﻿ Normal distribution calculator

# Normal Distribution Calculator

The normal distribution (also known as the Gaussian distribution), is the most widely used distribution in statistical analyses.
This is generally because many natural processes are naturally distributed or have a very similar spread.
Some examples of normally distributed data includes height, weight and error in measurements.

The Normal distribution has a symmetric "Bell Curve" structure. more data exist around the center, which is the average, and as further the value is from the center the less likely it occurs.

Usually, when adding independent random variables, the result trends toward the normal distribution (CLT - The Central Limit Theorem)
You can calculate the values of any normal distribution based on the standard normal distribution (normal distribution with mean equals zero and standard deviation equals one)
when X distributes normally, mean equals μ and standard deviation equals σ, Z=(x-μ)/σ distribute as the standard normal distribution

Calculate normal distribution or inverse normal distribution
 digits: 1 2 3 4 5 6 7 6 9 10 example digits=2: (0.001234 => 0.0012) μ: Mean(average) σ: Standard Deviation X: p(x≤X): Choose X or P(x≤X) Choose X and fill X value to get the cumulative probability P(x≤X) based on X, or choose P(x≤X) and fill the cumulative probability to get the value X based on the probability

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## Z table

Calculate p(x≤X)
For negative x values: p(x ≤ -X) = 1 - p(x ≤ X)

For any practical use you can use the above calculator instead of the Z table. The below z table is a bit more accurate.
Examples: Row 4 (0.3) and column 3 (0.02). 0.32 = 0.30 + 0.02
P(x ≤ 0.32) = 0.6255158347
P(x ≤ -0.32) = 1- P(x ≤ 0.32) = 0.3744841653

