Recreate arc sine Conversion Table A-16

John Quensen

June 3, 2017

Introduction

Table A-16 in the appendix of Snedecore and Cochran (1967) was my introduction to the arc sine conversion of percentages to angles. The table was actually reprinted from Bliss (1937), and tabulated values of arc sine(sqrt(percentage/100)). For example, the tabulated value for 15% is 22.79. Angles in R are in radians, not degrees. Thus the arc_sine function in this package does not return the values in Table A-16, but rather the corresponding angle in radians. For example, arc_sine(15) = 0.3976994. To get the angle in degrees, you can convert from radians to degrees with the function rad2deg. Thus rad2deg(arc_sine(15)) = 22.786498.

Just for kicks, the following code reproduces Table A-16:

part1 <- matrix(data = NA, nrow = 10, ncol = 10)
for (i in 1:10) {
  for (j in 1:10) {
    x <- rad2deg(arc_sine(((i-1)/10)+((j-1)/100)))
    part1[i,j] <- formatC(x, digits = 2, format = "f")
  }
}
col1 <- seq(from = 0, to = 0.9, by = 0.1)
part1 <- data.frame(col1, part1)
colnames(part1) <- c("%", "0", "1", "2", "3", "4", "5", "6", "7", "8", "9")

part2 <- matrix(data = NA, nrow = 98, ncol = 10)
for (i in 1:98) {
  for (j in 1:10) {
    x <- rad2deg(arc_sine(i+((j-1)/10)))
    part2[i,j] <-formatC(x, digits = 2, format = "f")
  }
}
col1 <- seq(from = 1, to = 98, by = 1)
part2 <- data.frame(col1, part2)
colnames(part2) <- c("%", "0", "1", "2", "3", "4", "5", "6", "7", "8", "9")

part3 <- matrix(data = NA, nrow = 11, ncol = 10)
for (i in 1:11) {
  for (j in 1:10) {
    m <- 99 + ((i-1)/10)
    n <- ((j-1)/100)
    if((m+n) <= 100) {
      x <- rad2deg(arc_sine(m+n))
       part3[i,j] <- formatC(x, digits = 2, format = "f")
    }
  }
}
col1 <- round(seq(from = 99, to = 100, by = 0.1), 1)
part3 <- data.frame(col1, part3)
colnames(part3) <- c("%", "0", "1", "2", "3", "4", "5", "6", "7", "8", "9")

a16 <- rbind(part1, part2, part3)

