 26.5 13.3 10.1 18.6 13.2 100.2
## 31 44.2 30.3 34.7 19.4 30.0 14.9 11.0 19.1 15.8 113.0
## 32 41.0 23.6 30.2 22.9 28.0 14.3 11.2 18.2 14.0 117.9
## 33 44.0 31.0 35.3 19.2 31.0 15.2 11.4 21.2 15.1 112.5
## 34 41.6 32.0 35.3 23.6 27.0 15.5 11.3 20.9 15.0 110.5
## 35 41.0 25.1 31.9 20.8 27.9 13.6 10.8 18.8 12.9 112.0
## 36 41.5 24.5 30.5 17.7 26.7 13.3 10.8 18.6 14.0 104.0
## 37 41.1 27.8 31.4 19.0 31.5 14.5 11.9 18.5 13.0 114.8
## 38 38.8 27.2 31.6 18.5 25.5 13.4 10.8 19.0 14.0 108.0
## 39 36.2 27.5 30.4 18.7 28.0 13.6 10.8 19.0 15.4 111.2
## 40 42.1 27.5 32.4 18.2 28.0 16.2 12.0 21.0 16.4 118.3
## 41 40.3 29.4 32.9 23.7 31.5 14.6 11.3 19.8 15.2 115.2
## 42 41.7 27.1 32.6 21.6 28.0 14.1 11.5 19.7 13.8 129.9
## 43 37.8 27.1 31.5 18.5 27.3 14.6 10.8 19.5 14.9 112.9
## 44 39.2 26.1 30.8 19.4 29.9 14.3 11.2 20.0 16.0 112.2
## 45 41.5 30.8 33.3 19.4 30.6 14.8 11.3 20.2 16.0 117.1
## 46 42.5 27.8 33.5 20.6 30.2 15.9 12.8 22.4 16.3 118.7
## 47 39.4 26.1 34.4 20.4 27.3 15.1 10.6 20.0 15.3 109.2
## 48 43.6 33.1 33.5 21.6 33.1 15.6 12.0 20.7 16.5 128.1
## 49 38.9 24.9 28.7 19.7 26.8 14.2 10.2 18.0 14.4 113.3
## 50 37.6 24.4 28.0 18.0 26.4 14.2 10.6 17.3 13.4 108.4
## 51 39.4 28.3 30.6 20.2 28.7 15.0 11.5 18.4 14.4 118.7
## 52 38.5 26.1 30.8 20.6 30.8 15.1 11.4 19.8 14.2 126.3
## 53 40.1 27.8 33.1 19.2 31.3 15.4 11.5 20.6 15.4 124.2
## 54 40.3 28.0 32.0 20.9 31.7 14.8 10.6 19.4 15.0 126.7
## 55 37.6 26.6 29.9 17.3 25.6 12.8 10.0 17.0 13.0 103.3
## 56 38.3 25.2 30.2 17.0 26.4 13.2 10.4 18.8 13.0 101.2
## 57 39.7 28.6 32.1 19.1 27.1 13.4 10.0 18.2 14.8 104.3
## 58 42.2 29.0 33.7 22.5 30.4 15.6 12.0 19.8 16.2 113.2
## 59 41.1 30.4 35.1 23.2 32.6 15.5 11.6 21.5 15.4 121.9
## 60 40.5 29.3 33.7 19.6 29.8 13.8 11.7 19.7 14.4 113.1
## 61 41.5 28.6 30.4 20.8 26.9 14.8 11.2 20.7 16.5 108.5
## 62 43.4 32.4 36.4 20.3 32.1 15.6 12.0 20.8 16.3 113.9
## 63 43.5 26.0 31.6 19.1 30.9 14.3 11.4 19.5 14.6 112.6
## 64 41.3 27.1 32.4 17.5 27.6 14.1 10.8 20.2 15.5 110.2
## 65 40.3 29.5 33.3 18.4 26.2 14.0 11.0 19.4 14.8 108.7
## 66 36.3 29.2 33.0 20.0 29.0 14.1 11.7 20.4 14.3 104.0
## 67 39.9 28.3 32.0 18.3 31.4 13.5 11.4 18.9 14.4 115.2
## 68 39.8 28.8 33.0 19.7 28.7 12.4 10.7 18.5 13.2 111.9
## 69 43.5 33.2 34.0 23.9 34.3 15.8 12.0 18.6 13.2 127.0
## 70 41.2 26.6 30.6 19.5 28.0 13.1 10.4 19.0 13.8 111.2
## 71 44.0 28.4 32.0 22.5 29.7 14.9 10.9 21.0 14.8 122.0
## 72 41.8 28.5 31.6 21.6 31.5 13.3 10.3 18.9 14.3 114.5
## 73 42.9 27.5 30.3 18.9 29.6 12.6 10.4 19.2 13.8 109.5
## 74 38.7 24.6 28.5 18.3 29.8 14.0 11.2 18.9 13.6 110.8
## 75 41.4 26.4 32.3 18.6 31.3 14.9 11.5 18.9 14.6 118.8
## 76 39.6 27.5 30.2 19.2 28.9 13.5 10.4 19.3 14.2 108.0
## 77 40.5 27.5 32.3 19.4 28.8 12.6 10.6 18.4 14.0 114.3
## 78 34.1 28.1 30.1 21.8 25.8 12.9 9.9 18.6 12.3 105.4
## 79 43.5 28.8 34.0 20.6 29.0 14.3 10.5 19.8 14.2 115.0
## 80 44.1 29.2 35.3 23.6 30.9 15.8 12.5 20.2 15.2 119.5
## 81 42.2 32.6 36.6 22.4 34.5 14.1 11.1 18.2 13.9 130.0
## 82 42.2 30.1 31.4 21.2 29.7 14.0 11.6 21.6 14.1 113.3
## 83 43.0 26.5 31.6 20.6 29.5 13.4 10.4 18.8 13.6 113.2
## 84 39.8 28.7 33.3 19.3 29.2 13.5 11.6 19.5 14.6 106.9
## 85 37.7 29.7 32.7 20.2 28.8 13.3 11.1 18.3 13.2 113.8
## 86 39.6 27.9 33.3 20.2 29.5 12.6 10.7 18.5 12.9 117.3
## 87 43.2 26.3 30.5 19.7 30.6 14.4 12.3 20.2 13.6 124.2
## 88 44.3 28.2 32.2 21.2 31.8 14.2 11.6 20.0 14.4 123.0
## 89 43.3 28.2 33.0 19.4 31.6 13.8 11.1 17.8 13.2 117.8
## 90 42.8 27.5 31.5 19.2 31.8 14.1 11.1 19.1 14.7 118.8
## 91 41.5 30.0 33.4 19.1 29.4 14.8 11.0 19.8 13.8 112.0
## 92 42.0 27.6 32.2 19.7 29.4 13.9 10.0 18.7 13.8 113.0
## 93 41.2 27.1 29.8 20.1 31.0 12.9 11.6 18.8 13.5 116.0
## 94 43.8 29.5 31.2 18.2 29.5 13.1 10.3 19.1 13.2 112.8
## 95 46.2 31.0 36.0 25.0 33.1 14.6 12.0 20.9 15.1 125.0
## 96 40.4 28.6 31.4 19.8 27.6 13.9 10.1 20.0 13.4 108.3
## 97 40.8 27.1 29.4 17.8 29.4 13.3 10.4 18.5 12.8 108.2
## 98 43.9 27.0 33.5 22.3 31.0 13.2 10.4 19.1 13.1 113.0
## 99 44.2 27.9 32.0 21.6 32.9 14.3 11.0 21.1 14.9 115.0
## 100 41.6 28.0 35.0 24.2 31.0 13.4 11.2 20.6 14.4 123.0
## 101 38.1 30.1 33.2 21.6 31.3 14.2 12.3 19.2 15.2 120.2
## 102 42.0 28.0 33.0 18.1 28.4 14.3 11.1 20.2 15.2 114.0
## 103 37.0 27.3 31.1 18.2 25.0 13.2 10.5 18.7 13.4 102.9
## 104 41.6 27.5 32.0 18.1 29.5 13.8 10.7 19.0 13.9 112.5
## 105 40.1 19.4 28.0 17.1 26.8 13.0 10.6 16.9 12.6 104.5
## 106 38.7 25.2 28.8 19.1 25.6 13.0 10.2 17.9 13.5 111.3
## 107 37.4 29.9 33.5 22.3 30.8 14.4 11.5 20.5 16.8 117.2
## 108 41.7 28.0 32.9 19.4 29.7 14.6 11.0 19.5 15.3 112.8
## 109 38.0 27.1 28.3 18.2 25.9 13.8 11.0 18.9 14.8 104.8
## 110 40.5 24.9 29.7 19.0 30.2 14.4 11.8 19.5 14.9 117.7
## 111 35.6 28.5 29.4 17.7 25.2 14.0 10.8 19.1 15.0 107.7
## 112 43.6 30.2 32.4 21.8 33.1 15.2 11.3 19.8 15.2 125.2
## 113 37.6 24.4 28.3 17.7 24.7 12.9 10.8 18.0 14.3 109.1
## 114 41.1 31.7 34.2 22.8 34.0 13.8 11.8 19.4 15.4 122.6
## 115 42.1 30.6 34.0 22.1 30.6 15.0 11.4 