Error Function Chart
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the error function is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial erfc function table differential equations. It is also called the Gauss error function or probability integral. The error function is defined as: Error Function Table The following is the error function and complementary error function table that shows the values of erf(x) and erfc(x) for x ranging from 0 to 3.5 with increment of 0.01. xerf(x)erfc(x)0.00.01.00.010.0112834160.9887165840.020.0225645750.9774354250.030.0338412220.9661587780.040.0451111060.9548888940.050.0563719780.9436280220.060.0676215940.9323784060.070.078857720.921142280.080.0900781260.9099218740.090.1012805940.8987194060.10.1124629160.8875370840.110.1236228960.8763771040.120.1347583520.8652416480.130.1458671150.8541328850.140.1569470330.8430529670.150.1679959710.8320040290.160.1790118130.8209881870.170.1899924610.8100075390.180.2009358390.7990641610.190.2118398920.7881601080.20.2227025890.7772974110.210.2335219230.7664780770.220.2442959120.7557040880.230.25502260.74497740.240.2657000590.7342999410.250.276326390.723673610.260.2868997230.7131002770.270.2974182190.7025817810.280.3078800680.6921199320.290.3182834960.6817165040.30.3286267590.6713732410.310.338908150.661091850.320.3491259950.6508740050.330.3592786550.6407213450.340.3693645290.6306354710.350.3793820540.6206179460.360.3893297010.6106702990.370.3992059840.6007940160.380.4090094530.5909905470.390.41873870.58126130.40.4283923550.5716076450.410.437969090.562030910.420.4474676180.5525323820.430.4568866950.5431133050.440.4662251150.5337748850.450.475481720.524518280.460.484655390.515344610.470.4937450510.5062549490.480.5027496710.4972503290.490.5116682610.4883317390.50.5204998780.4795001220.510.529243620.470756380.520.537898630.462101370.530.5464640970.4535359030.540.554939250.445060750.550.5633233660.4366766340.560.5716157640.4283842360.570.5798158060.4201841940.580.58792290.41207710.590.5959364970.4040635030.60.6038560910.3961439090.610.6116812190.3883187810.620.6194114620.3805885380.630.6270464430.3729535570.640.6345858290.3654141710.650.6420293270.3579706730.660.6493766880.3506233120.670.6566277020.3433722980.680.6637822030.3362177970.690.6708400620.3291599380.70.6778011940.3221988060.710.684665550.315334450.720.
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In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations above the mean. If the underlying random variable is y, https://en.wikipedia.org/wiki/Q-function then the proper argument to the tail probability is derived as: x = y − μ σ {\displaystyle x={\frac {y-\mu }{\sigma }}} which expresses the number of standard deviations away from the mean. Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.[3] Because of its relation to the cumulative distribution function of the normal distribution, the error function Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics. Contents 1 Definition and basic properties 2 Values 3 Generalization to high dimensions 4 References Definition and basic properties[edit] Formally, the Q-function is defined as Q ( x ) = 1 2 π ∫ x ∞ exp ( − u 2 2 ) d u error function table . {\displaystyle Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }\exp \left(-{\frac {u^{2}}{2}}\right)\,du.} Thus, Q ( x ) = 1 − Q ( − x ) = 1 − Φ ( x ) , {\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,} where Φ ( x ) {\displaystyle \Phi (x)} is the cumulative distribution function of the normal Gaussian distribution. The Q-function can be expressed in terms of the error function, or the complementary error function, as[2] Q ( x ) = 1 2 ( 2 π ∫ x / 2 ∞ exp ( − t 2 ) d t ) = 1 2 − 1 2 erf ( x 2 ) -or- = 1 2 erfc ( x 2 ) . {\displaystyle {\begin{aligned}Q(x)&={\frac {1}{2}}\left({\frac {2}{\sqrt {\pi }}}\int _{x/{\sqrt {2}}}^{\infty }\exp \left(-t^{2}\right)\,dt\right)\\&={\frac {1}{2}}-{\frac {1}{2}}\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)~~{\text{ -or-}}\\&={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right).\end{aligned}}} An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:[4] Q ( x ) = 1 π ∫ 0 π 2 exp ( − x 2 2 sin 2 θ ) d θ . {\displaystyle Q(x)={\frac {1}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{2\sin ^{2}\theta }}\right)d\theta .} This expression is valid only for
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