knitr::kable(a16, align = "l", caption = "Table of Angles Corresponding to Percentages, after Bliss(1937).")
Table of Angles Corresponding to Percentages, after Bliss(1937).
% 0 1 2 3 4 5 6 7 8 9
0.0 0.00 0.57 0.81 0.99 1.15 1.28 1.40 1.52 1.62 1.72
0.1 1.81 1.90 1.99 2.07 2.14 2.22 2.29 2.36 2.43 2.50
0.2 2.56 2.63 2.69 2.75 2.81 2.87 2.92 2.98 3.03 3.09
0.3 3.14 3.19 3.24 3.29 3.34 3.39 3.44 3.49 3.53 3.58
0.4 3.63 3.67 3.72 3.76 3.80 3.85 3.89 3.93 3.97 4.01
0.5 4.05 4.10 4.14 4.17 4.21 4.25 4.29 4.33 4.37 4.41
0.6 4.44 4.48 4.52 4.55 4.59 4.62 4.66 4.70 4.73 4.76
0.7 4.80 4.83 4.87 4.90 4.93 4.97 5.00 5.03 5.07 5.10
0.8 5.13 5.16 5.20 5.23 5.26 5.29 5.32 5.35 5.38 5.41
0.9 5.44 5.47 5.50 5.53 5.56 5.59 5.62 5.65 5.68 5.71
1.0 5.74 6.02 6.29 6.55 6.80 7.03 7.27 7.49 7.71 7.92
2.0 8.13 8.33 8.53 8.72 8.91 9.10 9.28 9.46 9.63 9.80
3.0 9.97 10.14 10.30 10.47 10.63 10.78 10.94 11.09 11.24 11.39
4.0 11.54 11.68 11.83 11.97 12.11 12.25 12.38 12.52 12.66 12.79
5.0 12.92 13.05 13.18 13.31 13.44 13.56 13.69 13.81 13.94 14.06
6.0 14.18 14.30 14.42 14.54 14.65 14.77 14.89 15.00 15.12 15.23
7.0 15.34 15.45 15.56 15.68 15.79 15.89 16.00 16.11 16.22 16.32
8.0 16.43 16.54 16.64 16.74 16.85 16.95 17.05 17.15 17.26 17.36
9.0 17.46 17.56 17.66 17.76 17.85 17.95 18.05 18.15 18.24 18.34
10.0 18.43 18.53 18.63 18.72 18.81 18.91 19.00 19.09 19.19 19.28
11.0 19.37 19.46 19.55 19.64 19.73 19.82 19.91 20.00 20.09 20.18
12.0 20.27 20.36 20.44 20.53 20.62 20.70 20.79 20.88 20.96 21.05
13.0 21.13 21.22 21.30 21.39 21.47 21.56 21.64 21.72 21.81 21.89
14.0 21.97 22.06 22.14 22.22 22.30 22.38 22.46 22.54 22.63 22.71
15.0 22.79 22.87 22.95 23.03 23.11 23.18 23.26 23.34 23.42 23.50
16.0 23.58 23.66 23.73 23.81 23.89 23.97 24.04 24.12 24.20 24.27
17.0 24.35 24.43 24.50 24.58 24.65 24.73 24.80 24.88 24.95 25.03
18.0 25.10 25.18 25.25 25.33 25.40 25.47 25.55 25.62 25.70 25.77
19.0 25.84 25.91 25.99 26.06 26.13 26.21 26.28 26.35 26.42 26.49
20.0 26.57 26.64 26.71 26.78 26.85 26.92 26.99 27.06 27.13 27.20
21.0 27.27 27.35 27.42 27.49 27.56 27.62 27.69 27.76 27.83 27.90
22.0 27.97 28.04 28.11 28.18 28.25 28.32 28.39 28.45 28.52 28.59
23.0 28.66 28.73 28.79 28.86 28.93 29.00 29.06 29.13 29.20 29.27
24.0 29.33 29.40 29.47 29.53 29.60 29.67 29.73 29.80 29.87 29.93
25.0 30.00 30.07 30.13 30.20 30.26 30.33 30.40 30.46 30.53 30.59
26.0 30.66 30.72 30.79 30.85 30.92 30.98 31.05 31.11 31.18 31.24
27.0 31.31 31.37 31.44 31.50 31.56 31.63 31.69 31.76 31.82 31.88
28.0 31.95 32.01 32.08 32.14 32.20 32.27 32.33 32.39 32.46 32.52
29.0 32.58 32.65 32.71 32.77 32.83 32.90 32.96 33.02 33.09 33.15
30.0 33.21 33.27 33.34 33.40 33.46 33.52 33.58 33.65 33.71 33.77