20.2 15.4 117.3
## 116 40.5 28.3 32.4 19.4 27.8 13.4 11.0 19.0 14.5 109.1
## 117 40.9 28.5 31.3 21.1 29.7 14.3 11.7 19.0 15.6 116.7
## 118 43.0 30.6 33.8 23.3 35.3 15.6 12.0 21.6 16.4 124.9
## 119 40.5 27.8 31.1 21.8 30.6 15.0 11.6 20.4 15.2 118.6
## 120 41.9 25.4 30.2 14.4 26.8 12.6 9.8 18.8 13.6 108.8
## 121 42.1 28.5 33.1 20.2 30.6 15.6 12.2 19.7 15.6 121.9
## 122 43.8 29.2 32.6 18.7 30.4 14.6 11.7 20.0 15.2 117.3
## 123 42.1 28.5 31.7 19.4 28.0 14.0 11.3 19.0 14.4 115.0
## 124 43.4 32.0 36.2 23.5 35.6 16.1 12.6 23.0 16.3 134.8
## 125 38.7 26.8 31.5 21.4 27.8 13.8 10.8 18.2 13.3 113.8
## 126 39.6 28.7 32.4 18.2 28.3 15.2 11.8 19.6 14.8 119.4
## 127 43.4 30.6 32.9 21.6 28.3 15.0 12.0 20.5 17.2 117.9
## 128 40.5 29.7 31.7 22.1 32.6 15.2 11.3 21.2 15.2 127.3
## 129 40.3 30.4 34.2 21.1 34.0 13.6 12.0 19.2 13.8 118.8
## 130 44.2 30.6 33.8 22.1 32.4 15.3 11.5 20.9 16.5 122.4
## 131 41.3 26.8 32.2 21.4 31.1 13.6 11.0 19.1 15.0 111.5
## 132 39.8 25.6 31.3 23.5 32.0 14.0 11.2 21.2 16.4 113.2
## 133 41.3 29.0 32.2 25.2 30.8 14.4 11.0 19.7 15.8 115.3
## 134 38.9 27.5 32.9 22.5 33.3 14.6 11.0 20.5 15.3 122.5
## 135 41.1 25.6 29.9 23.3 25.2 14.1 10.7 19.0 15.1 114.4
## 136 41.5 30.6 35.8 21.1 28.0 15.0 11.8 21.0 15.6 112.8
## 137 38.5 27.8 31.7 19.7 26.4 13.1 11.0 18.4 14.8 112.2
## 138 39.4 29.7 33.1 23.0 30.4 14.2 11.6 20.4 15.0 119.4
## 139 40.9 26.1 27.5 20.2 28.0 13.2 10.4 18.6 14.8 108.4
## 140 41.1 23.0 29.4 21.8 30.6 15.0 10.8 19.3 14.5 122.4
## 141 43.6 28.0 32.4 27.5 33.5 14.6 11.7 21.4 15.1 128.8
## 142 39.8 29.0 34.9 22.5 28.3 14.3 11.7 19.8 15.4 118.0
## 143 42.1 27.8 31.7 20.2 28.7 14.3 11.5 19.6 15.6 116.5
## 144 41.1 27.1 33.8 24.9 29.4 14.4 12.4 18.0 15.1 120.4
## 145 44.2 30.4 36.5 21.6 31.5 15.4 11.6 20.4 15.4 123.1
## 146 38.9 26.8 31.5 20.4 29.0 13.6 10.8 18.9 15.2 117.2
## 147 40.1 28.7 32.2 18.0 29.4 15.2 11.8 20.7 15.4 113.0
## 148 38.7 26.8 31.5 18.0 27.8 12.9 10.4 18.0 14.3 109.4
## 149 35.6 26.4 30.8 19.2 29.4 14.6 11.5 19.6 15.3 105.7
## 150 40.5 26.8 30.6 21.4 32.4 15.0 11.8 20.4 15.8 119.7
## 151 41.1 25.4 32.0 21.6 28.7 14.3 12.4 19.6 14.3 118.0
## 152 38.7 26.1 29.2 18.2 24.9 13.6 10.4 17.6 14.2 104.3
## 153 38.9 27.1 30.4 20.4 28.7 14.8 11.7 19.4 14.6 114.1
## 154 39.4 29.7 33.1 22.3 31.5 15.6 12.0 19.5 14.8 119.0
## 155 37.6 27.8 32.2 20.2 31.3 14.6 11.0 21.5 15.8 113.8
## 156 39.8 25.9 31.3 19.4 29.2 14.3 11.2 18.7 14.3 118.0
## 157 40.3 27.3 30.4 20.4 29.0 15.0 11.3 19.1 14.6 116.0
## 158 40.3 25.2 29.2 18.7 28.5 13.2 10.2 18.9 14.3 108.3
## 159 43.8 30.4 34.9 24.0 33.3 15.4 11.6 17.8 14.6 129.2
## 160 41.1 27.8 32.9 18.0 26.8 14.6 11.2 19.5 15.8 109.2
## 161 41.7 30.6 34.7 24.9 32.0 14.9 10.8 18.9 14.1 124.3
## 162 45.4 30.8 35.6 20.9 29.7 15.2 10.8 18.6 15.0 122.6
## 163 43.0 30.8 34.7 22.1 32.2 16.0 13.2 19.5 16.1 122.4
## 164 41.1 25.4 30.4 19.2 29.9 14.8 10.6 18.7 15.2 114.9
## 165 41.1 29.2 31.5 19.7 29.9 14.8 11.0 18.0 15.0 111.1
## 166 45.2 32.2 36.0 22.5 33.5 15.8 11.3 20.5 14.8 126.0
## 167 39.8 34.7 34.7 23.5 30.2 14.8 10.6 20.6 14.3 115.5
## 168 39.6 28.7 32.0 20.2 32.9 14.3 11.5 19.6 15.1 124.7
## 169 42.3 30.2 34.4 25.4 32.2 15.2 10.7 18.8 14.8 125.3
## 170 40.9 29.0 32.2 20.2 29.2 13.8 10.4 18.4 14.4 118.2
## 171 39.8 28.0 34.2 23.3 31.5 14.0 11.0 19.7 13.8 124.3
## 172 42.3 27.1 31.7 17.3 29.4 14.4 11.6 21.2 15.8 118.3
## 173 42.3 26.8 32.0 25.4 28.7 15.0 11.2 18.4 14.4 122.3
## 174 40.5 28.7 30.4 22.1 29.2 14.3 11.2 18.5 14.5 116.5
## 175 42.7 28.5 36.7 23.7 30.8 15.8 12.9 19.3 16.0 123.0
## 176 43.6 30.8 33.3 20.4 29.7 14.3 10.9 19.6 15.4 117.6
## 177 42.5 31.3 33.1 19.4 32.0 14.8 11.3 20.1 15.5 117.5
## 178 41.3 30.8 33.3 22.5 28.3 13.0 10.5 19.7 13.4 117.3
## 179 41.3 24.7 35.4 22.5 30.2 14.8 11.3 20.4 16.0 114.5
## 180 42.3 26.6 33.3 23.3 28.7 14.2 11.1 19.5 14.3 122.9
## 181 36.9 25.9 31.7 19.9 27.3 14.8 10.6 19.4 14.3 115.5
## 182 40.3 28.5 35.1 19.2 32.0 14.8 12.0 21.2 16.2 116.1
## 183 40.1 26.4 32.0 21.4 32.6 14.8 12.0 20.0 15.2 121.6
## 184 43.2 31.3 34.0 23.0 32.6 15.7 11.5 20.5 15.2 131.6
## 185 45.0 29.0 33.3 25.4 30.8 15.4 11.0 18.8 15.0 124.5
## 186 42.1 29.7 35.6 23.5 31.3 14.3 11.0 18.2 14.8 115.6
## 187 40.9 28.7 33.5 23.7 30.2 15.8 11.6 19.3 16.8 117.0
## 188 41.3 27.3 32.2 20.2 28.3 13.8 11.4 18.0 13.0 108.6
## 189 42.1 29.7 32.9 24.9 30.8 15.3 11.5 19.2 14.6 120.2
## 190 45.2 29.7 33.8 19.9 32.0 15.5 11.8 19.6 16.1 116.2
## 191 42.1 31.1 34.9 22.1 31.3 15.2 11.0 19.0 14.4 115.4
## 192 41.2 29.8 32.2 22.4 29.8 15.6 11.2 19.6 14.8 120.0
## 193 41.7 29.2 33.8 19.2 29.7 13.8 10.7 18.6 14.2 110.3
## 194 43.0 29.4 33.8 24.7 31.5 14.2 11.2 19.4 14.2 119.7
## 195 41.7 28.5 33.8 23.5 32.9 15.6 12.0 18.4 16.3 123.5
## 196 38.0 29.7 34.0 22.8 27.3 13.1 10.8 17.3 15.5 107.8
## 197 41.9 27.8 33.3 19.0 28.7 15.1 11.3 19.2 14.9 118.7
## 198 40.7 28.0 35.3 19.4 31.7 14.4 10.7 18.6 14.4 120.7
## 199 41.3 29.7 34.7 22.8 32.0 15.5 11.2 18.4 