21.8 The Gaussian distribution 617
so thatF(a)is the probability that the variable has valuex 1 < 1 a, and
(21.41)
for unit total probability. The probability that the variable has a value in the interval
a 1 < 1 x 1 < 1 bis then
(21.42)
and this is the area under the curve betweenx 1 = 1 aandx 1 = 1 b.
The integral in (21.40) cannot be evaluated by the methods of the calculus
described in Chapters 5 and 6, but extensive tabulations are found in most statistics
texts. These tabulations are of the auxiliary function
(21.43)
obtained from (21.40) by means of the substitution. Then
and
(21.44)
It is found from these tabulations that the probabilities that xlies withinσ, 2 σ,and
3 σof the mean are
P(μ 1 − 1 σ 1 < 1 x 1 < 1 μ 1 + 1 σ) = 1 0.6827
P(μ 1 − 12 σ 1 < 1 x 1 < 1 μ 1 + 12 σ) 1 = 1 0.9545 (21.45)
P(μ 1 − 13 σ 1 < 1 x 1 < 1 μ 1 + 13 σ) 1 = 1 0.9973
Therefore, about 68% of observed values are expected to lie within one standard
deviation of the mean (Figure 21.8), 95% within two standard deviations, and almost
all within three standard deviations.
Pa x b
ba
()<< =
−
−
−
ΦΦ
μ
σ
μ
σ
Fx
x
()=
−
Φ
μ
σ
z
x
=
−μ
σ
Φ()zedz
zz=
−−1
2
22π
Z
∞Pa x b Fb Fa e dx
abx()()()
()<< = − =
−1
2
222σ
μσπ
Z
Fx() ()→= xdx=
−∞
∞∞Z ρ 1
.......
........
.......
........................x
μ
μ− 3 σμ− 2 σμ−σ μ+σμ+ 2 σμ+3σ68%
16% 16%
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.............................
............................Figure 21.8