正規分布において±1σの範囲に存在する割合は0.68となる．では割合が0.1とか0.5になるようにσ値に係数を掛けたい場合，その係数をいくつにすれば良いのだろう？ということで細かく計算した．計算はMaxima 5.12.0．
±aσの範囲に存在する割合をbとすると，aとbは以下の表に従う．
aを0から0.1づつ変化させた
| a |
b |
| 0.0 |
0.00000000000000 |
| 0.1 |
0.07965567455406 |
| 0.2 |
0.15851941887821 |
| 0.3 |
0.23582284437791 |
| 0.4 |
0.31084348322065 |
| 0.5 |
0.38292492254803 |
| 0.6 |
0.45149376449985 |
| 0.7 |
0.51607269555385 |
| 0.8 |
0.57628920283321 |
| 0.9 |
0.63187974930648 |
| 1.0 |
0.68268949213709 |
| 1.1 |
0.72866787810723 |
| 1.2 |
0.76986065955658 |
| 1.3 |
0.80639903082878 |
| 1.4 |
0.83848668153246 |
| 1.5 |
0.86638559746228 |
| 1.6 |
0.89040141660088 |
| 1.7 |
0.91086907448291 |
| 1.8 |
0.92813936177415 |
| 1.9 |
0.94256688036800 |
| 2.0 |
0.95449973610364 |
| 2.1 |
0.96427115887437 |
| 2.2 |
0.97219310497300 |
| 2.3 |
0.97855177995665 |
| 2.4 |
0.98360492815081 |
| 2.5 |
0.98758066934845 |
| 2.6 |
0.99067762395256 |
| 2.7 |
0.99306605239392 |
| 2.8 |
0.99488973933914 |
| 2.9 |
0.99626837339923 |
| 3.0 |
0.99730020393674 |
| 3.1 |
0.99806479357356 |
| 3.2 |
0.99862572412417 |
| 3.3 |
0.99903315171523 |
| 3.4 |
0.99932614146865 |
| 3.5 |
0.99953474184193 |
| 3.6 |
0.99968178281968 |
| 3.7 |
0.99978440053305 |
| 3.8 |
0.99985530391215 |
| 3.9 |
0.99990380731196 |
| 4.0 |
0.99993665751633 |
| 4.1 |
0.99995868498617 |
| 4.2 |
0.99997330850197 |
| 4.3 |
0.99998292018906 |
| 4.4 |
0.99998917491218 |
| 4.5 |
0.99999320465375 |
| 4.6 |
0.99999577509060 |
| 4.7 |
0.99999739838509 |
| 4.8 |
0.99999841334370 |
| 4.9 |
0.99999904163345 |
| 5.0 |
0.99999942669686 |
| 5.1 |
0.99999966034652 |
| 5.2 |
0.99999980071147 |
| 5.3 |
0.99999988419732 |
| 5.4 |
0.99999993335910 |
| 5.5 |
0.99999996202088 |
| 5.6 |
0.99999997856482 |
| 5.7 |
0.99999998801926 |
| 5.8 |
0.99999999336851 |
| 5.9 |
0.99999999636498 |
| 6.0 |
0.99999999802682 |
| 6.1 |
0.99999999893932 |
| 6.2 |
0.99999999943537 |
| 6.3 |
0.99999999970235 |
| 6.4 |
0.99999999984462 |
| 6.5 |
0.99999999991968 |
| 6.6 |
0.99999999995888 |
| 6.7 |
0.99999999997916 |
| 6.8 |
0.99999999998954 |
| 6.9 |
0.99999999999480 |
| 7.0 |
0.99999999999744 |
| 7.1 |
0.99999999999875 |
| 7.2 |
0.99999999999940 |
| 7.3 |
0.99999999999971 |
| 7.4 |
0.99999999999986 |
| 7.5 |
0.99999999999994 |
| 7.6 |
0.99999999999997 |
| 7.7 |
0.99999999999999 |
|
aを0から2まで0.01づつ変化させた
| a |
b |
| 0.00 |
0.00000000000000 |
| 0.01 |
0.00797871262926 |
| 0.02 |
0.01595662743380 |
| 0.03 |
0.02393294682823 |
| 0.04 |
0.03190687370566 |
| 0.05 |
0.03987761167675 |
| 0.06 |
0.04784436530821 |
| 0.07 |
0.05580634036104 |
| 0.08 |
0.06376274402798 |
| 0.09 |
0.07171278517034 |
| 0.10 |
0.07965567455406 |
| 0.11 |
0.08759062508463 |
