Definition
1∞2
e−t /2 dt
Good Approximation (good for programming in calculator)
Q(x)=√
2π x
1 1 −x2/2 Q(x)≈ √√e,
(1−a)x+a x2+b 2π where a = 1/π, b = 2π
Simple Upper Bound
Q ( x ) < 21 e − x 2 / 2
Relation to Error Functions
1x √
Q(x) = 2 erfc √2 , Property
erfc(x) = 2Q(x
2)
Q(−x) = 1 − Q(x) x 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.00 5.0000e-01 0.10 4.6017e-01 0.20 4.2074e-01 0.30 3.8209e-01 0.40 3.4458e-01 0.50 3.0854e-01 0.60 2.7425e-01 0.70 2.4196e-01 0.80 2.1186e-01 0.90 1.8406e-01 1.00 1.5866e-01 1.10 1.3567e-01 1.20 1.1507e-01 1.30 9.6800e-02 1.40 8.0757e-02 1.50 6.6807e-02 1.60 5.4799e-02 1.70 4.4565e-02 1.80 3.5930e-02 1.90 2.8717e-02
4.9601e-01 4.5620e-01 4.1683e-01 3.7828e-01 3.4090e-01 3.0503e-01 2.7093e-01 2.3885e-01 2.0897e-01 1.8141e-01 1.5625e-01 1.3350e-01 1.1314e-01 9.5098e-02 7.9270e-02 6.5522e-02 5.3699e-02 4.3633e-02 3.5148e-02 2.8067e-02
4.9202e-01 4.5224e-01 4.1294e-01 3.7448e-01 3.3724e-01 3.0153e-01 2.6763e-01 2.3576e-01 2.0611e-01 1.7879e-01 1.5386e-01 1.3136e-01 1.1123e-01 9.3418e-02 7.7804e-02 6.4255e-02 5.2616e-02 4.2716e-02 3.4380e-02 2.7429e-02
4.8803e-01 4.4828e-01 4.0905e-01 3.7070e-01 3.3360e-01 2.9806e-01 2.6435e-01 2.3270e-01 2.0327e-01 1.7619e-01 1.5151e-01 1.2924e-01 1.0935e-01 9.1759e-02 7.6359e-02 6.3008e-02 5.1551e-02 4.1815e-02 3.3625e-02 2.6803e-02
4.8405e-01 4.4433e-01 4.0517e-01 3.6693e-01 3.2997e-01 2.9460e-01 2.6109e-01 2.2965e-01 2.0045e-01 1.7361e-01 1.4917e-01 1.2714e-01 1.0749e-01 9.0123e-02 7.4934e-02 6.1780e-02 5.0503e-02 4.0930e-02 3.2884e-02 2.6190e-02
4.8006e-01 4.4038e-01 4.0129e-01 3.6317e-01 3.2636e-01 2.9116e-01 2.5785e-01 2.2663e-01 1.9766e-01 1.7106e-01 1.4686e-01 1.2507e-01 1.0565e-01 8.8508e-02 7.3529e-02 6.0571e-02 4.9471e-02 4.0059e-02 3.2157e-02 2.5588e-02
4.7608e-01 4.3644e-01 3.9743e-01 3.5942e-01 3.2276e-01 2.8774e-01 2.5463e-01 2.2363e-01 1.9489e-01 1.6853e-01 1.4457e-01 1.2302e-01 1.0383e-01 8.6915e-02 7.2145e-02 5.9380e-02 4.8457e-02 3.9204e-02 3.1443e-02 2.4998e-02
4.7210e-01 4.3251e-01 3.9358e-01 3.5569e-01 3.1918e-01 2.8434e-01 2.5143e-01 2.2065e-01 1.9215e-01 1.6602e-01 1.4231e-01 1.2100e-01 1.0204e-01 8.5343e-02 7.0781e-02 5.8208e-02 4.7460e-02 3.8364e-02 3.0742e-02 2.4419e-02
4.6812e-01 4.2858e-01 3.8974e-01 3.5197e-01 3.1561e-01 2.8096e-01 2.4825e-01 2.1770e-01 1.8943e-01 1.6354e-01 1.4007e-01 1.1900e-01 1.0027e-01 8.3793e-02 6.9437e-02 5.7053e-02 4.6479e-02 3.7538e-02 3.0054e-02 2.3852e-02
