I’ve recently been studing for a professional exam that does not allow computers or advanced calculators. Some of the subject matter will require use of a few statistical distributions which can be very time-consuming to calculate manually. In lieu of access to statistical functions, you are allowed to bring books and some reference sheets. I wanted to see if I could recreate these distribution tables in R.

Generating distribution values is very simple in R. The d, q, p, and r functions are all that you will need to fill in the table values. The table below is for the t distribution.

alpha <- c(.1, .05, .025, .01, .005)
v <- c(seq(1, 30, by = 1), Inf)
tTable <- sapply((1-alpha), function(x) qt(x, df = v))
colnames(tTable) <- alpha

I’m using the xtable package here to format my table. It has some handy options to parse math symbols and notations with its sanitize.text print options and allows more customization of table style than the kable function in knitr. I’m setting some of style options below.

options(xtable.type = 'html',
        xtable.caption.placement = "top",
        xtable.include.rownames = FALSE,
        xtable.html.table.attributes= list("border='2' cellpadding='15' cellpadding='5' rules ='all' width ='100%'"))

When rendering the table, you can use LaTeX notation within the data of the table and have it parsed into math notation.

library(xtable)

tTable <- data.frame("$v \\big\\backslash \\alpha$" = paste0("", v, ""),
                     tTable,
                     stringsAsFactors = FALSE,
                     check.names = FALSE)
tTable[[1]][tTable[[1]] == "Inf"] <- "$\\infty$" 
tTable <- xtable(tTable,
                      caption = "Table 3: Percentiles of the *t* Distribution",
                      align = rep("c", ncol(tTable)+1),
                      digits = 4)
print(tTable)
Table 3: Percentiles of the *t* Distribution
`\(v \big\backslash \alpha\)` 0.1 0.05 0.025 0.01 0.005
1 3.0777 6.3138 12.7062 31.8205 63.6567
2 1.8856 2.9200 4.3027 6.9646 9.9248
3 1.6377 2.3534 3.1824 4.5407 5.8409
4 1.5332 2.1318 2.7764 3.7469 4.6041
5 1.4759 2.0150 2.5706 3.3649 4.0321
6 1.4398 1.9432 2.4469 3.1427 3.7074
7 1.4149 1.8946 2.3646 2.9980 3.4995
8 1.3968 1.8595 2.3060 2.8965 3.3554
9 1.3830 1.8331 2.2622 2.8214 3.2498
10 1.3722 1.8125 2.2281 2.7638 3.1693
11 1.3634 1.7959 2.2010 2.7181 3.1058
12 1.3562 1.7823 2.1788 2.6810 3.0545
13 1.3502 1.7709 2.1604 2.6503 3.0123
14 1.3450 1.7613 2.1448 2.6245 2.9768
15 1.3406 1.7531 2.1314 2.6025 2.9467
16 1.3368 1.7459 2.1199 2.5835 2.9208
17 1.3334 1.7396 2.1098 2.5669 2.8982
18 1.3304 1.7341 2.1009 2.5524 2.8784
19 1.3277 1.7291 2.0930 2.5395 2.8609
20 1.3253 1.7247 2.0860 2.5280 2.8453
21 1.3232 1.7207 2.0796 2.5176 2.8314
22 1.3212 1.7171 2.0739 2.5083 2.8188
23 1.3195 1.7139 2.0687 2.4999 2.8073
24 1.3178 1.7109 2.0639 2.4922 2.7969
25 1.3163 1.7081 2.0595 2.4851 2.7874
26 1.3150 1.7056 2.0555 2.4786 2.7787
27 1.3137 1.7033 2.0518 2.4727 2.7707
28 1.3125 1.7011 2.0484 2.4671 2.7633
29 1.3114 1.6991 2.0452 2.4620 2.7564
30 1.3104 1.6973 2.0423 2.4573 2.7500
`\(\infty\)` 1.2816 1.6449 1.9600 2.3263 2.5758

I also wanted to include a plot of the distribution and describe what the values are visually. ?mathplot provides a guide to writing math notation in expression to show math notation in plots. More information on how the math notation will look is available here.

