- Take the Galton dataset and find the mean, standard deviation and
correlation between the parental and child heights.
library(UsingR)
## Warning: package 'UsingR' was built under R version 4.2.3
## Loading required package: MASS
## Loading required package: HistData
## Warning: package 'HistData' was built under R version 4.2.3
## Loading required package: Hmisc
## Warning: package 'Hmisc' was built under R version 4.2.3
##
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
##
## format.pval, units
data("galton")
head(galton)
## child parent
## 1 61.7 70.5
## 2 61.7 68.5
## 3 61.7 65.5
## 4 61.7 64.5
## 5 61.7 64.0
## 6 62.2 67.5
summary(galton)
## child parent
## Min. :61.70 Min. :64.00
## 1st Qu.:66.20 1st Qu.:67.50
## Median :68.20 Median :68.50
## Mean :68.09 Mean :68.31
## 3rd Qu.:70.20 3rd Qu.:69.50
## Max. :73.70 Max. :73.00
cor(galton )
## child parent
## child 1.0000000 0.4587624
## parent 0.4587624 1.0000000
- Center the parent and child variables and verify that the centered
variable means are 0.
galton
## child parent
## 1 61.7 70.5
## 2 61.7 68.5
## 3 61.7 65.5
## 4 61.7 64.5
## 5 61.7 64.0
## 6 62.2 67.5
## 7 62.2 67.5
## 8 62.2 67.5
## 9 62.2 66.5
## 10 62.2 66.5
## 11 62.2 66.5
## 12 62.2 64.5
## 13 63.2 70.5
## 14 63.2 69.5
## 15 63.2 68.5
## 16 63.2 68.5
## 17 63.2 68.5
## 18 63.2 68.5
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## 39 63.2 64.5
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## 43 63.2 64.0
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## 45 64.2 69.5
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## 763 70.2 68.5
## 764 70.2 68.5
## 765 70.2 67.5
## 766 70.2 67.5
## 767 70.2 67.5
## 768 70.2 67.5
## 769 70.2 67.5
## 770 70.2 67.5
## 771 70.2 67.5
## 772 70.2 67.5
## 773 70.2 67.5
## 774 70.2 67.5
## 775 70.2 67.5
## 776 70.2 67.5
## 777 70.2 67.5
## 778 70.2 67.5
## 779 70.2 67.5
## 780 70.2 67.5
## 781 70.2 67.5
## 782 70.2 67.5
## 783 70.2 67.5
## 784 70.2 66.5
## 785 70.2 66.5
## 786 70.2 66.5
## 787 70.2 66.5
## 788 70.2 65.5
## 789 70.2 65.5
## 790 70.2 65.5
## 791 70.2 65.5
## 792 70.2 65.5
## 793 71.2 72.5
## 794 71.2 72.5
## 795 71.2 71.5
## 796 71.2 71.5
## 797 71.2 71.5
## 798 71.2 71.5
## 799 71.2 70.5
## 800 71.2 70.5
## 801 71.2 70.5
## 802 71.2 70.5
## 803 71.2 70.5
## 804 71.2 70.5
## 805 71.2 70.5
## 806 71.2 69.5
## 807 71.2 69.5
## 808 71.2 69.5
## 809 