31.0 33.83 33.90 33.96 34.02 34.08 34.14 34.20 34.27 34.33 34.39
32.0 34.45 34.51 34.57 34.63 34.70 34.76 34.82 34.88 34.94 35.00
33.0 35.06 35.12 35.18 35.24 35.30 35.37 35.43 35.49 35.55 35.61
34.0 35.67 35.73 35.79 35.85 35.91 35.97 36.03 36.09 36.15 36.21
35.0 36.27 36.33 36.39 36.45 36.51 36.57 36.63 36.69 36.75 36.81
36.0 36.87 36.93 36.99 37.05 37.11 37.17 37.23 37.29 37.35 37.41
37.0 37.46 37.52 37.58 37.64 37.70 37.76 37.82 37.88 37.94 38.00
38.0 38.06 38.12 38.17 38.23 38.29 38.35 38.41 38.47 38.53 38.59
39.0 38.65 38.70 38.76 38.82 38.88 38.94 39.00 39.06 39.11 39.17
40.0 39.23 39.29 39.35 39.41 39.47 39.52 39.58 39.64 39.70 39.76
41.0 39.82 39.87 39.93 39.99 40.05 40.11 40.16 40.22 40.28 40.34
42.0 40.40 40.45 40.51 40.57 40.63 40.69 40.74 40.80 40.86 40.92
43.0 40.98 41.03 41.09 41.15 41.21 41.27 41.32 41.38 41.44 41.50
44.0 41.55 41.61 41.67 41.73 41.78 41.84 41.90 41.96 42.02 42.07
45.0 42.13 42.19 42.25 42.30 42.36 42.42 42.48 42.53 42.59 42.65
46.0 42.71 42.76 42.82 42.88 42.94 42.99 43.05 43.11 43.17 43.22
47.0 43.28 43.34 43.39 43.45 43.51 43.57 43.62 43.68 43.74 43.80
48.0 43.85 43.91 43.97 44.03 44.08 44.14 44.20 44.26 44.31 44.37
49.0 44.43 44.48 44.54 44.60 44.66 44.71 44.77 44.83 44.89 44.94
50.0 45.00 45.06 45.11 45.17 45.23 45.29 45.34 45.40 45.46 45.52
51.0 45.57 45.63 45.69 45.74 45.80 45.86 45.92 45.97 46.03 46.09
52.0 46.15 46.20 46.26 46.32 46.38 46.43 46.49 46.55 46.61 46.66
53.0 46.72 46.78 46.83 46.89 46.95 47.01 47.06 47.12 47.18 47.24
54.0 47.29 47.35 47.41 47.47 47.52 47.58 47.64 47.70 47.75 47.81
55.0 47.87 47.93 47.98 48.04 48.10 48.16 48.22 48.27 48.33 48.39
56.0 48.45 48.50 48.56 48.62 48.68 48.73 48.79 48.85 48.91 48.97
57.0 49.02 49.08 49.14 49.20 49.26 49.31 49.37 49.43 49.49 49.55
58.0 49.60 49.66 49.72 49.78 49.84 49.89 49.95 50.01 50.07 50.13
59.0 50.18 50.24 50.30 50.36 50.42 50.48 50.53 50.59 50.65 50.71
60.0 50.77 50.83 50.89 50.94 51.00 51.06 51.12 51.18 51.24 51.30
61.0 51.35 51.41 51.47 51.53 51.59 51.65 51.71 51.77 51.83 51.88
62.0 51.94 52.00 52.06 52.12 52.18 52.24 52.30 52.36 52.42 52.48
63.0 52.54 52.59 52.65 52.71 52.77 52.83 52.89 52.95 53.01 53.07
64.0 53.13 53.19 53.25 53.31 53.37 53.43 53.49 53.55 53.61 53.67
65.0 53.73 53.79 53.85 53.91 53.97 54.03 54.09 54.15 54.21 54.27
66.0 54.33 54.39 54.45 54.51 54.57 54.63 54.70 54.76 54.82 54.88
67.0 54.94 55.00 55.06 55.12 55.18 55.24 55.30 55.37 55.43 55.49
68.0 55.55 55.61 55.67 55.73 55.80 55.86 55.92 55.98 56.04 56.10
69.0 56.17 56.23 56.29 56.35 56.42 56.48 56.54 56.60 56.66 56.73
70.0 56.79 56.85 56.91 56.98 57.04 57.10 57.17 57.23 57.29 57.35
71.0 57.42 57.48 57.54 57.61 57.67 57.73 57.80 57.86 57.92 57.99