library(ggplot2)

x <- seq(-3.5, 3.5, by = .001)
ggplot() +
  geom_ribbon(aes(x = seq(qnorm(.95), max(x), by = .001),
                  ymin = 0,
                  ymax = dnorm(seq(qnorm(.95), max(x), by = .001))),
              fill = 'gray') +
  geom_vline(xintercept = 0, color = "black") +
  geom_hline(yintercept = 0, color = "black") +
  stat_function(data = data.frame(x=x),aes(x = x),  
                fun = dnorm, 
                size = 1) +
  geom_segment(aes(x = qnorm(.95),
                   y = 0,
                   xend = qnorm(.95),
                   yend = dnorm(qnorm(.95))
  ),
  
  color = 'black') +
  theme_void() +
  theme(plot.margin = unit(c(.1,.1,.1,.1), "cm")) +
  annotate(geom = 'text',
           x = c(qnorm(.95)),
           y = c(-.04),
           label = c(expression(t[paste(alpha,',',v)])),
           size = 4,
           parse = TRUE) +
  coord_cartesian(ylim = c(-.06, dnorm(qnorm(.5))))

Below, is the code for the rest of the tables. I’ve also placed a printable version here.






\(\text{Normal Distribution}\)

x <- seq(-3.5, 3.5, by = .001)
ggplot() +
  geom_ribbon(aes(x = seq(min(x), qnorm(.95), by = .001),
                  ymin = 0,
                  ymax = dnorm(seq(min(x), qnorm(.95), by = .001))),
              fill = 'gray') +
  geom_vline(xintercept = 0, color = "black") +
  geom_hline(yintercept = 0, color = "black") +
  stat_function(data = data.frame(x=x),aes(x = x),  
                fun = dnorm, 
                size = 1) +
  geom_segment(aes(x = qnorm(.95),
                   y = 0,
                   xend = qnorm(.95),
                   yend = dnorm(qnorm(.95))
  ),
  
  color = 'black') +
  theme_void() +
  theme(plot.margin = unit(c(.1,.1,.1,.1), "cm")) +
  annotate(geom = 'text',
           x = c(qnorm(.95)),
           y = c(-.04),
           label = c(expression(z[alpha])),
           size = 4,
           parse = TRUE) +
  coord_cartesian(ylim = c(-.06, dnorm(qnorm(.5))))


normalTable <- matrix(round(pnorm(seq(0, 3.49, by = .01)), 4), 
                      ncol = 10,
                      byrow = TRUE)
colnames(normalTable) <- seq(0, .09, by = .01)
z <- sprintf("%01.1f", seq(0, 3.4, by = .1))
normalTable <- data.frame("$z$" = paste0("", z, ""),
                          normalTable,
                          stringsAsFactors = FALSE)
colnames(normalTable) <- c("z", seq(0, .09, by = .01))
normalTable <- xtable(normalTable,
                      caption = "Table 2: Cumulative Probabilities of the Standard Normal Distribution, $X \\sim N(0,1)$",
                      align = rep("c", ncol(normalTable)+1),
                      digits = 4)
print(normalTable)
Table 2: Cumulative Probabilities of the Standard Normal Distribution, `\(X \sim N(0,1)\)`
z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

\(t \text{ Distribution}\)

x <- seq(-3.5, 3.5, by = .001)
ggplot() +
  geom_ribbon(aes(x = seq(qnorm(.95), max(x), by = .001),
                  ymin = 0,
                  ymax = dnorm(seq(qnorm(.95), max(x), by = .001))),
              fill = 'gray') +
  geom_vline(xintercept = 0, color = "black") +
  geom_hline(yintercept = 0, color = "black") +
  stat_function(data = data.frame(x=x),aes(x = x),  
                fun = dnorm, 
                size = 1) +
  geom_segment(aes(x = qnorm(.95),
                   y = 0,
                   xend = qnorm(.95),
                   yend = dnorm(qnorm(.95))
  ),
  
