A sample of
![$N N(0, 1)$](img1.png)
observations is ranked from largest (rank 1) to smallest (rank
![$N$](img2.png)
). The expected values of the order statistics (the ranked values) are given. Only the expected values for the upper half of the order statistics are given since the expected values are symmetric about zero. The column headings give the size of the sample and the row headings the rank of the order statistic.
Table 1:
Expected values of normal order statistics.
|
Sample size |
Rank |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
1 |
.56419 |
.34628 |
1.02938 |
1.16296 |
1.26721 |
1.35218 |
1.42360 |
1.48501 |
1.53875 |
1.58644 |
1.62923 |
1.66799 |
1.70338 |
2 |
|
.00000 |
.29701 |
.49502 |
.64176 |
.75737 |
.85222 |
.93230 |
1.00136 |
1.06192 |
1.11573 |
1.16408 |
1.20790 |
3 |
|
|
|
.00000 |
.20155 |
.35271 |
.47282 |
.57197 |
.65606 |
.72884 |
.79284 |
.84983 |
.90113 |
4 |
|
|
|
|
|
.00000 |
.15251 |
.27453 |
.37576 |
.46198 |
.53684 |
.60285 |
.66176 |
5 |
|
|
|
|
|
|
|
.00000 |
.12267 |
.22489 |
.31225 |
.38833 |
.45557 |
6 |
|
|
|
|
|
|
|
|
|
.00000 |
.10259 |
.19052 |
.26730 |
7 |
|
|
|
|
|
|
|
|
|
|
|
.00000 |
.08816 |
|
Sample size |
Rank |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
1 |
1.73591 |
1.76599 |
1.79394 |
1.82003 |
1.84448 |
1.86748 |
1.88917 |
1.90969 |
1.92916 |
1.94767 |
1.96531 |
1.98216 |
1.99827 |
2 |
1.24794 |
1.28474 |
1.31878 |
1.35041 |
1.37994 |
1.40760 |
1.43362 |
1.45816 |
1.48137 |
1.50338 |
1.52430 |
1.54423 |
1.56326 |
3 |
.94769 |
.99027 |
1.02946 |
1.06573 |
1.09945 |
1.13095 |
1.16047 |
1.18824 |
1.21445 |
1.23924 |
1.26275 |
1.28511 |
1.30641 |
4 |
.71488 |
.76317 |
.80738 |
.84812 |
.88586 |
.92098 |
.95380 |
.98459 |
1.01356 |
1.04091 |
1.06679 |
1.09135 |
1.11471 |
5 |
.51570 |
.57001 |
.61946 |
.66479 |
.70661 |
.74538 |
.78150 |
.81527 |
.84697 |
.87682 |
.90501 |
.93171 |
.95705 |
6 |
.33530 |
.39622 |
.45133 |
.50158 |
.54771 |
.59030 |
.62982 |
.66667 |
.70115 |
.73354 |
.76405 |
.79289 |
.82021 |
7 |
.16530 |
.23375 |
.29519 |
.35084 |
.40164 |
.44833 |
.49148 |
.53157 |
.56896 |
.60299 |
.63690 |
.66794 |
.69727 |
8 |
.00000 |
.07729 |
.14599 |
.20774 |
.26374 |
.31493 |
.36203 |
.40559 |
.44609 |
.48391 |
.51935 |
.55267 |
.58411 |
9 |
|
|
.00000 |
.06880 |
.13072 |
.18696 |
.23841 |
.28579 |
.32965 |
.37047 |
.40860 |
.44436 |
.47801 |
10 |
|
|
|
|
.00000 |
.06200 |
.11836 |
.16997 |
.21755 |
.26163 |
.30268 |