15.6 118.7
## 200 40.9 28.7 32.9 21.6 30.6 15.0 11.0 18.7 13.8 118.0
## 201 41.7 26.8 32.0 19.7 27.8 14.0 10.5 18.4 14.6 105.0
## 202 41.3 31.1 34.9 26.4 30.2 15.0 11.6 18.8 15.8 110.6
## 203 41.1 27.8 32.0 21.1 30.6 14.8 11.4 17.6 14.2 123.1
## 204 43.4 27.3 34.7 19.9 31.3 14.0 11.2 20.2 14.3 121.0
## 205 40.7 27.8 34.0 21.1 29.4 15.6 12.0 21.2 16.4 111.7
## 206 42.5 25.2 30.6 20.9 30.4 15.3 11.4 18.9 13.8 111.0
## 207 40.3 28.3 30.6 18.2 29.2 12.9 10.6 20.2 14.2 112.0
## 208 39.8 27.8 32.9 22.3 29.7 16.6 11.8 20.8 15.3 115.9
## 209 39.8 28.3 30.4 19.7 30.2 12.9 11.0 18.6 12.7 121.0
## 210 40.5 29.9 34.9 21.4 32.9 14.5 11.7 20.4 15.0 123.6
## 211 41.1 26.8 32.4 20.2 31.1 14.4 11.8 20.4 15.2 120.0
## 212 42.5 29.4 34.2 23.5 34.7 15.1 11.8 21.8 15.8 128.7
## 213 42.5 29.4 34.4 19.9 34.0 14.5 11.0 21.3 14.4 124.5
## 214 38.5 24.4 30.4 18.0 29.9 14.3 10.1 18.3 13.2 112.4
## 215 43.8 29.2 35.6 19.9 28.3 14.8 12.8 20.7 14.3 112.2
## 216 41.5 29.0 33.3 21.4 32.4 15.3 11.0 20.6 15.0 127.0
## 217 40.3 27.8 33.5 20.6 29.9 15.3 11.2 20.4 13.8 115.3
## 218 40.3 30.2 32.2 20.6 29.2 14.5 11.5 20.0 16.9 111.5
## 219 42.1 27.1 32.2 20.2 30.6 13.7 10.8 18.9 14.3 118.3
## 220 42.1 31.3 35.1 20.6 31.1 16.7 12.0 21.0 15.1 118.0
## 221 38.9 29.4 33.3 24.2 33.5 14.2 12.8 20.6 15.5 127.0
## 222 44.4 24.0 30.2 20.6 32.0 14.2 11.2 20.2 14.5 124.4
## 223 40.9 24.4 28.7 18.7 29.0 14.3 11.4 17.8 14.6 112.4
## 224 42.1 28.7 33.5 25.2 29.4 14.3 11.5 18.0 13.8 125.8
## 225 43.0 27.1 33.5 25.2 31.5 15.2 11.8 19.3 15.1 131.7
## 226 38.0 27.1 32.2 20.6 28.0 13.9 10.3 19.1 14.0 117.3
## 227 37.1 24.2 30.6 17.7 27.8 13.8 11.0 17.8 14.6 107.6
## 228 41.1 24.0 29.4 21.6 30.8 13.6 11.7 18.8 14.6 122.4
## 229 40.9 24.4 30.6 19.7 29.9 13.8 11.5 19.0 15.1 117.0
## 230 38.9 27.1 31.7 21.6 29.7 15.0 12.2 20.5 15.2 119.0
## 231 40.9 28.3 34.0 23.7 28.5 14.3 12.0 19.0 14.3 123.5
## 232 39.2 26.8 34.0 23.7 30.8 14.2 11.2 19.1 14.5 121.4
## 233 41.7 26.6 31.5 19.9 29.9 14.6 11.2 19.2 14.8 112.3
## 234 39.4 26.6 31.7 24.0 31.1 13.8 11.8 20.0 15.8 113.2
## 235 40.1 26.4 32.0 21.8 30.2 15.8 12.4 20.7 15.8 121.7
## 236 38.9 25.6 32.9 21.1 29.0 15.6 10.6 20.2 14.8 116.6
## 237 38.9 26.4 31.7 21.6 27.8 14.4 11.3 19.6 14.8 117.8
## 238 41.7 26.4 31.1 20.2 28.3 14.6 10.2 18.4 15.3 118.0
## 239 43.2 26.8 32.6 22.1 32.9 15.4 12.0 20.5 16.8 131.1
## 240 40.5 29.2 33.5 23.7 31.1 15.2 11.3 20.0 16.1 121.4
## 241 39.2 23.5 29.9 19.7 29.0 15.4 11.5 19.0 14.8 116.3
## 242 40.9 25.4 32.0 20.6 30.2 16.1 11.5 19.3 15.8 121.8
## 243 41.7 27.3 31.5 21.8 29.7 14.9 11.8 18.9 13.6 118.2
## 244 43.8 32.2 38.0 25.4 32.0 16.0 10.7 21.0 16.8 126.3
## 245 41.9 28.0 33.1 26.4 29.9 15.6 11.5 21.2 15.9 121.0
## 246 43.0 27.8 34.2 21.4 31.5 14.3 11.1 21.0 14.8 123.1
## 247 41.5 28.5 33.5 19.7 29.4 14.5 10.5 19.4 15.3 114.9
## che.gi wai.gi nav.gi hip.gi thi.gi bic.gi for.gi kne.gi cal.gi ank.gi
## 1 89.5 71.5 74.5 93.5 51.5 32.5 26.0 34.5 36.5 23.5
## 2 97.0 79.0 86.5 94.8 51.5 34.4 28.0 36.5 37.5 24.5
## 3 97.5 83.2 82.9 95.0 57.3 33.4 28.8 37.0 37.3 21.9
## 4 97.0 77.8 78.8 94.0 53.0 31.0 26.2 37.0 34.8 23.0
## 5 97.5 80.0 82.5 98.5 55.4 32.0 28.4 37.7 38.6 24.4
## 6 99.9 82.5 80.1 95.3 57.5 33.0 28.0 36.6 36.1 23.5
## 7 106.9 82.0 84.0 101.0 60.9 42.4 32.3 40.1 40.3 23.6
## 8 102.5 76.8 80.5 98.0 56.0 34.1 28.0 39.2 36.7 22.5
## 9 91.0 68.5 69.0 89.5 50.0 33.0 26.0 35.5 35.0 22.0
## 10 93.5 77.5 81.5 99.8 59.8 36.5 29.2 38.3 38.6 22.2
## 11 97.7 81.9 81.0 98.4 60.5 34.6 27.9 38.9 40.1 23.2
## 12 99.5 82.6 82.5 95.0 58.5 38.5 30.4 39.0 38.4 24.3
## 13 103.0 85.0 94.5 103.0 59.0 33.5 29.0 40.5 40.0 26.0
## 14 99.6 85.6 89.2 98.0 59.1 35.6 29.0 35.8 36.0 21.5
## 15 101.5 78.0 89.5 95.0 57.0 36.0 29.0 34.5 35.0 22.0
## 16 104.1 82.0 84.0 97.0 56.0 34.5 29.5 39.0 35.7 24.0
## 17 100.0 88.3 93.5 105.0 65.8 37.0 28.8 40.9 41.7 24.2
## 18 93.8 73.6 74.9 90.1 54.1 31.2 26.9 36.4 35.6 22.0
## 19 98.5 78.5 86.0 94.5 55.0 34.5 28.5 38.0 36.5 23.0
## 20 104.0 87.3 88.0 101.1 59.5 37.0 30.5 39.8 42.0 26.5
## 21 100.0 92.0 91.0 98.0 57.5 32.0 27.6 37.5 35.2 21.0
## 22 100.0 80.0 83.7 99.5 57.0 37.0 30.0 37.5 35.5 23.0
## 23 99.0 74.5 75.9 92.2 53.4 31.2 26.9 36.2 33.3 23.5
## 24 101.0 90.6 89.6 101.2 59.5 37.0 28.3 35.4 40.6 22.9
## 25 101.6 81.4 81.6 98.8 61.3 39.4 31.9 38.5 41.2 22.8
## 26 108.8 89.5 89.5 106.0 59.5 37.5 30.1 39.9 41.5 23.5
## 27 100.0 85.0 94.5 105.0 62.0 35.5 28.5 38.0 40.0 23.5
## 28 88.0 73.5 77.7 97.0 56.3 32.5 27.8 39.0 38.2 23.5
## 29 88.7 75.8 83.0 89.0 52.6 31.2 26.5 37.0 37.4 21.5
## 30 84.5 74.0 81.0 93.5 50.5 27.5 24.8 34.0 32.8 21.0
## 31 93.6 77.5 82.1 95.0 56.5 32.8 26.2 37.6 36.3 21.0
## 32 105.0 74.0 72.0 90.0 54.2 34.1 28.6 36.2 36.6 22.4
## 33 90.9 74.0 78.8 96.4 51.8 29.8 27.0 36.4 34.6 22.9
## 34 91.2 82.0 89.5 100.0 57.5 32.8 28.0 40.7 40.1 24.3
## 35 98.4 73.0 83.0 95.4 56.3 36.4 27.5 37.2 34.5 21.8
## 36 85.0 70.5 84.0 90.0 50.0 29.0 