71.2 69.5
## 810 71.2 69.5
## 811 71.2 69.5
## 812 71.2 69.5
## 813 71.2 69.5
## 814 71.2 69.5
## 815 71.2 69.5
## 816 71.2 69.5
## 817 71.2 69.5
## 818 71.2 69.5
## 819 71.2 69.5
## 820 71.2 69.5
## 821 71.2 69.5
## 822 71.2 69.5
## 823 71.2 69.5
## 824 71.2 69.5
## 825 71.2 69.5
## 826 71.2 68.5
## 827 71.2 68.5
## 828 71.2 68.5
## 829 71.2 68.5
## 830 71.2 68.5
## 831 71.2 68.5
## 832 71.2 68.5
## 833 71.2 68.5
## 834 71.2 68.5
## 835 71.2 68.5
## 836 71.2 68.5
## 837 71.2 68.5
## 838 71.2 68.5
## 839 71.2 68.5
## 840 71.2 68.5
## 841 71.2 68.5
## 842 71.2 68.5
## 843 71.2 68.5
## 844 71.2 67.5
## 845 71.2 67.5
## 846 71.2 67.5
## 847 71.2 67.5
## 848 71.2 67.5
## 849 71.2 67.5
## 850 71.2 67.5
## 851 71.2 67.5
## 852 71.2 67.5
## 853 71.2 67.5
## 854 71.2 67.5
## 855 71.2 65.5
## 856 71.2 65.5
## 857 72.2 73.0
## 858 72.2 72.5
## 859 72.2 72.5
## 860 72.2 72.5
## 861 72.2 72.5
## 862 72.2 72.5
## 863 72.2 72.5
## 864 72.2 72.5
## 865 72.2 71.5
## 866 72.2 71.5
## 867 72.2 71.5
## 868 72.2 71.5
## 869 72.2 71.5
## 870 72.2 71.5
## 871 72.2 71.5
## 872 72.2 71.5
## 873 72.2 71.5
## 874 72.2 70.5
## 875 72.2 70.5
## 876 72.2 70.5
## 877 72.2 70.5
## 878 72.2 69.5
## 879 72.2 69.5
## 880 72.2 69.5
## 881 72.2 69.5
## 882 72.2 69.5
## 883 72.2 69.5
## 884 72.2 69.5
## 885 72.2 69.5
## 886 72.2 69.5
## 887 72.2 69.5
## 888 72.2 69.5
## 889 72.2 68.5
## 890 72.2 68.5
## 891 72.2 68.5
## 892 72.2 68.5
## 893 72.2 67.5
## 894 72.2 67.5
## 895 72.2 67.5
## 896 72.2 67.5
## 897 72.2 65.5
## 898 73.2 73.0
## 899 73.2 73.0
## 900 73.2 73.0
## 901 73.2 72.5
## 902 73.2 72.5
## 903 73.2 71.5
## 904 73.2 71.5
## 905 73.2 70.5
## 906 73.2 70.5
## 907 73.2 70.5
## 908 73.2 69.5
## 909 73.2 69.5
## 910 73.2 69.5
## 911 73.2 69.5
## 912 73.2 68.5
## 913 73.2 68.5
## 914 73.2 68.5
## 915 73.7 72.5
## 916 73.7 72.5
## 917 73.7 72.5
## 918 73.7 72.5
## 919 73.7 71.5
## 920 73.7 71.5
## 921 73.7 70.5
## 922 73.7 70.5
## 923 73.7 70.5
## 924 73.7 69.5
## 925 73.7 69.5
## 926 73.7 69.5
## 927 73.7 69.5
## 928 73.7 69.5
x = galton$parent
y = galton$child
x[1 : 10]
## [1] 70.5 68.5 65.5 64.5 64.0 67.5 67.5 67.5 66.5 66.5
y[1 : 10]
## [1] 61.7 61.7 61.7 61.7 61.7 62.2 62.2 62.2 62.2 62.2
xc = x - mean(x)
xc = x - mean(x, na.rm = TRUE)
mean(xc)
## [1] 9.775954e-16
- Rescale the parent and child variables and verify that the scaled
variable standard deviations are 1.