  color = 'black') +
  theme_void() +
  theme(plot.margin = unit(c(.1,.1,.1,.1), "cm")) +
  annotate(geom = 'text',
           x = c(qnorm(.95)),
           y = c(-.04),
           label = c(expression(t[paste(alpha,',',v)])),
           size = 4,
           parse = TRUE) +
  coord_cartesian(ylim = c(-.06, dnorm(qnorm(.5))))


alpha <- c(.1, .05, .025, .01, .005)
v <- c(seq(1, 30, by = 1), Inf)
tTable <- sapply((1-alpha), function(x) qt(x, df = v))
colnames(tTable) <- alpha
tTable <- data.frame("$v \\big\\backslash \\alpha$" = paste0("", v, ""),
                     tTable,
                     stringsAsFactors = FALSE,
                     check.names = FALSE)
tTable[[1]][tTable[[1]] == "Inf"] <- "$\\infty$" 
tTable <- xtable(tTable,
                      caption = "Table 3: Percentiles of the *t* Distribution",
                      align = rep("c", ncol(tTable)+1),
                      digits = 4)
print(tTable)
Table 3: Percentiles of the *t* Distribution
`\(v \big\backslash \alpha\)` 0.1 0.05 0.025 0.01 0.005
1 3.0777 6.3138 12.7062 31.8205 63.6567
2 1.8856 2.9200 4.3027 6.9646 9.9248
3 1.6377 2.3534 3.1824 4.5407 5.8409
4 1.5332 2.1318 2.7764 3.7469 4.6041
5 1.4759 2.0150 2.5706 3.3649 4.0321
6 1.4398 1.9432 2.4469 3.1427 3.7074
7 1.4149 1.8946 2.3646 2.9980 3.4995
8 1.3968 1.8595 2.3060 2.8965 3.3554
9 1.3830 1.8331 2.2622 2.8214 3.2498
10 1.3722 1.8125 2.2281 2.7638 3.1693
11 1.3634 1.7959 2.2010 2.7181 3.1058
12 1.3562 1.7823 2.1788 2.6810 3.0545
13 1.3502 1.7709 2.1604 2.6503 3.0123
14 1.3450 1.7613 2.1448 2.6245 2.9768
15 1.3406 1.7531 2.1314 2.6025 2.9467
16 1.3368 1.7459 2.1199 2.5835 2.9208
17 1.3334 1.7396 2.1098 2.5669 2.8982
18 1.3304 1.7341 2.1009 2.5524 2.8784
19 1.3277 1.7291 2.0930 2.5395 2.8609
20 1.3253 1.7247 2.0860 2.5280 2.8453
21 1.3232 1.7207 2.0796 2.5176 2.8314
22 1.3212 1.7171 2.0739 2.5083 2.8188
23 1.3195 1.7139 2.0687 2.4999 2.8073
24 1.3178 1.7109 2.0639 2.4922 2.7969
25 1.3163 1.7081 2.0595 2.4851 2.7874
26 1.3150 1.7056 2.0555 2.4786 2.7787
27 1.3137 1.7033 2.0518 2.4727 2.7707
28 1.3125 1.7011 2.0484 2.4671 2.7633
29 1.3114 1.6991 2.0452 2.4620 2.7564
30 1.3104 1.6973 2.0423 2.4573 2.7500
`\(\infty\)` 1.2816 1.6449 1.9600 2.3263 2.5758

\(\chi^2 \text{ Distribution}\)

x <- seq(0, 30, .01)
v <- 10
ggplot() +
  geom_ribbon(aes(x = seq(qchisq(.95, df = v), max(x), by = .001),
                  ymin = 0,
                  ymax = dchisq(seq(qchisq(.95, df = v), max(x), by = .001),
                                df = v)
                  ),
              fill = 'gray') +
  geom_vline(xintercept = 0, color = "black") +
  geom_hline(yintercept = 0, color = "black") +
  stat_function(data = data.frame(x=x),aes(x = x),  
                fun = dchisq, 
                args = list(df = v),
                size = 1) +
  geom_segment(aes(x = qchisq(.95, df = v),
                   y = 0,
                   xend = qchisq(.95, df = v),
                   yend = dchisq(qchisq(.95, df = v), df = v)
  ),
  
  color = 'black') +
  theme_void() +
  theme(plot.margin = unit(c(.1,.1,.1,.1), "cm")) +
  annotate(geom = 'text',
           x = c(qchisq(.95, df = v)),
           y = c(-.015),
           label = c(expression( paste(Chi^2)[paste(alpha,',','v')])),
           size = 4,
           parse = TRUE) +
  coord_cartesian(ylim = c(-.02, dchisq(qchisq(.5, df = v), df = v)))