26.0 36.0 34.5 21.5
## 37 97.2 75.0 77.2 91.3 49.5 31.0 26.1 36.3 35.1 21.0
## 38 91.5 72.1 79.2 91.0 54.9 29.5 24.5 36.1 37.2 22.9
## 39 91.2 78.8 78.0 93.2 55.8 31.9 27.4 36.4 35.1 23.0
## 40 101.1 77.5 78.0 97.0 55.0 37.7 29.9 38.3 39.6 23.3
## 41 104.3 91.5 93.2 103.9 62.0 36.3 29.0 36.7 39.4 23.1
## 42 110.8 84.9 83.0 102.6 66.4 42.3 30.9 37.0 37.7 22.6
## 43 96.3 79.1 78.3 97.1 60.1 35.5 28.8 36.9 38.2 23.4
## 44 102.7 77.9 77.9 90.7 56.7 35.4 28.3 35.6 35.5 22.9
## 45 103.9 91.7 89.4 101.8 61.0 35.7 29.4 37.7 40.0 22.2
## 46 105.6 86.6 87.3 103.9 63.2 37.8 29.7 39.0 40.2 24.3
## 47 95.8 84.7 84.0 101.4 60.0 35.0 28.5 38.4 37.9 23.2
## 48 111.2 90.3 93.5 108.7 66.9 40.2 32.4 39.2 40.1 25.7
## 49 100.0 79.7 87.1 98.4 61.1 36.3 28.6 34.5 36.1 21.4
## 50 91.6 73.1 75.4 86.5 50.6 30.8 26.1 31.7 33.6 20.3
## 51 108.0 79.8 82.5 94.8 58.3 39.8 29.6 34.2 38.1 21.1
## 52 109.6 81.6 86.5 100.9 61.7 39.5 31.7 38.8 36.5 22.7
## 53 105.7 76.8 83.4 98.0 56.8 37.9 30.9 35.9 38.3 23.4
## 54 109.1 85.9 90.4 100.9 61.3 40.1 30.0 36.8 38.6 21.9
## 55 88.8 73.3 77.9 85.7 46.9 30.5 24.8 31.1 30.5 19.0
## 56 86.1 69.9 67.4 84.1 50.8 31.5 26.6 32.8 36.3 20.0
## 57 91.3 72.7 83.2 91.4 51.2 27.8 26.0 34.8 34.7 21.1
## 58 100.6 82.7 83.5 98.0 55.8 33.1 28.0 37.9 39.1 23.2
## 59 105.5 90.1 89.2 104.5 62.7 36.3 29.6 38.4 42.4 25.3
## 60 97.5 82.9 83.6 95.8 52.6 34.5 27.0 35.2 35.2 21.4
## 61 94.4 77.9 79.0 91.7 57.1 31.2 27.5 36.6 37.5 21.6
## 62 99.6 92.5 96.2 103.4 58.5 34.5 28.4 38.4 38.0 22.4
## 63 98.1 77.8 77.2 90.0 52.4 33.2 26.4 34.2 36.0 21.8
## 64 93.6 72.7 77.3 91.7 51.9 32.1 27.4 33.5 33.8 21.1
## 65 93.4 75.0 79.2 94.0 53.8 34.2 27.9 36.1 36.2 22.0
## 66 92.0 76.0 83.0 93.0 54.5 29.5 26.0 37.0 34.5 22.8
## 67 99.2 82.7 84.2 93.0 56.6 32.4 27.6 35.8 36.3 21.8
## 68 97.6 80.0 85.7 97.4 57.8 33.8 28.6 36.2 37.4 22.0
## 69 108.8 107.1 107.2 108.3 67.0 39.6 30.6 40.0 39.6 24.6
## 70 91.9 76.2 78.1 90.0 52.0 30.7 25.8 34.8 32.6 21.0
## 71 105.2 90.2 88.6 100.2 60.8 35.7 29.4 39.2 39.1 24.5
## 72 98.3 89.4 87.4 97.7 54.8 31.0 26.0 36.4 35.6 21.6
## 73 92.5 80.9 78.5 96.0 59.0 31.5 26.3 36.1 39.0 21.2
## 74 92.5 73.5 76.4 92.0 53.1 30.6 27.1 36.0 36.0 23.8
## 75 101.6 70.9 76.7 95.3 56.0 36.0 28.6 36.0 34.0 22.0
## 76 94.6 76.1 78.0 86.3 52.4 28.6 23.9 34.5 37.9 22.7
## 77 92.5 81.0 85.2 92.5 54.7 32.3 26.8 35.8 37.6 21.1
## 78 88.2 72.0 72.0 85.5 50.2 28.6 24.8 34.9 35.1 20.1
## 79 91.0 76.8 80.0 94.5 54.6 33.2 28.0 37.5 35.6 22.1
## 80 106.0 86.0 92.0 103.0 60.6 34.0 29.8 38.8 39.5 23.6
## 81 115.0 98.5 106.6 116.5 67.8 35.8 27.2 38.0 41.2 23.3
## 82 100.0 79.0 82.5 98.5 62.1 34.0 28.8 39.6 40.8 25.9
## 83 94.7 77.5 80.5 92.0 54.2 30.9 26.6 36.5 35.8 21.3
## 84 92.5 75.2 80.2 91.6 49.6 29.2 26.1 36.2 35.7 22.1
## 85 96.7 82.0 82.2 92.7 54.6 32.0 27.1 35.6 36.4 20.7
## 86 97.4 79.6 80.8 95.0 54.2 32.6 27.4 36.5 38.0 21.6
## 87 106.7 75.2 77.8 94.5 57.4 36.7 29.9 38.1 36.0 22.6
## 88 106.2 88.6 88.3 100.5 63.4 36.9 29.4 38.4 38.6 23.1
## 89 103.6 81.5 83.3 91.8 55.0 33.0 27.8 35.4 36.5 21.9
## 90 98.3 79.9 82.4 87.5 54.4 33.5 27.3 36.8 37.9 22.3
## 91 87.8 73.5 77.5 94.9 53.5 34.3 28.5 36.5 35.2 22.0
## 92 99.8 80.3 80.8 93.0 55.4 33.3 28.0 36.0 37.8 20.3
## 93 104.6 81.5 85.0 92.0 54.1 33.0 28.0 35.1 35.2 21.1
## 94 86.5 74.0 76.5 91.3 53.5 30.5 26.1 36.6 38.6 21.2
## 95 110.0 104.0 99.0 111.7 63.2 37.5 29.0 41.2 39.3 24.6
## 96 93.2 76.2 83.8 92.8 55.2 31.2 26.2 36.8 37.7 22.7
## 97 90.0 76.5 77.7 91.2 54.2 33.1 27.2 35.5 35.3 21.5
## 98 98.4 81.0 80.5 96.2 56.0 32.0 27.4 37.0 35.5 24.0
## 99 107.2 88.8 86.8 100.0 61.0 34.6 27.9 38.0 39.4 23.2
## 100 108.3 94.0 98.0 108.2 66.8 35.6 27.3 39.5 43.0 25.3
## 101 105.7 83.4 86.5 101.1 61.3 34.7 29.4 39.4 41.8 24.0
## 102 88.5 77.0 79.0 93.0 51.7 33.5 27.9 38.4 38.5 22.5
## 103 79.3 75.4 78.0 88.6 50.0 25.6 22.7 33.8 32.5 21.2
## 104 90.9 80.3 80.8 92.8 53.9 32.5 28.0 36.5 35.0 21.0
## 105 90.2 68.0 67.0 81.5 49.5 27.0 23.6 34.0 34.5 20.9
## 106 91.6 80.6 78.0 91.3 55.0 30.7 25.3 35.5 34.0 20.8
## 107 105.2 88.6 94.7 94.7 58.3 36.9 28.8 40.3 39.7 26.3
## 108 97.0 81.1 88.2 93.9 53.5 33.7 28.6 35.0 37.3 23.1
## 109 85.3 70.8 84.9 89.4 55.8 28.7 25.5 38.5 34.2 21.3
## 110 99.6 73.3 82.1 89.3 55.4 36.3 32.5 34.3 34.3 22.3
## 111 94.3 75.9 79.1 92.6 54.4 33.2 27.9 34.8 35.5 23.0
## 112 111.8 86.2 93.5 96.3 59.1 36.3 28.0 38.3 34.7 23.0
## 113 94.3 75.0 82.2 88.0 53.8 36.3 28.9 34.5 33.5 23.0
## 114 106.1 101.0 99.7 105.5 60.2 38.6 30.3 39.5 39.4 25.6
## 115 102.0 90.0 92.3 102.3 60.0 34.6 29.7 38.2 38.3 23.7
## 116 94.8 75.0 79.6 91.6 49.4 30.0 26.5 31.7 30.2 16.4
## 117 102.9 75.9 77.0 93.4 55.0 35.2 28.7 37.0 37.7 24.5
## 118 115.8 96.0 95.9 103.6 62.2 38.2 30.1 41.2 39.4 25.1
## 119 105.4 84.0 90.4 94.8 57.6 38.7 30.2 38.6 38.2 22.8
## 120 87.1 67.1 80.4 85.9 46.8 30.3 25.4 32.7 32.1 20.0
## 121 104.1 82.5 90.1 98.4 57.7 37.9 31.6 37.8 38.3 24.6
## 122 95.8 