| 0.12 |
0.09551685204117 |
| 0.13 |
0.10343357330912 |
| 0.14 |
0.11134000961181 |
| 0.15 |
0.11923538474049 |
| 0.16 |
0.12711892578287 |
| 0.17 |
0.13498986335008 |
| 0.18 |
0.14284743180180 |
| 0.19 |
0.15069086946959 |
| 0.20 |
0.15851941887821 |
| 0.21 |
0.16633232696488 |
| 0.22 |
0.17412884529643 |
| 0.23 |
0.18190823028401 |
| 0.24 |
0.18966974339559 |
| 0.25 |
0.19741265136585 |
| 0.26 |
0.20513622640352 |
| 0.27 |
0.21283974639608 |
| 0.28 |
0.22052249511159 |
| 0.29 |
0.22818376239775 |
| 0.30 |
0.23582284437791 |
| 0.31 |
0.24343904364404 |
| 0.32 |
0.25103166944664 |
| 0.33 |
0.25860003788131 |
| 0.34 |
0.26614347207206 |
| 0.35 |
0.27366130235124 |
| 0.36 |
0.28115286643598 |
| 0.37 |
0.28861750960109 |
| 0.38 |
0.29605458484833 |
| 0.39 |
0.30346345307196 |
| 0.40 |
0.31084348322065 |
| 0.41 |
0.31819405245535 |
| 0.42 |
0.32551454630350 |
| 0.43 |
0.33280435880908 |
| 0.44 |
0.34006289267881 |
| 0.45 |
0.34728955942416 |
| 0.46 |
0.35448377949930 |
| 0.47 |
0.36164498243489 |
| 0.48 |
0.36877260696755 |
| 0.49 |
0.37586610116522 |
| 0.50 |
0.38292492254803 |
| 0.51 |
0.38994853820496 |
| 0.52 |
0.39693642490607 |
| 0.53 |
0.40388806921025 |
| 0.54 |
0.41080296756860 |
| 0.55 |
0.41768062642331 |
| 0.56 |
0.42452056230195 |
| 0.57 |
0.43132230190735 |
| 0.58 |
0.43808538220287 |
| 0.59 |
0.44480935049307 |
| 0.60 |
0.45149376449985 |
| 0.61 |
0.45813819243399 |
| 0.62 |
0.46474221306203 |
| 0.63 |
0.47130541576864 |
| 0.64 |
0.47782740061428 |
| 0.65 |
0.48430777838827 |
| 0.66 |
0.49074617065733 |
| 0.67 |
0.49714220980938 |
| 0.68 |
0.50349553909286 |
| 0.69 |
0.50980581265138 |
| 0.70 |
0.51607269555385 |
| 0.71 |
0.52229586382003 |
| 0.72 |
0.52847500444150 |
| 0.73 |
0.53460981539821 |
| 0.74 |
0.54070000567042 |
| 0.75 |
0.54674529524626 |
| 0.76 |
0.55274541512480 |
| 0.77 |
0.55870010731470 |
| 0.78 |
0.56460912482853 |
| 0.79 |
0.57047223167273 |
| 0.80 |
0.57628920283321 |
| 0.81 |
0.58205982425680 |
| 0.82 |
0.58778389282837 |
| 0.83 |
0.59346121634386 |
| 0.84 |
0.59909161347910 |
| 0.85 |
0.60467491375462 |
| 0.86 |
0.61021095749638 |
| 0.87 |
0.61569959579261 |
| 0.88 |
0.62114069044658 |
| 0.89 |
0.62653411392565 |
| 0.90 |
0.63187974930648 |
| 0.91 |
0.63717749021641 |
| 0.92 |
0.64242724077126 |
| 0.93 |
0.64762891550948 |
| 0.94 |
0.65278243932275 |
| 0.95 |
0.65788774738304 |
| 0.96 |
0.66294478506632 |
| 0.97 |
0.66795350787294 |
| 0.98 |
0.67291388134462 |
| 0.99 |
0.67782588097834 |
| 1.00 |
0.68268949213709 |
| 1.01 |
0.68750470995749 |
| 1.02 |
0.69227153925453 |
| 1.03 |
0.69698999442331 |
| 1.04 |
0.70166009933804 |
| 1.05 |
0.70628188724821 |
| 1.06 |