.34105 |
.37706 |
11 |
|
|
|
|
|
|
.00000 |
.05642 |
.10813 |
.15583 |
.20006 |
.24128 |
.27983 |
12 |
|
|
|
|
|
|
|
|
.00000 |
.05176 |
.09953 |
.14387 |
.18520 |
13 |
|
|
|
|
|
|
|
|
|
|
.00000 |
.04781 |
.09220 |
14 |
|
|
|
|
|
|
|
|
|
|
|
|
.00000 |
|
Sample size |
Rank |
28 |
29 |
30 |
31 |
32 |
33 |
34 |
35 |
36 |
37 |
38 |
39 |
|
1 |
2.01371 |
2.02852 |
2.04276 |
2.05646 |
2.06967 |
2.08241 |
2.09471 |
2.10661 |
2.11812 |
2.12928 |
2.14009 |
2.15059 |
|
2 |
1.58145 |
1.59888 |
1.61560 |
1.63166 |
1.64712 |
1.66200 |
1.67636 |
1.69023 |
1.70362 |
1.71659 |
1.72914 |
1.74131 |
|
3 |
1.32674 |
1.34619 |
1.36481 |
1.38268 |
1.39985 |
1.41637 |
1.43228 |
1.44762 |
1.46244 |
1.47676 |
1.49061 |
1.50402 |
|
4 |
1.13697 |
1.15822 |
1.17855 |
1.19803 |
1.21672 |
1.23468 |
1.25196 |
1.26860 |
1.28466 |
1.30016 |
1.31514 |
1.32964 |
|
5 |
.98115 |
1.00414 |
1.02609 |
1.04709 |
1.06721 |
1.08652 |
1.10509 |
1.12295 |
1.14016 |
1.15677 |
1.17280 |
1.18830 |
|
6 |
.84615 |
.87084 |
.89439 |
.91688 |
.93841 |
.95905 |
.97886 |
.99790 |
1.01624 |
1.03390 |
1.05095 |
1.06741 |
|
7 |
.72508 |
.75150 |
.77666 |
.80066 |
.82359 |
.84555 |
.86660 |
.88681 |
.90625 |
.92496 |
.94300 |
.96041 |
|
8 |
.61385 |
.64205 |
.66885 |
.69438 |
.71875 |
.74204 |
.76435 |
.78574 |
.80629 |
.82605 |
.84508 |
.86343 |
|
9 |
.50977 |
.53982 |
.56834 |
.59545 |
.62129 |
.64596 |
.66954 |
.69214 |
.71382 |
.73465 |
.75468 |
.77398 |
|
10 |
.41096 |
.44298 |
.47329 |
.50206 |
.52943 |
.55552 |
.58043 |
.60427 |
.62710 |
.64902 |
.67009 |
.69035 |
|
11 |
.31603 |
.35013 |
.38235 |
.41287 |
.44185 |
.46942 |
.49572 |
.52084 |
.54488 |
.56793 |
.59005 |
.61131 |
|
12 |
.22389 |
.26023 |
.29449 |
.32686 |
.35755 |
.38669 |
.41444 |
.44091 |
.46620 |
.49042 |
.51363 |
.53592 |
|
13 |
.13361 |
.17240 |
.20885 |
.24322 |
.27573 |
.30654 |
.33582 |
.36371 |
.39032 |
.41576 |
.44012 |
.46348 |
|
14 |
.04442 |
.08588 |
.12473 |
.16126 |
.19572 |
.22832 |
.25924 |
.28863 |
.31663 |
.34336 |
.36892 |
.39340 |
|
15 |
|
.00000 |
.04148 |
.08037 |
.11695 |
.15147 |
.18415 |
.21515 |
.24463 |
.27272 |
.29954 |
.32520 |
|
16 |
|
|
|
.00000 |
.03890 |
.07552 |
.11009 |
.14282 |
.17388 |
.20342 |
.23159 |
.25849 |
|
17 |
|
|
|
|
|
.00000 |
.03663 |
.07123 |
.10399 |
.13509 |
.16469 |
.19292 |
|
18 |
|
|
|
|
|
|
|
.00000 |
.03461 |
.06739 |
.09853 |
.12817 |
|
19 |
|
|
|
|
|
|
|
|
|
.00000 |
.03280 |
.06395 |
|
20 |
|
|
|
|
|
|
|
|
|
|
|
.00000 |
|