83.7 84.2 98.4 56.2 33.7 28.0 38.0 39.6 25.8
## 123 98.6 76.7 85.8 93.3 56.0 35.7 27.6 34.7 34.6 20.6
## 124 118.7 105.2 105.0 115.5 69.9 39.4 32.1 42.2 47.7 27.0
## 125 100.9 90.6 93.3 97.7 58.0 34.8 28.0 34.1 35.8 22.2
## 126 98.5 85.7 92.9 98.6 55.5 35.3 28.7 39.3 35.9 23.0
## 127 96.9 82.5 90.8 94.9 54.4 32.8 28.7 39.2 37.0 27.5
## 128 110.7 94.7 92.0 101.3 60.1 37.2 30.9 40.5 40.0 24.2
## 129 108.0 105.2 103.4 108.1 60.5 38.0 30.2 36.9 37.7 21.6
## 130 109.0 86.0 90.2 98.0 59.5 40.0 31.2 38.3 39.0 25.8
## 131 104.2 90.9 92.7 100.2 51.8 30.1 26.8 38.1 36.4 23.2
## 132 100.5 92.4 92.4 97.0 50.9 32.9 29.0 37.7 37.7 23.4
## 133 105.7 96.5 98.2 97.4 54.3 31.9 28.5 37.7 39.3 24.5
## 134 112.4 98.4 101.5 107.9 67.4 39.2 30.5 42.6 40.7 25.3
## 135 96.2 76.7 83.5 93.9 50.4 32.1 27.7 36.1 32.9 23.2
## 136 97.5 94.8 98.2 98.6 48.3 31.1 27.0 37.7 36.8 24.6
## 137 90.9 80.1 79.8 91.3 56.2 32.9 27.2 36.2 33.0 23.0
## 138 108.4 97.4 103.7 105.3 55.6 36.6 28.4 38.2 36.6 22.9
## 139 94.3 73.7 74.5 88.2 52.3 29.6 26.2 35.2 36.2 21.2
## 140 109.1 76.1 90.1 93.3 51.7 37.0 30.6 36.8 37.7 23.6
## 141 115.0 95.6 101.9 107.9 64.6 37.1 30.0 41.8 39.6 24.7
## 142 104.4 101.0 98.9 103.3 54.4 38.1 29.8 39.7 41.8 25.0
## 143 100.1 84.5 84.5 94.4 54.7 33.9 28.6 38.5 37.6 25.0
## 144 108.4 98.0 101.8 101.5 56.9 38.2 29.9 37.7 39.2 24.9
## 145 107.3 101.6 103.8 110.0 57.8 34.9 28.9 40.3 40.0 23.7
## 146 101.8 87.8 90.2 98.4 55.6 33.1 28.4 38.2 37.7 24.5
## 147 94.5 80.0 85.0 95.0 52.0 31.5 26.5 36.9 36.4 22.9
## 148 91.7 81.8 82.9 98.3 56.3 31.0 25.7 35.0 33.0 22.0
## 149 95.9 84.4 86.8 99.0 55.0 30.5 26.4 36.1 38.4 21.3
## 150 110.5 85.0 83.5 95.7 59.0 39.2 29.9 37.9 37.7 23.8
## 151 104.0 90.0 86.0 96.0 52.5 33.5 29.1 36.0 36.9 23.0
## 152 86.8 72.9 73.4 89.5 51.0 29.8 24.8 32.6 33.1 22.1
## 153 106.7 81.0 80.2 93.7 54.8 35.5 30.6 36.9 37.3 22.7
## 154 102.5 86.5 89.0 97.0 57.0 34.0 28.4 38.0 37.0 22.5
## 155 103.0 93.9 98.6 103.6 60.5 34.4 28.5 40.9 40.8 24.6
## 156 95.0 77.0 78.0 93.0 52.0 32.6 28.4 34.4 34.4 20.0
## 157 99.0 75.0 75.0 90.0 50.6 32.0 27.3 33.8 34.0 22.0
## 158 89.7 80.6 80.8 90.0 55.5 28.9 25.0 34.6 37.4 23.0
## 159 111.5 100.5 107.3 109.5 61.8 37.4 31.6 41.0 39.7 25.4
## 160 94.1 81.2 84.0 91.6 51.5 33.0 27.0 35.2 35.5 23.1
## 161 118.3 103.4 106.2 108.5 60.5 35.4 29.7 42.3 40.8 24.8
## 162 106.5 90.3 101.1 101.6 57.2 35.4 28.6 40.4 37.8 24.9
## 163 110.4 98.0 98.0 99.6 56.7 36.4 29.2 40.9 42.1 26.1
## 164 102.3 86.5 87.7 91.9 55.0 35.0 28.9 38.3 37.8 24.0
## 165 98.5 77.9 87.3 90.8 50.8 35.0 28.4 35.5 35.0 21.0
## 166 111.6 89.1 95.1 104.8 62.7 37.9 31.2 41.1 41.2 27.7
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## 168 110.4 85.3 82.9 96.5 57.0 39.0 29.8 36.8 36.0 21.6
## 169 114.0 98.5 103.8 108.1 61.3 37.2 31.4 41.9 42.1 26.4
## 170 101.8 79.5 90.1 95.3 54.8 34.2 28.5 36.6 36.2 22.8
## 171 109.6 94.9 94.7 104.3 59.0 35.9 27.8 37.7 36.8 23.2
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## 173 104.3 88.4 89.6 98.8 54.8 35.5 29.7 37.7 37.0 23.7
## 174 104.2 84.2 84.0 93.2 55.0 33.0 25.4 35.6 36.4 22.8
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## 177 103.0 92.1 91.3 103.8 56.6 33.3 27.7 37.1 37.4 22.6
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## 179 99.0 88.7 91.0 100.0 57.5 34.0 28.3 40.9 38.8 26.4
## 180 100.3 83.9 89.4 103.9 59.8 36.1 29.4 37.0 36.5 24.3
## 181 100.2 79.5 88.7 95.3 52.5 34.6 25.8 35.6 35.1 21.8
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## 184 110.1 90.7 91.9 101.7 58.0 36.8 29.0 36.9 38.9 24.2
## 185 107.0 88.8 97.5 103.8 61.0 36.7 28.6 38.4 39.5 24.4
## 186 105.6 103.6 100.7 100.6 55.3 33.6 26.9 37.8 37.9 23.9
## 187 103.6 98.5 99.9 103.6 57.5 33.7 29.0 38.7 36.7 24.3
## 188 97.1 82.9 88.1 91.2 51.7 30.7 25.7 33.9 33.4 21.2
## 189 109.8 90.5 92.3 95.2 52.4 35.8 28.7 32.5 36.5 23.7
## 190 103.3 84.5 94.5 98.2 53.7 32.5 27.8 36.0 36.3 22.0
## 191 103.7 86.0 93.8 97.1 53.1 33.9 27.3 35.7 36.6 22.6
## 192 108.5 84.2 88.9 97.5 58.8 36.6 29.9 34.2 34.8 22.0
## 193 97.8 85.7 89.5 94.9 51.7 32.2 26.4 34.4 32.6 22.0
## 194 112.7 112.1 105.9 106.3 56.9 35.7 27.8 37.3 36.6 22.7
## 195 111.4 99.7 102.9 105.8 57.8 36.5 30.5 39.0 41.2 25.7
## 196 95.1 84.7 92.9 96.7 54.6 32.8 25.3 36.6 35.0 23.5
## 197 100.2 80.3 92.4 96.4 53.8 35.7 29.2 38.4 37.0 25.6
## 198 106.3 98.6 111.7 118.7 70.0 37.1 27.8 37.5 39.2 25.7
## 199 109.5 90.0 96.2 104.3 63.5 39.4 29.9 37.4 37.3 24.3
## 200 106.3 86.6 93.9 95.9 53.6 34.4 28.6 35.0 34.1 21.7
## 201 91.5 80.2 85.7 89.2 48.5 28.7 25.0 35.4 32.3 23.0
## 202 104.9 109.2 104.4 101.7 56.4 34.0 29.2 39.3 38.5 24.3
## 203 109.6 88.4 92.0 95.1 52.5 39.4 29.8 34.5 34.0 22.0
## 204 102.5 81.0 84.5 105.0 56.0 31.5 26.5 38.5 36.9 21.3
## 205 101.2 88.8 90.8 101.3 62.5 35.2 28.9 40.6 39.2 23.0
## 206 98.7 78.1 78.6 92.0 56.5 35.7 29.2 36.4 37.0 22.0
## 207 96.0 81.5 80.5 89.5 52.0 30.3 25.0 36.0 35.0 22.4
## 208 