Z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
00.50.50398935630.50797831370.51196647340.51595343690.51993880580.52392218270.52790317020.5318813720.5358563926
0.10.53982783730.54379531250.5477584260.55171678670.55567000480.55961769240.56355946290.56749493170.57142371590.5753454347
0.20.57925970940.58316616350.58706442260.59095411510.59483487170.59870632570.60256811320.60641987320.61026124760.6140918812
0.30.61791142220.62171952180.62551583470.62930001890.6330717360.63683065120.64057643320.64430875480.64802729240.6517317265
0.40.65542174160.65909702620.66275727320.66640217940.67003144630.67364477970.67724188970.68082249120.68438630350.6879330506
0.50.69146246130.69497426910.69846821250.70194403460.70540148380.70884031320.71226028120.7156611510.71904269110.7224046752
0.60.72574688220.72906909620.73237110650.73565270790.73891370030.74215388920.74537308530.74857110490.75174776950.7549029063
0.70.75803634780.76114793190.76423750220.76730490770.77035000280.77337264760.77637270760.77935005370.78230456240.7852361158
0.80.78814460140.79102991210.79389194640.79673060820.79954580670.80233745690.80510547870.80784979790.81057034520.813267057
0.90.81593987470.81858874510.82121362040.82381445780.82639121970.82894387370.83147239250.83397675390.83645694070.8389129405
10.84134474610.8437523550.84613576960.84849499720.85083004970.85314094360.85542770030.85769034560.85992890990.862143428
1.10.86433393910.86650048680.8686431190.87076188780.87285684940.87492806440.87697559690.87899951560.88099989250.882976804
1.20.88493032980.88686055360.88876756260.89065144760.89251230290.89435022630.89616531890.89795768490.8997274320.901474671
1.30.90319951540.90490208220.9065824910.90824086430.90987732750.91149200860.91308503810.91465654920.91620667760.9177355613
1.40.91924334080.92073015850.92219615950.92364149050.92506630050.92647074040.9278549630.9292191230.93056337670.931887882
1.50.93319279870.93447828790.93574451220.93699163550.93821982330.9394292420.94062005940.94179244440.94294656680.9440825975
1.60.94520070830.94630107190.94738386150.94844925150.94949741650.9505285320.95154277370.95254031820.95352134210.9544860227
1.70.95543453720.95636706350.95728377920.95818486240.9590704910.95994084310.96079609670.96163642960.96246201970.9632730443
1.80.96406968090.96485210640.96562049760.96637503060.96711588130.96784322520.9685572370.96925809110.9699459610.97062102
1.90.97128344020.97193339330.97257105030.97319658110.97381015510.97441194050.97500210490.97558081470.97614823570.9767045322
20.97724986810.97778440560.97830830620.97882173040.97932483710.97981778460.98030072960.98077382780.98123723360.9816911001
2.10.98213557940.98257082210.98299697740.98341419330.98382261660.98422239260.98461366520.9849965770.98537126920.9857378816
2.20.98609655250.98644741890.98679061620.98712627860.98745453860.98777552730.98808937460.98839620850.98869615580.9889893417
2.30.989275890.98955592290.98982956130.99009692440.99035813010.99061329450.99086253250.99110595740.9913436810.9915758136
2.40.99180246410.99202373970.99223974640.99245058860.9926563690.99285718930.99305314920.99324434740.99343088090.9936128452
2.50.99379033470.99396344190.99413225830.99429687370.99445737660.9946138540.99476639180.99491507430.99505998420.9952012034
2.60.9953388120.99547288890.99560351170.99573075660.99585469860.99597541150.99609296740.99620743770.9963188920.996427399
2.70.99653302620.99663583960.99673590420.99683328370.99692804080.99702023680.99710993190.99719718540.99728205510.9973645979
2.80.99744486970.9975229250.99759881750.99767259980.99774432330.99781403850.9978817950.9979476410.99801162410.9980737909
2.90.99813418670.99819285620.99824984310.998305190.99835893880.99841113040.99846180480.99851100130.99855875810.9986051128
30.9986501020.99869376160.99873612660.99877723130.99881710930.99885579320.9988933150.99892970610.9989649970.9989992175
3.10.99903239680.99906456330.99909574480.99912596850.99915526080.99918364770.99921115430.99923780530.99926362470.999288636
3.20.99931286210.99933632510.9993590470.99938104890.99940235150.9994229750.99944293890.99946226260.99948096460.9994990631
3.30.99951657590.99953352010.99954991280.99956577010.99958110810.99959594220.99961028760.99962415910.99963757090.9996505369
3.40.99966307070.99967518560.99968689430.99969820940.99970914290.99971970670.99972991230.99973977080.99974929310.9997584897
3.50.99976737090.99977594670.99978422660.99979222020.99979993650.99980738440.99981457260.99982150940.99982820290.999834661
3.60.99984089140.99984690150.99985269850.99985828940.9998636810.99986887980.99987389240.99987872480.9998833830.999887873
3.70.99989220030.99989637040.99990038860.99990426010.99990798990.99991158270.99991504330.99991837620.99992158580.9999246764
3.80.9999276520.99993051660.99993327420.99993592840.99993848280.99994094110.99994330650.99994558230.99994777180.9999498779
3.90.99995190370.99995385190.99995572550.99995752710.99995925920.99996092440.99996252510.99996406370.99996554240.9999669634
40.99996832880.99996964060.99997090090.99997211160.99997327440.99997439120.99997546360.99997649340.99997748210.9999784313
4.10.99997934250.9999802170.99998105640.99998186180.99998263470.99998337620.99998408760.999984770.99998542450.9999860523
4.20.99998665430.99998723150.99998778490.99998831540.9999888240.99998931150.99998977870.99999022640.99999065530.9999910663

## Inverse Z table

#### Z1-α=-Zα

For any practical use you can use the above calculator instead of the inverse Z table. Just choose P(x≤X) and insert the relevant α The below inverse z table is a bit more accurate.
Example: (row number 6)
P( x ≤ -1.644854 ) = 0.05,  P( x ≤ 1.644854 ) = 0.95,  P( x ≤ -1.959964 ) = 0.025,  P( x ≤ 1.959964 ) = 0.975

α Zα Z1-α Zα/2 Z1-α/2
0.001-3.0902323.090232-3.2905273.290527
0.0025-2.8070342.807034-3.0233413.023341
0.005-2.5758292.575829-2.8070342.807034
0.01-2.3263482.326348-2.5758292.575829
0.025-1.9599641.959964-2.2414032.241403
0.05-1.6448541.644854-1.9599641.959964
0.1-1.2815521.281552-1.6448541.644854
0.15-1.0364331.036433-1.4395311.439531
0.2-0.8416210.841621-1.2815521.281552
0.25-0.674490.67449-1.1503491.150349
0.3-0.5244010.524401-1.0364331.036433
0.35-0.385320.38532-0.9345890.934589
0.4-0.2533470.253347-0.8416210.841621