正規分布における±xσの範囲に存在する割合を細かく計算してみた

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

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