104.2 87.6 89.6 100.5 60.7 36.5 31.1 40.7 38.4 22.9
## 209 103.0 92.5 88.5 94.0 55.0 36.0 28.0 36.5 37.0 21.8
## 210 107.0 97.0 95.5 104.5 56.6 34.4 28.9 40.0 38.5 24.0
## 211 101.0 82.0 82.0 98.0 59.0 35.4 29.0 38.0 42.0 23.6
## 212 112.1 99.5 97.8 107.4 60.5 35.5 28.8 42.9 38.7 25.7
## 213 103.4 93.8 92.7 112.2 66.2 34.8 30.1 45.7 43.6 26.9
## 214 100.3 79.1 80.4 94.3 51.0 32.6 27.2 36.1 32.7 22.5
## 215 95.7 85.8 94.7 106.8 60.6 33.6 27.3 37.5 41.8 26.2
## 216 108.0 90.8 92.4 100.3 56.5 36.8 29.3 37.5 36.3 23.4
## 217 101.0 85.4 86.9 101.8 57.0 34.0 27.1 39.4 37.1 23.7
## 218 98.6 91.6 102.1 106.7 57.8 32.3 27.9 41.8 41.5 27.7
## 219 101.6 79.7 82.0 98.4 58.1 36.5 31.0 38.0 38.1 25.3
## 220 99.0 86.1 90.8 101.3 56.0 33.5 28.3 37.7 38.3 25.2
## 221 116.7 113.2 102.9 107.9 57.7 37.4 28.9 37.8 37.4 24.1
## 222 105.0 79.5 85.8 92.7 51.8 36.2 27.0 33.9 28.9 20.3
## 223 95.1 85.4 84.1 94.3 52.7 31.2 25.9 37.4 34.4 24.5
## 224 106.7 97.9 94.7 104.6 60.8 36.2 28.7 37.7 39.4 22.6
## 225 116.6 94.7 93.1 103.3 58.0 42.3 31.9 39.9 39.6 25.5
## 226 103.5 86.1 90.5 96.7 55.2 34.5 27.7 37.3 34.8 22.5
## 227 90.4 84.9 83.7 97.9 51.8 28.0 25.2 37.5 36.0 21.3
## 228 107.5 92.2 88.6 97.5 53.7 34.3 27.6 36.0 37.5 23.3
## 229 100.8 83.7 82.5 97.7 53.3 32.5 27.1 35.4 37.0 22.7
## 230 104.2 97.8 93.6 102.5 58.0 35.8 29.4 39.0 38.1 23.0
## 231 104.9 98.6 99.2 103.3 55.3 35.0 29.3 35.9 36.0 23.5
## 232 112.8 89.5 96.9 100.9 54.9 38.5 28.7 35.9 33.6 22.1
## 233 95.7 82.9 89.6 94.8 51.9 32.2 25.3 33.4 32.8 21.4
## 234 104.9 90.1 90.8 96.6 55.0 32.9 26.5 35.6 37.9 22.5
## 235 109.1 78.9 91.0 98.1 55.7 38.5 30.1 37.2 36.8 23.4
## 236 105.4 80.7 87.4 95.5 56.8 38.3 28.6 35.0 34.3 23.3
## 237 104.3 94.3 99.4 100.4 59.1 35.0 26.7 35.5 35.9 22.6
## 238 100.5 70.2 79.3 92.0 51.7 32.1 26.7 33.2 34.9 23.7
## 239 111.8 83.6 92.7 101.5 59.5 40.4 31.7 36.7 38.3 25.5
## 240 109.5 91.9 96.5 103.3 57.7 36.5 29.1 34.6 38.8 25.1
## 241 101.2 71.8 82.3 87.6 50.1 34.2 29.2 34.1 33.4 23.1
## 242 103.3 85.0 90.8 97.9 55.2 35.2 29.4 34.9 37.3 23.5
## 243 101.6 85.7 91.0 95.9 50.9 34.0 28.4 35.0 34.3 21.1
## 244 103.1 96.5 99.0 111.8 62.3 34.8 27.5 41.7 37.0 24.3
## 245 104.6 82.4 85.7 99.9 63.3 38.6 32.0 38.4 39.8 25.4
## 246 104.3 86.3 87.8 103.3 59.7 36.4 30.4 39.3 42.0 27.7
## 247 95.9 83.2 88.0 100.6 57.8 34.0 28.2 36.3 39.6 25.2
## wri.gi age wgt hgt sex
## 1 16.5 21 65.6 174.0 1
## 2 17.0 23 71.8 175.3 1
## 3 16.9 28 80.7 193.5 1
## 4 16.6 23 72.6 186.5 1
## 5 18.0 22 78.8 187.2 1
## 6 16.9 21 74.8 181.5 1
## 7 18.8 26 86.4 184.0 1
## 8 18.0 27 78.4 184.5 1
## 9 16.5 23 62.0 175.0 1
## 10 16.9 21 81.6 184.0 1
## 11 16.2 23 76.6 180.0 1
## 12 18.2 22 83.6 177.8 1
## 13 18.0 20 90.0 192.0 1
## 14 16.6 26 74.6 176.0 1
## 15 16.5 23 71.0 174.0 1
## 16 17.5 22 79.6 184.0 1
## 17 17.8 30 93.8 192.7 1
## 18 17.1 22 70.0 171.5 1
## 19 18.5 29 72.4 173.0 1
## 20 18.8 22 85.9 176.0 1
## 21 17.3 22 78.8 176.0 1
## 22 18.0 20 77.8 180.5 1
## 23 16.0 22 66.2 172.7 1
## 24 17.0 24 86.4 176.0 1
## 25 18.2 26 81.8 173.5 1
## 26 19.0 24 89.6 178.0 1
## 27 16.7 21 82.8 180.3 1
## 28 17.8 24 76.4 180.3 1
## 29 16.5 23 63.2 164.5 1
## 30 15.6 19 60.9 173.0 1
## 31 16.6 23 74.8 183.5 1
## 32 17.2 25 70.0 175.5 1
## 33 17.0 23 72.4 188.0 1
## 34 17.2 23 84.1 189.2 1
## 35 16.9 23 69.1 172.8 1
## 36 17.0 20 59.5 170.0 1
## 37 17.0 22 67.2 182.0 1
## 38 16.0 24 61.3 170.0 1
## 39 16.8 22 68.6 177.8 1
## 40 18.7 24 80.1 184.2 1
## 41 17.6 21 87.8 186.7 1
## 42 17.5 23 84.7 171.4 1
## 43 16.7 24 73.4 172.7 1
## 44 17.0 35 72.1 175.3 1
## 45 17.9 29 82.6 180.3 1
## 46 18.9 25 88.7 182.9 1
## 47 16.8 23 84.1 188.0 1
## 48 17.7 20 94.1 177.2 1
## 49 16.3 25 74.9 172.1 1
## 50 15.8 29 59.1 167.0 1
## 51 16.9 23 75.6 169.5 1
## 52 17.9 23 86.2 174.0 1
## 53 17.3 36 75.3 172.7 1
## 54 16.4 25 87.1 182.2 1
## 55 15.0 24 55.2 164.1 1
## 56 15.8 20 57.0 163.0 1
## 57 15.3 52 61.4 171.5 1
## 58 17.7 50 76.8 184.2 1
## 59 17.5 46 86.8 174.0 1
## 60 16.5 51 72.2 174.0 1
## 61 16.1 28 71.6 177.0 1
## 62 17.2 48 84.8 186.0 1
## 63 17.5 35 68.2 167.0 1
## 64 16.9 23 66.1 171.8 1
## 65 17.5 23 72.0 182.0 1
## 66 17.4 62 64.6 167.0 1
## 67 16.7 21 74.8 177.8 1
## 68 17.2 26 70.0 164.5 1
## 69 17.7 33 101.6 192.0 1
## 70 16.1 36 63.2 175.5 1
## 71 17.9 41 79.1 171.2 1
## 72 16.6 40 78.9 181.6 1
## 73 15.6 27 67.7 167.4 1
## 74 17.5 27 66.0 181.1 1
## 75 17.0 23 68.2 177.0 1
## 76 15.8 31 63.9 174.5 1
## 77 15.8 26 72.0 177.5 1
## 78 15.4 23 56.8 170.5 1
## 79 16.3 24 74.5 182.4 1
## 80 18.0 24 90.9 197.1 1
## 81 16.0 34 93.0 180.1 1
## 82 18.0 21 80.9 175.5 1
## 83 16.5 25 72.7 180.6 1
## 84 16.7 34 68.0 184.4 1
## 85 16.2 31 70.9 175.5 1
## 86 16.6 40 72.5 180.6 1
## 87 17.3 21 72.5 177.0 1
## 88 17.3 33 83.4 177.1 1
## 89 16.4 25 75.5 181.6 1
## 90 16.6 29 73.0 176.5 1
## 91 17.0 27 70.2 175.0 1
## 92 16.5 44 73.4 174.0 1
## 93 16.3 26 70.5 165.1 1
## 94 16.0 22 68.9 