0.71085540067218 |
| 1.07 |
0.71538069128812 |
| 1.08 |
0.71985781982246 |
| 1.09 |
0.72428685593593 |
| 1.10 |
0.72866787810723 |
| 1.11 |
0.73300097351451 |
| 1.12 |
0.73728623791454 |
| 1.13 |
0.74152377551996 |
| 1.14 |
0.74571369887440 |
| 1.15 |
0.74985612872570 |
| 1.16 |
0.75395119389731 |
| 1.17 |
0.75799903115796 |
| 1.18 |
0.76199978508960 |
| 1.19 |
0.76595360795378 |
| 1.20 |
0.76986065955658 |
| 1.21 |
0.77372110711205 |
| 1.22 |
0.77753512510433 |
| 1.23 |
0.78130289514862 |
| 1.24 |
0.78502460585083 |
| 1.25 |
0.78870045266629 |
| 1.26 |
0.79233063775740 |
| 1.27 |
0.79591536985036 |
| 1.28 |
0.79945486409112 |
| 1.29 |
0.80294934190050 |
| 1.30 |
0.80639903082878 |
| 1.31 |
0.80980416440952 |
| 1.32 |
0.81316498201306 |
| 1.33 |
0.81648172869944 |
| 1.34 |
0.81975465507110 |
| 1.35 |
0.82298401712520 |
| 1.36 |
0.82617007610583 |
| 1.37 |
0.82931309835607 |
| 1.38 |
0.83241335516997 |
| 1.39 |
0.83547112264466 |
| 1.40 |
0.83848668153246 |
| 1.41 |
0.84146031709322 |
| 1.42 |
0.84439231894691 |
| 1.43 |
0.84728298092652 |
| 1.44 |
0.85013260093135 |
| 1.45 |
0.85294148078070 |
| 1.46 |
0.85570992606821 |
| 1.47 |
0.85843824601663 |
| 1.48 |
0.86112675333334 |
| 1.49 |
0.86377576406655 |
| 1.50 |
0.86638559746228 |
| 1.51 |
0.86895657582217 |
| 1.52 |
0.87148902436213 |
| 1.53 |
0.87398327107204 |
| 1.54 |
0.87643964657638 |
| 1.55 |
0.87885848399588 |
| 1.56 |
0.88124011881041 |
| 1.57 |
0.88358488872289 |
| 1.58 |
0.88589313352449 |
| 1.59 |
0.88816519496106 |
| 1.60 |
0.89040141660088 |
| 1.61 |
0.89260214370376 |
| 1.62 |
0.89476772309150 |
| 1.63 |
0.89689850301982 |
| 1.64 |
0.89899483305179 |
| 1.65 |
0.90105706393270 |
| 1.66 |
0.90308554746655 |
| 1.67 |
0.90508063639411 |
| 1.68 |
0.90704268427256 |
| 1.69 |
0.90897204535690 |
| 1.70 |
0.91086907448291 |
| 1.71 |
0.91273412695194 |
| 1.72 |
0.91456755841734 |
| 1.73 |
0.91636972477281 |
| 1.74 |
0.91814098204239 |
| 1.75 |
0.91988168627237 |
| 1.76 |
0.92159219342503 |
| 1.77 |
0.92327285927426 |
| 1.78 |
0.92492403930297 |
| 1.79 |
0.92654608860255 |
| 1.80 |
0.92813936177415 |
| 1.81 |
0.92970421283192 |
| 1.82 |
0.93124099510822 |
| 1.83 |
0.93275006116074 |
| 1.84 |
0.93423176268167 |
| 1.85 |
0.93568645040877 |
| 1.86 |
0.93711447403849 |
| 1.87 |
0.93851618214107 |
| 1.88 |
0.93989192207760 |
| 1.89 |
0.94124203991918 |
| 1.90 |
0.94256688036800 |
| 1.91 |
0.94386678668045 |
| 1.92 |
0.94514210059233 |
| 1.93 |
0.94639316224589 |
| 1.94 |
0.94762031011909 |
| 1.95 |
0.94882388095672 |
| 1.96 |
0.95000420970356 |
| 1.97 |
0.95116162943955 |
| 1.98 |
0.95229647131698 |
| 1.99 |
0.95340906449958 |
| 2.00 |
0.95449973610364 |
|