72.0 58.05 58.12 58.18 58.24 58.31 58.37 58.44 58.50 58.56 58.63
73.0 58.69 58.76 58.82 58.89 58.95 59.02 59.08 59.15 59.21 59.28
74.0 59.34 59.41 59.47 59.54 59.60 59.67 59.74 59.80 59.87 59.93
75.0 60.00 60.07 60.13 60.20 60.27 60.33 60.40 60.47 60.53 60.60
76.0 60.67 60.73 60.80 60.87 60.94 61.00 61.07 61.14 61.21 61.27
77.0 61.34 61.41 61.48 61.55 61.61 61.68 61.75 61.82 61.89 61.96
78.0 62.03 62.10 62.17 62.24 62.31 62.38 62.44 62.51 62.58 62.65
79.0 62.73 62.80 62.87 62.94 63.01 63.08 63.15 63.22 63.29 63.36
80.0 63.43 63.51 63.58 63.65 63.72 63.79 63.87 63.94 64.01 64.09
81.0 64.16 64.23 64.30 64.38 64.45 64.53 64.60 64.67 64.75 64.82
82.0 64.90 64.97 65.05 65.12 65.20 65.27 65.35 65.42 65.50 65.57
83.0 65.65 65.73 65.80 65.88 65.96 66.03 66.11 66.19 66.27 66.34
84.0 66.42 66.50 66.58 66.66 66.74 66.82 66.89 66.97 67.05 67.13
85.0 67.21 67.29 67.37 67.46 67.54 67.62 67.70 67.78 67.86 67.94
86.0 68.03 68.11 68.19 68.28 68.36 68.44 68.53 68.61 68.70 68.78
87.0 68.87 68.95 69.04 69.12 69.21 69.30 69.38 69.47 69.56 69.64
88.0 69.73 69.82 69.91 70.00 70.09 70.18 70.27 70.36 70.45 70.54
89.0 70.63 70.72 70.81 70.91 71.00 71.09 71.19 71.28 71.37 71.47
90.0 71.57 71.66 71.76 71.85 71.95 72.05 72.15 72.24 72.34 72.44
91.0 72.54 72.64 72.74 72.85 72.95 73.05 73.15 73.26 73.36 73.46
92.0 73.57 73.68 73.78 73.89 74.00 74.11 74.21 74.32 74.44 74.55
93.0 74.66 74.77 74.88 75.00 75.11 75.23 75.35 75.46 75.58 75.70
94.0 75.82 75.94 76.06 76.19 76.31 76.44 76.56 76.69 76.82 76.95
95.0 77.08 77.21 77.34 77.48 77.62 77.75 77.89 78.03 78.17 78.32
96.0 78.46 78.61 78.76 78.91 79.06 79.22 79.37 79.53 79.70 79.86
97.0 80.03 80.20 80.37 80.54 80.72 80.90 81.09 81.28 81.47 81.67
98.0 81.87 82.08 82.29 82.51 82.73 82.97 83.20 83.45 83.71 83.98
99.0 84.26 84.29 84.32 84.35 84.38 84.41 84.44 84.47 84.50 84.53
99.1 84.56 84.59 84.62 84.65 84.68 84.71 84.74 84.77 84.80 84.84
99.2 84.87 84.90 84.93 84.97 85.00 85.03 85.07 85.10 85.13 85.17
99.3 85.20 85.24 85.27 85.30 85.34 85.38 85.41 85.45 85.48 85.52
99.4 85.56 85.59 85.63 85.67 85.71 85.75 85.79 85.83 85.86 85.90
99.5 85.95 85.99 86.03 86.07 86.11 86.15 86.20 86.24 86.28 86.33
99.6 86.37 86.42 86.47 86.51 86.56 86.61 86.66 86.71 86.76 86.81
99.7 86.86 86.91 86.97 87.02 87.08 87.13 87.19 87.25 87.31 87.37
99.8 87.44 87.50 87.57 87.64 87.71 87.78 87.86 87.93 88.01 88.10
99.9 88.19 88.28 88.38 88.48 88.60 88.72 88.85 89.01 89.19 89.43
100.0 90.00 NA NA NA NA NA NA NA NA NA

References

Bliss, C. J. 1937. Plant Protection, No. 12, Leningrad.

Snedecore, G. W., and W. G. Cochran. 1967. Statistical Methods. Sixth Edition. The Iowas State University Press, Ames, Iowa, USA.