177.0 1
## 95 18.6 37 102.3 192.0 1
## 96 16.2 38 68.4 176.5 1
## 97 16.2 20 65.9 169.4 1
## 98 16.5 21 75.7 182.1 1
## 99 17.0 24 84.5 179.8 1
## 100 17.7 45 87.7 175.3 1
## 101 17.8 25 86.4 184.9 1
## 102 17.5 22 73.2 177.3 1
## 103 14.6 29 53.9 167.4 1
## 104 16.4 37 72.0 178.1 1
## 105 16.0 20 55.5 168.9 1
## 106 16.4 20 58.4 157.2 1
## 107 18.1 32 83.2 180.3 1
## 108 16.7 23 72.7 170.2 1
## 109 15.6 25 64.1 177.8 1
## 110 18.4 27 72.3 172.7 1
## 111 16.3 21 65.0 165.1 1
## 112 16.1 27 86.4 186.7 1
## 113 17.1 25 65.0 165.1 1
## 114 18.3 38 88.6 174.0 1
## 115 18.0 44 84.1 175.3 1
## 116 16.1 27 66.8 185.4 1
## 117 17.3 37 75.5 177.8 1
## 118 17.7 28 93.2 180.3 1
## 119 18.2 33 82.7 180.3 1
## 120 15.3 25 58.0 177.8 1
## 121 18.1 21 79.5 177.8 1
## 122 17.6 30 78.6 177.8 1
## 123 16.0 26 71.8 177.8 1
## 124 19.2 27 116.4 177.8 1
## 125 16.3 33 72.2 163.8 1
## 126 17.4 29 83.6 188.0 1
## 127 17.9 27 85.5 198.1 1
## 128 17.6 34 90.9 175.3 1
## 129 18.3 42 85.9 166.4 1
## 130 19.5 29 89.1 190.5 1
## 131 16.3 41 75.0 166.4 1
## 132 17.0 43 77.7 177.8 1
## 133 17.1 43 86.4 179.7 1
## 134 17.5 29 90.9 172.7 1
## 135 17.0 27 73.6 190.5 1
## 136 18.4 62 76.4 185.4 1
## 137 16.8 33 69.1 168.9 1
## 138 18.1 45 84.5 167.6 1
## 139 16.2 30 64.5 175.3 1
## 140 18.5 20 69.1 170.2 1
## 141 18.2 22 108.6 190.5 1
## 142 19.6 51 86.4 177.8 1
## 143 17.7 34 80.9 190.5 1
## 144 17.6 44 87.7 177.8 1
## 145 18.1 46 94.5 184.2 1
## 146 17.1 34 80.2 176.5 1
## 147 17.0 32 72.0 177.8 1
## 148 15.5 28 71.4 180.3 1
## 149 16.4 31 72.7 171.4 1
## 150 17.1 29 84.1 172.7 1
## 151 18.0 42 76.8 172.7 1
## 152 16.3 29 63.6 177.8 1
## 153 16.8 31 80.9 177.8 1
## 154 17.4 30 80.9 182.9 1
## 155 17.1 27 85.5 170.2 1
## 156 16.5 25 68.6 167.6 1
## 157 17.0 24 67.7 175.3 1
## 158 16.4 33 66.4 165.1 1
## 159 18.4 45 102.3 185.4 1
## 160 16.3 37 70.5 181.6 1
## 161 17.1 44 95.9 172.7 1
## 162 17.9 34 84.1 190.5 1
## 163 19.5 55 87.3 179.1 1
## 164 17.3 43 71.8 175.3 1
## 165 16.6 24 65.9 170.2 1
## 166 18.1 22 95.9 193.0 1
## 167 16.8 38 91.4 171.4 1
## 168 17.3 24 81.8 177.8 1
## 169 17.9 29 96.8 177.8 1
## 170 17.3 25 69.1 167.6 1
## 171 16.5 37 82.7 167.6 1
## 172 18.2 30 75.5 180.3 1
## 173 18.3 26 79.5 182.9 1
## 174 15.9 35 73.6 176.5 1
## 175 18.8 29 91.8 186.7 1
## 176 18.0 30 84.1 188.0 1
## 177 17.1 37 85.9 188.0 1
## 178 17.1 34 81.8 177.8 1
## 179 18.1 28 82.5 174.0 1
## 180 18.7 27 80.5 177.8 1
## 181 16.3 32 70.0 171.4 1
## 182 17.3 28 81.8 185.4 1
## 183 17.5 22 84.1 185.4 1
## 184 17.9 44 90.5 188.0 1
## 185 16.8 25 91.4 188.0 1
## 186 17.1 49 89.1 182.9 1
## 187 17.7 54 85.0 176.5 1
## 188 16.7 49 69.1 175.3 1
## 189 17.5 60 73.6 175.3 1
## 190 17.7 42 80.5 188.0 1
## 191 17.5 52 82.7 188.0 1
## 192 18.1 23 86.4 175.3 1
## 193 16.4 33 67.7 170.5 1
## 194 16.4 46 92.7 179.1 1
## 195 18.8 43 93.6 177.8 1
## 196 17.1 56 70.9 175.3 1
## 197 17.9 21 75.0 182.9 1
## 198 15.9 18 93.2 170.8 1
## 199 17.2 21 93.2 188.0 1
## 200 17.1 45 77.7 180.3 1
## 201 16.2 22 61.4 177.8 1
## 202 17.4 55 94.1 185.4 1
## 203 17.0 42 75.0 168.9 1
## 204 16.6 29 83.6 185.4 1
## 205 17.5 40 85.5 180.3 1
## 206 17.2 24 73.9 174.0 1
## 207 16.7 62 66.8 167.6 1
## 208 17.9 26 87.3 182.9 1
## 209 17.5 35 72.3 160.0 1
## 210 17.0 37 88.6 180.3 1
## 211 17.5 34 75.5 167.6 1
## 212 17.1 25 101.4 186.7 1
## 213 17.0 30 91.1 175.3 1
## 214 17.9 32 67.3 175.3 1
## 215 17.8 27 77.7 175.9 1
## 216 16.7 42 81.8 175.3 1
## 217 16.6 44 75.5 179.1 1
## 218 18.1 46 84.5 181.6 1
## 219 17.7 19 76.6 177.8 1
## 220 17.4 43 85.0 182.9 1
## 221 17.1 28 102.5 177.8 1
## 222 15.8 39 77.3 184.2 1
## 223 17.1 30 71.8 179.1 1
## 224 17.5 36 87.9 176.5 1
## 225 18.9 48 94.3 188.0 1
## 226 16.4 48 70.9 174.0 1
## 227 15.1 53 64.5 167.6 1
## 228 16.7 45 77.3 170.2 1
## 229 16.6 39 72.3 167.6 1
## 230 17.0 43 87.3 188.0 1
## 231 18.1 65 80.0 174.0 1
## 232 17.1 45 82.3 176.5 1
## 233 16.3 37 73.6 180.3 1
## 234 17.8 55 74.1 167.6 1
## 235 19.6 33 85.9 188.0 1
## 236 17.3 25 73.2 180.3 1
## 237 17.3 35 76.3 167.6 1
## 238 18.4 28 65.9 183.0 1
## 239 19.4 26 90.9 183.0 1
## 240 18.9 43 89.1 179.1 1
## 241 18.3 30 62.3 170.2 1
## 242 17.3 26 82.7 177.8 1
## 243 16.3 51 79.1 179.1 1
## 244 16.7 30 98.2 190.5 1
## 245 18.1 24 84.1 177.8 1
## 246 18.4 35 83.2 180.3 1
## 247 17.0 37 83.2 180.3 1- Make a histogram of men’s heights and a histogram of women’s heights. How would you compare the various aspects of the two distributions?
hist(mdims$hgt)
hist(fdims$hgt)
fivenum(mdims$hgt)## [1] 157.20 172.90 177.80 182.65 198.10fivenum(fdims$hgt)## [1] 147.2 160.0 164.5 169.5 182.9#The Tukey five-number summary shows the median, range and inter-quartile range for the men's and women's heights.The normal distribution
In your description of the distributions, did you use words like bell-shaped or normal? It’s tempting to say so when faced with a unimodal symmetric distribution.