Table 4: Percentiles of the `\(\chi^2\)` Distribution
`\(v \big\backslash \alpha\)` 0.995 0.99 0.975 0.95 0.9 0.75 0.5 0.25 0.1 0.05 0.025 0.01 0.005 0.001
1 0.00 0.00 0.00 0.00 0.02 0.10 0.45 1.32 2.71 3.84 5.02 6.63 7.88 10.83
2 0.01 0.02 0.05 0.10 0.21 0.58 1.39 2.77 4.61 5.99 7.38 9.21 10.60 13.82
3 0.07 0.11 0.22 0.35 0.58 1.21 2.37 4.11 6.25 7.81 9.35 11.34 12.84 16.27
4 0.21 0.30 0.48 0.71 1.06 1.92 3.36 5.39 7.78 9.49 11.14 13.28 14.86 18.47
5 0.41 0.55 0.83 1.15 1.61 2.67 4.35 6.63 9.24 11.07 12.83 15.09 16.75 20.52
6 0.68 0.87 1.24 1.64 2.20 3.45 5.35 7.84 10.64 12.59 14.45 16.81 18.55 22.46
7 0.99 1.24 1.69 2.17 2.83 4.25 6.35 9.04 12.02 14.07 16.01 18.48 20.28 24.32
8 1.34 1.65 2.18 2.73 3.49 5.07 7.34 10.22 13.36 15.51 17.53 20.09 21.95 26.12
9 1.73 2.09 2.70 3.33 4.17 5.90 8.34 11.39 14.68 16.92 19.02 21.67 23.59 27.88
10 2.16 2.56 3.25 3.94 4.87 6.74 9.34 12.55 15.99 18.31 20.48 23.21 25.19 29.59
11 2.60 3.05 3.82 4.57 5.58 7.58 10.34 13.70 17.28 19.68 21.92 24.72 26.76 31.26
12 3.07 3.57 4.40 5.23 6.30 8.44 11.34 14.85 18.55 21.03 23.34 26.22 28.30 32.91
13 3.57 4.11 5.01 5.89 7.04 9.30 12.34 15.98 19.81 22.36 24.74 27.69 29.82 34.53
14 4.07 4.66 5.63 6.57 7.79 10.17 13.34 17.12 21.06 23.68 26.12 29.14 31.32 36.12
15 4.60 5.23 6.26 7.26 8.55 11.04 14.34 18.25 22.31 25.00 27.49 30.58 32.80 37.70
16 5.14 5.81 6.91 7.96 9.31 11.91 15.34 19.37 23.54 26.30 28.85 32.00 34.27 39.25
17 5.70 6.41 7.56 8.67 10.09 12.79 16.34 20.49 24.77 27.59 30.19 33.41 35.72 40.79
18 6.26 7.01 8.23 9.39 10.86 13.68 17.34 21.60 25.99 28.87 31.53 34.81 37.16 42.31
19 6.84 7.63 8.91 10.12 11.65 14.56 18.34 22.72 27.20 30.14 32.85 36.19 38.58 43.82
20 7.43 8.26 9.59 10.85 12.44 15.45 19.34 23.83 28.41 31.41 34.17 37.57 40.00 45.31
21 8.03 8.90 10.28 11.59 13.24 16.34 20.34 24.93 29.62 32.67 35.48 38.93 41.40 46.80
22 8.64 9.54 10.98 12.34 14.04 17.24 21.34 26.04 30.81 33.92 36.78 40.29 42.80 48.27
23 9.26 10.20 11.69 13.09 14.85 18.14 22.34 27.14 32.01 35.17 38.08 41.64 44.18 49.73
24 9.89 10.86 12.40 13.85 15.66 19.04 23.34 28.24 33.20 36.42 39.36 42.98 45.56 51.18
25 10.52 11.52 13.12 14.61 16.47 19.94 24.34 29.34 34.38 37.65 40.65 44.31 46.93 52.62
30 13.79 14.95 16.79 18.49 20.60 24.48 29.34 34.80 40.26 43.77 46.98 50.89 53.67 59.70
40 20.71 22.16 24.43 26.51 29.05 33.66 39.34 45.62 51.81 55.76 59.34 63.69 66.77 73.40
50 27.99 29.71 32.36 34.76 37.69 42.94 49.33 56.33 63.17 67.50 71.42 76.15 79.49 86.66
60 35.53 37.48 40.48 43.19 46.46 52.29 59.33 66.98 74.40 79.08 83.30 88.38 91.95 99.61
70 43.28 45.44 48.76 51.74 55.33 61.70 69.33 77.58 85.53 90.53 95.02 100.43 104.21 112.32
80 51.17 53.54 57.15 60.39 64.28 71.14 79.33 88.13 96.58 101.88 106.63 112.33 116.32 124.84
90 59.20 61.75 65.65 69.13 73.29 80.62 89.33 98.65 107.57 113.15 118.14 124.12 128.30 137.21
100 67.33 70.06 74.22 77.93 82.36 90.13 99.33 109.14 118.50 124.34 129.56 135.81 140.17 149.45