4.6414e-01 4.2465e-01 3.8591e-01 3.4827e-01 3.1207e-01 2.7760e-01 2.4510e-01 2.1476e-01 1.8673e-01 1.6109e-01 1.3786e-01 1.1702e-01 9.8525e-02 8.2264e-02 6.8112e-02 5.5917e-02 4.5514e-02 3.6727e-02 2.9379e-02 2.3295e-02
Q-Function
x 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
2.00 2.2750e-02 2.10 1.7864e-02 2.20 1.3903e-02 2.30 1.0724e-02 2.40 8.1975e-03 2.50 6.2097e-03 2.60 4.6612e-03 2.70 3.4670e-03 2.80 2.5551e-03 2.90 1.8658e-03 3.00 1.3499e-03 3.10 9.6760e-04 3.20 6.8714e-04 3.30 4.8342e-04 3.40 3.3693e-04 3.50 2.3263e-04 3.60 1.5911e-04 3.70 1.0780e-04 3.80 7.2348e-05 3.90 4.8096e-05 4.00 3.1671e-05 4.10 2.0658e-05 4.20 1.3346e-05 4.30 8.5399e-06 4.40 5.4125e-06 4.50 3.3977e-06 4.60 2.1125e-06 4.70 1.3008e-06 4.80 7.9333e-07 4.90 4.7918e-07 5.00 2.8665e-07 5.10 1.6983e-07 5.20 9.9644e-08 5.30 5.7901e-08 5.40 3.3320e-08 5.50 1.8990e-08 5.60 1.0718e-08 5.70 5.9904e-09 5.80 3.3157e-09 5.90 1.8175e-09
2.2216e-02 1.7429e-02 1.3553e-02 1.0444e-02 7.9763e-03 6.0366e-03 4.5271e-03 3.3642e-03 2.4771e-03 1.8071e-03 1.3062e-03 9.3544e-04 6.6367e-04 4.6648e-04 3.2481e-04 2.2405e-04 1.5310e-04 1.0363e-04 6.9483e-05 4.6148e-05 3.0359e-05 1.9783e-05 1.2769e-05 8.1627e-06 5.1685e-06 3.2414e-06 2.0133e-06 1.2386e-06 7.5465e-07 4.5538e-07 2.7215e-07 1.6108e-07 9.4420e-08 5.4813e-08 3.1512e-08 1.7942e-08 1.0116e-08 5.6488e-09 3.1236e-09 1.7105e-09
2.1692e-02 1.7003e-02 1.3209e-02 1.0170e-02 7.7603e-03 5.8677e-03 4.3965e-03 3.2641e-03 2.4012e-03 1.7502e-03 1.2639e-03 9.0426e-04 6.4095e-04 4.5009e-04 3.1311e-04 2.1577e-04 1.4730e-04 9.9611e-05 6.6726e-05 4.4274e-05 2.9099e-05 1.8944e-05 1.2215e-05 7.8015e-06 4.9350e-06 3.0920e-06 1.9187e-06 1.1792e-06 7.1779e-07 4.3272e-07 2.5836e-07 1.5277e-07 8.9462e-08 5.1884e-08 2.9800e-08 1.6950e-08 9.5479e-09 5.3262e-09 2.9424e-09 1.6097e-09
2.1178e-02 1.6586e-02 1.2874e-02 9.9031e-03 7.5494e-03 5.7031e-03 4.2692e-03 3.1667e-03 2.3274e-03 1.6948e-03 1.2228e-03 8.7403e-04 6.1895e-04 4.3423e-04 3.0179e-04 2.0778e-04 1.4171e-04 9.5740e-05 6.4072e-05 4.2473e-05 2.7888e-05 1.8138e-05 1.1685e-05 7.4555e-06 4.7117e-06 2.9492e-06 1.8283e-06 1.1226e-06 6.8267e-07 4.1115e-07 2.4524e-07 1.4487e-07 8.4755e-08 4.9106e-08 2.8177e-08 1.6012e-08 9.0105e-09 5.0215e-09 2.7714e-09 1.5147e-09