To see how accurate that description is, we can plot a normal distribution curve on top of a histogram to see how closely the data follow a normal distribution. This normal curve should have the same mean and standard deviation as the data. We’ll be working with women’s heights, so let’s store them as a separate object and then calculate some statistics that will be referenced later.
fhgtmean <- mean(fdims$hgt)
fhgtsd <- sd(fdims$hgt)Next we make a density histogram to use as the backdrop and use the lines function to overlay a normal probability curve. The difference between a frequency histogram and a density histogram is that while in a frequency histogram the heights of the bars add up to the total number of observations, in a density histogram the areas of the bars add up to 1. The area of each bar can be calculated as simply the height times the width of the bar. Using a density histogram allows us to properly overlay a normal distribution curve over the histogram since the curve is a normal probability density function.Frequency and density histograms both display the same exact shape; they only differ in their y-axis. You can verify this by comparing the frequency histogram you constructed earlier and the density histogram created by the commands below.
hist(fdims$hgt, probability = TRUE)
x <- 140:190
y <- dnorm(x = x, mean = fhgtmean, sd = fhgtsd)
lines(x = x, y = y, col = "blue")
After plotting the density histogram with the first command, we create the x- and y-coordinates for the normal curve. We chose the x range as 140 to 190 in order to span the entire range of fheight. To create y, we use dnorm to calculate the density of each of those x-values in a distribution that is normal with mean fhgtmean and standard deviation fhgtsd. The final command draws a curve on the existing plot (the density histogram) by connecting each of the points specified by x and y. The argument col simply sets the color for the line to be drawn. If we left it out, the line would be drawn in black.
The top of the curve is cut off because the limits of the x- and y-axes are set to best fit the histogram. To adjust the y-axis you can add a third argument to the histogram function: ylim = c(0, 0.06).
- Based on the this plot, does it appear that the data follow a nearly normal distribution?
Yes, based on this plot, the data does appear to follow a nearly normal distribution.
Evaluating the normal distribution
Eyeballing the shape of the histogram is one way to determine if the data appear to be nearly normally distributed, but it can be frustrating to decide just how close the histogram is to the curve. An alternative approach involves constructing a normal probability plot, also called a normal Q-Q plot for “quantile-quantile”.
qqnorm(fdims$hgt)
qqline(fdims$hgt)
A data set that is nearly normal will result in a probability plot where the points closely follow the line. Any deviations from normality leads to deviations of these points from the line. The plot for female heights shows points that tend to follow the line but with some errant points towards the tails. We’re left with the same problem that we encountered with the histogram above: how close is close enough?
A useful way to address this question is to rephrase it as: what do probability plots look like for data that I know came from a normal distribution? We can answer this by simulating data from a normal distribution using rnorm.
sim_norm <- rnorm(n = length(fdims$hgt), mean = fhgtmean, sd = fhgtsd)The first argument indicates how many numbers you’d like to generate, which we specify to be the same number of heights in the fdims data set using the length function. The last two arguments determine the mean and standard deviation of the normal distribution from which the simulated sample will be generated. We can take a look at the shape of our simulated data set, sim_norm, as well as its normal probability plot.
- Make a normal probability plot of
sim_norm. Do all of the points fall on the line? How does this plot compare to the probability plot for the real data?
qqnorm(sim_norm)
qqline(sim_norm)
Even better than comparing the original plot to a single plot generated from a normal distribution is to compare it to many more plots using the following function. It may be helpful to click the zoom button in the plot window.
qqnormsim(fdims$hgt)
- Does the normal probability plot for
fdims$hgtlook similar to the plots created for the simulated data? That is, do plots provide evidence that the female heights are nearly normal?
Yes, the normal probability plot for female heights does look similar to the plots created for the simulated data, which provides further evidence that the female heights are nearly normal.
- Using the same technique, determine whether or not female weights appear to come from a normal distribution.
#fdims$wgt
fivenum(fdims$wgt)## [1] 42.0 54.5 59.0 65.7 105.2fwgtmean<-mean(fdims$wgt)
fwgtsd<-sd(fdims$wgt)
hist(fdims$wgt, probability = TRUE)
x <- 40:110
y <- dnorm(x = x, mean = fwgtmean, sd = fwgtsd)
lines(x = x, y = y, col = "blue")
qqnorm(fdims$wgt)
qqline(fdims$wgt)
sim_norm_fwgt <- rnorm(n = length(fdims$wgt), mean = fwgtmean, sd = fwgtsd)
#qqnorm(sim_norm_fwgt)
#qqline(sim_norm_fwgt)
qqnormsim(fdims$wgt)
cat("Based on the above, it seems that female weights do not come from a nearly normal distribution. They seem to come from a right-skewed distribution.")## Based on the above, it seems that female weights do not come from a nearly normal distribution. They seem to come from a right-skewed distribution.Normal probabilities
Okay, so now you have a slew of tools to judge whether or not a variable is normally distributed. Why should we care?
It turns out that statisticians know a lot about the normal distribution. Once we decide that a random variable is approximately normal, we can answer all sorts of questions about that variable related to probability. Take, for example, the question of, “What is the probability that a randomly chosen young adult female is taller than 6 feet (about 182 cm)?” (The study that published this data set is clear to point out that the sample was not random and therefore inference to a general population is not suggested. We do so here only as an exercise.)
If we assume that female heights are normally distributed (a very close approximation is also okay), we can find this probability by calculating a Z score and consulting a Z table (also called a normal probability table). In R, this is done in one step with the function pnorm.
1 - pnorm(q = 182, mean = fhgtmean, sd = fhgtsd)## [1] 0.004434387Note that the function pnorm gives the area under the normal curve below a given value, q, with a given mean and standard deviation. Since we’re interested in the probability that someone is taller than 182 cm, we have to take one minus that probability.
Assuming a normal distribution has allowed us to calculate a theoretical probability. If we want to calculate the probability empirically, we simply need to determine how many observations fall above 182 then divide this number by the total sample size.
sum(fdims$hgt > 182) / length(fdims$hgt)## [1] 0.003846154Although the probabilities are not exactly the same, they are reasonably close. The closer that your distribution is to being normal, the more accurate the theoretical probabilities will be.
- Write out two probability questions that you would like to answer; one regarding female heights and one regarding female weights. Calculate those probabilities using both the theoretical normal distribution as well as the empirical distribution (four probabilities in all). Which variable, height or weight, had a closer agreement between the two methods?
What is the probability that a young adult female is shorter than 150 cm?
What is the probability that a young adult female is heavier than 65 kgs?
pnorm(q = 150, mean = fhgtmean, sd = fhgtsd)## [1] 0.01152955sum(fdims$hgt < 150) / length(fdims$hgt)## [1] 0.011538461 - pnorm(q = 65, mean = fwgtmean, sd = fwgtsd)## [1] 0.3236397sum(fdims$wgt > 65) / length(fdims$wgt)## [1] 0.2615385As expected, female heights tend to follow the theoretical normal distribution more closely, while female weights show a greater deviation, on account of the right-skewed nature of the data. The actual data shows a longer right tail than the theoretical normal distribution.
On Your Own
Now let’s consider some of the other variables in the body dimensions data set. Using the figures at the end of the exercises, match the histogram to its normal probability plot. All of the variables have been standardized (first subtract the mean, then divide by the standard deviation), so the units won’t be of any help. If you are uncertain based on these figures, generate the plots in R to check.
a. The histogram for female biiliac (pelvic) diameter (
bii.di) belongs to normal probability plot letter B__.b. The histogram for female elbow diameter (
elb.di) belongs to normal probability plot letter C.c. The histogram for general age (
age) belongs to normal probability plot letter D__.d. The histogram for female chest depth (
che.de) belongs to normal probability plot letter A__.Note that normal probability plots C and D have a slight stepwise pattern.
Why do you think this is the case?This is probably because these two measurements are more discrete in nature rather than continuous. For example, age is expressed in years, so the data “jumps” from year to year.
As you can see, normal probability plots can be used both to assess normality and visualize skewness. Make a normal probability plot for female knee diameter (
kne.di). Based on this normal probability plot, is this variable left skewed, symmetric, or right skewed? Use a histogram to confirm your findings.
#fdims$kne.di
fivenum(fdims$kne.di)## [1] 15.7 17.3 18.0 18.7 24.3fknemean<-mean(fdims$kne.di)
fknesd<-sd(fdims$kne.di)
hist(fdims$kne.di, probability = TRUE)
x <- 10:28
y <- dnorm(x = x, mean = fknemean, sd = fknesd)
lines(x = x, y = y, col = "blue")
qqnorm(fdims$kne.di)
qqline(fdims$kne.di)
sim_norm_fkne <- rnorm(n = length(fdims$kne.di), mean = fknemean, sd = fknesd)
#qqnorm(sim_norm_fkne)
#qqline(sim_norm_fkne)
qqnormsim(fdims$kne.di)
cat("Based on the above, it seems that female knee widths do not come from a nearly normal distribution. They seem to come from a right-skewed distribution.")## Based on the above, it seems that female knee widths do not come from a nearly normal distribution. They seem to come from a right-skewed distribution.
histQQmatch
This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel from a lab written by Mark Hansen of UCLA Statistics.