\(F(v_1, v_2) \text{ Distribution}\)

x <- seq(0, 5, .01)
v1 <- 10
v2 <- 10
ggplot() +
  geom_ribbon(aes(x = seq(qf(.95, df1 = v1, df2 = v2), max(x), by = .001),
                  ymin = 0,
                  ymax = df(seq(qf(.95, df1 = v1, df2 = v2), max(x), by = .001),
                                df1 = v1, df2 = v2)
                  ),
              fill = 'gray') +
  geom_vline(xintercept = 0, color = "black") +
  geom_hline(yintercept = 0, color = "black") +
  stat_function(data = data.frame(x=x),aes(x = x),  
                fun = df, 
                args = list(df1 = v1, df2 = v2),
                size = 1) +
  geom_segment(aes(x = qf(.95, df1 = v1, df2 = v2),
                   y = 0,
                   xend = qf(.95, df1 = v1, df2 = v2),
                   yend = df(qf(.95, df1 = v1, df2 = v2), df1 = v1, df2 = v2)
  ),
  
  color = 'black') +
  theme_void() +
  theme(plot.margin = unit(c(.1,.1,.1,.1), "cm")) +
  annotate(geom = 'text',
           x = c(qf(.95, df1 = v1, df2 = v2)),
           y = c(-.07),
           label = c(expression(F(paste(v1,',',v2)))),
           size = 4,
           parse = TRUE) +
  coord_cartesian(ylim = c(-.07, df(qf(.25, df1 = v1, df2 = v2), df1 = v1, df2 = v2)))


v1 <- c(seq(1, 10, by = 1), 12, 15, 20, 24, 30, 40, 60, 120, Inf)
v2 <- v1
fTable95 <- sapply(v1, function(x) round(qf(.95, x, v2), 2))
colnames(fTable95) <- v1
colnames(fTable95)[colnames(fTable95) == 'Inf'] <- "$\\infty$"
fTable95 <- data.frame("$v_2 \\big\\backslash v_1$" = paste0("", v2, ""),
                       fTable95,
                       stringsAsFactors = FALSE,
                       check.names = FALSE)
fTable95[[1]][fTable95[[1]] == "Inf"] <- "$\\infty$"                        
fTable95 <- xtable(fTable95,
                      caption = "Table 5: 95th Percentiles of the $F(v_1,v_2)$",
                      align = rep("c", ncol(fTable95)+1))

print(fTable95)
Table 5: 95th Percentiles of the `\(F(v_1,v_2)\)`
`\(v_2 \big\backslash v_1\)` 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 `\(\infty\)`
1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88 243.91 245.95 248.01 249.05 250.10 251.14 252.20 253.25 254.31
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84
24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39
120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25
`\(\infty\)` 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00