2.0675e-02 1.6177e-02 1.2545e-02 9.6419e-03 7.3436e-03 5.5426e-03 4.1453e-03 3.0720e-03 2.2557e-03 1.6411e-03 1.1829e-03 8.4474e-04 5.9765e-04 4.1889e-04 2.9086e-04 2.0006e-04 1.3632e-04 9.2010e-05 6.1517e-05 4.0741e-05 2.6726e-05 1.7365e-05 1.1176e-05 7.1241e-06 4.4979e-06 2.8127e-06 1.7420e-06 1.0686e-06 6.4920e-07 3.9061e-07 2.3277e-07 1.3737e-07 8.0288e-08 4.6473e-08 2.6640e-08 1.5124e-08 8.5025e-09 4.7338e-09 2.6100e-09 1.4251e-09
2.0182e-02 1.5778e-02 1.2224e-02 9.3867e-03 7.1428e-03 5.3861e-03 4.0246e-03 2.9798e-03 2.1860e-03 1.5889e-03 1.1442e-03 8.1635e-04 5.7703e-04 4.0406e-04 2.8029e-04 1.9262e-04 1.3112e-04 8.8417e-05 5.9059e-05 3.9076e-05 2.5609e-05 1.6624e-05 1.0689e-05 6.8069e-06 4.2935e-06 2.6823e-06 1.6597e-06 1.0171e-06 6.1731e-07 3.7107e-07 2.2091e-07 1.3024e-07 7.6050e-08 4.3977e-08 2.5185e-08 1.4283e-08 8.0224e-09 4.4622e-09 2.4579e-09 1.3407e-09
1.9699e-02 1.5386e-02 1.1911e-02 9.1375e-03 6.9469e-03 5.2336e-03 3.9070e-03 2.8901e-03 2.1182e-03 1.5382e-03 1.1067e-03 7.8885e-04 5.5706e-04 3.8971e-04 2.7009e-04 1.8543e-04 1.2611e-04 8.4957e-05 5.6694e-05 3.7475e-05 2.4536e-05 1.5912e-05 1.0221e-05 6.5031e-06 4.0980e-06 2.5577e-06 1.5810e-06 9.6796e-07 5.8693e-07 3.5247e-07 2.0963e-07 1.2347e-07 7.2028e-08 4.1611e-08 2.3807e-08 1.3489e-08 7.5686e-09 4.2057e-09 2.3143e-09 1.2612e-09
1.9226e-02 1.5003e-02 1.1604e-02 8.8940e-03 6.7557e-03 5.0849e-03 3.7926e-03 2.8028e-03 2.0524e-03 1.4890e-03 1.0703e-03 7.6219e-04 5.3774e-04 3.7584e-04 2.6023e-04 1.7849e-04 1.2128e-04 8.1624e-05 5.4418e-05 3.5936e-05 2.3507e-05 1.5230e-05 9.7736e-06 6.2123e-06 3.9110e-06 2.4386e-06 1.5060e-06 9.2113e-07 5.5799e-07 3.3476e-07 1.9891e-07 1.1705e-07 6.8212e-08 3.9368e-08 2.2502e-08 1.2737e-08 7.1399e-09 3.9636e-09 2.1790e-09 1.1863e-09
1.8763e-02 1.4629e-02 1.1304e-02 8.6563e-03 6.5691e-03 4.9400e-03 3.6811e-03 2.7179e-03 1.9884e-03 1.4412e-03 1.0350e-03 7.3638e-04 5.1904e-04 3.6243e-04 2.5071e-04 1.7180e-04 1.1662e-04 7.8414e-05 5.2228e-05 3.4458e-05 2.2518e-05 1.4575e-05 9.3447e-06 5.9340e-06 3.7322e-06 2.3249e-06 1.4344e-06 8.7648e-07 5.3043e-07 3.1792e-07 1.8872e-07 1.1094e-07 6.4592e-08 3.7243e-08 2.1266e-08 1.2026e-08 6.7347e-09 3.7350e-09 2.0513e-09 1.1157e-09
1.8309e-02 1.4262e-02 1.1011e-02 8.4242e-03 6.3872e-03 4.7988e-03 3.5726e-03 2.6354e-03 1.9262e-03 1.3949e-03 1.0008e-03 7.1136e-04 5.0094e-04 3.4946e-04 2.4151e-04 1.6534e-04 1.1213e-04 7.5324e-05 5.0122e-05 3.3037e-05 2.1569e-05 1.3948e-05 8.9337e-06 5.6675e-06 3.5612e-06 2.2162e-06 1.3660e-06 8.3391e-07 5.0418e-07 3.0190e-07 1.7903e-07 1.0515e-07 6.1158e-08 3.5229e-08 2.0097e-08 1.1353e-08 6.3520e-09 3.5193e-09 1.9310e-09 1.0492e-09