x / sd(x)
## [1] 39.44424 38.32525 36.64677 36.08728 35.80753 37.76576 37.76576 37.76576
## [9] 37.20626 37.20626 37.20626 36.08728 39.44424 38.88474 38.32525 38.32525
## [17] 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576
## [25] 37.76576 37.76576 37.20626 37.20626 37.20626 36.64677 36.64677 36.64677
## [33] 36.64677 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728 36.08728
## [41] 36.08728 36.08728 35.80753 35.80753 38.88474 38.88474 38.88474 38.88474
## [49] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [57] 38.88474 38.88474 38.88474 38.88474 38.32525 38.32525 38.32525 38.32525
## [65] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576
## [73] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [81] 37.76576 37.76576 37.76576 37.76576 37.76576 37.20626 37.20626 37.20626
## [89] 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728
## [97] 36.08728 36.08728 36.08728 35.80753 35.80753 35.80753 35.80753 40.00373
## [105] 39.44424 38.88474 38.88474 38.88474 38.88474 38.32525 38.32525 38.32525
## [113] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [121] 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576
## [129] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [137] 37.76576 37.76576 37.76576 37.76576 37.20626 37.20626 36.64677 36.64677
## [145] 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728 35.80753 40.00373
## [153] 40.00373 40.00373 39.44424 38.88474 38.88474 38.88474 38.88474 38.88474
## [161] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [169] 38.88474 38.88474 38.88474 38.88474 38.32525 38.32525 38.32525 38.32525
## [177] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [185] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [193] 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576
## [201] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [209] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [217] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [225] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [233] 37.76576 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [241] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [249] 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677 36.64677 36.64677
## [257] 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728 36.08728 36.08728
## [265] 36.08728 36.08728 35.80753 35.80753 40.00373 40.00373 40.00373 40.00373
## [273] 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474 38.88474 38.88474
## [281] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [289] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [297] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.32525 38.32525
## [305] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [313] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [321] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [329] 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576
## [337] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [345] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [353] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [361] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [369] 37.76576 37.76576 37.76576 37.20626 37.20626 37.20626 37.20626 37.20626
## [377] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [385] 37.20626 37.20626 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677
## [393] 36.64677 36.64677 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728
## [401] 36.08728 36.08728 36.08728 36.08728 35.80753 35.80753 40.56322 40.00373
## [409] 40.00373 40.00373 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424
## [417] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 38.88474 38.88474
## [425] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [433] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [441] 38.88474 38.88474 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [449] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [457] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [465] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [473] 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576 37.76576
## [481] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [489] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [497] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [505] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [513] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 36.64677 36.64677
## [521] 36.64677 36.64677 36.64677 36.64677 36.64677 35.80753 40.56322 40.56322
## [529] 40.00373 40.00373 40.00373 40.00373 40.00373 39.44424 39.44424 39.44424
## [537] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424
## [545] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 38.88474
## [553] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [561] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [569] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [577] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [585] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [593] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [601] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [609] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [617] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [625] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [633] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [641] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [649] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [657] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [665] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.20626 37.20626
## [673] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [681] 37.20626 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677 36.64677
## [689] 36.64677 36.64677 36.08728 36.08728 35.80753 40.56322 40.00373 40.00373
## [697] 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373
## [705] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424
## [713] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 38.88474 38.88474
## [721] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [729] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [737] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.32525
## [745] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [753] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [761] 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576 37.76576
## [769] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [777] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.20626
## [785] 37.20626 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677 36.64677
## [793] 40.56322 40.56322 40.00373 40.00373 40.00373 40.00373 39.44424 39.44424
## [801] 39.44424 39.44424 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474
## [809] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [817] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [825] 38.88474 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [833] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [841] 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576 37.76576 37.76576
## [849] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 36.64677 36.64677
## [857] 40.84297 40.56322 40.56322 40.56322 40.56322 40.56322 40.56322 40.56322
## [865] 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373
## [873] 40.00373 39.44424 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474
## [881] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [889] 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576 37.76576
## [897] 36.64677 40.84297 40.84297 40.84297 40.56322 40.56322 40.00373 40.00373
## [905] 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474 38.88474 38.32525
## [913] 38.32525 38.32525 40.56322 40.56322 40.56322 40.56322 40.00373 40.00373
## [921] 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474 38.88474 38.88474
xs = x / sd(x)
sd(xs)
## [1] 1
- Normalize the parental and child heights. Verify that the normalized
variables have mean 0 and standard deviation 1 and take the correlation
between them.
x[1 : 10]
## [1] 70.5 68.5 65.5 64.5 64.0 67.5 67.5 67.5 66.5 66.5
y[1 : 10]
## [1] 61.7 61.7 61.7 61.7 61.7 62.2 62.2 62.2 62.2 62.2
xn = (x - mean(x)) / sd(x)
yn = (y - mean(y)) / sd(y)
mean(xn)
## [1] 5.501733e-16
mean(yn)
## [1] 2.183943e-16
sd(xn)
## [1] 1
sd(yn)
## [1] 1
cor(xn, yn)
## [1] 0.4587624
cor(x, y)
## [1] 0.4587624
xn[1]
## [1] 1.226302
yn[1]
## [1] -2.53718