See legend defining symbols at bottom of table. The statistics for some other tests have their own articles, including the Wald test and the likelihood ratio test .
統計量
公式
Assumptions or notes
z-検定
z
=
x
¯
−
μ
0
(
σ
/
n
)
{\displaystyle z={\frac {{\overline {x}}-\mu _{0}}{(\sigma /{\sqrt {n}})}}}
(正規母集団 or n > 30) and σが既知.(z is the distance from the mean in relation to 平均からの標準偏差). For non-normal distributions it is possible to calculate a minimum proportion of a population that falls within k standard deviations for any k (see: Chebyshev's inequality
t-検定
t
=
x
¯
−
μ
0
(
s
/
n
)
,
{\displaystyle t={\frac {{\overline {x}}-\mu _{0}}{(s/{\sqrt {n}})}},}
d
f
=
n
−
1
{\displaystyle df=n-1\ }
(Normal population or n > 30) and σ unknown
One-sample chi-square test
χ
2
=
(
n
−
1
)
s
2
σ
0
2
{\displaystyle \chi ^{2}={\frac {(n-1)s^{2}}{\sigma _{0}^{2}}}}
One of the following
• All expected counts are at least 5
• All expected counts are > 1 and no more that 20% of expected counts are less than 5
*Two-sample F test for equality of variances
F
=
s
1
2
s
2
2
{\displaystyle F={\frac {s_{1}^{2}}{s_{2}^{2}}}}
Arrange so
s
1
2
{\displaystyle s_{1}^{2}}
>
s
2
2
{\displaystyle s_{2}^{2}}
0 for
F
>
F
(
α
/
2
,
n
1
−
1
,
n
2
−
1
)
{\displaystyle F>F(\alpha /2,n_{1}-1,n_{2}-1)}
[ 1]
Definition of symbols
α
{\displaystyle \alpha }
probability of Type I error (rejecting a null hypothesis when it is in fact true)
n
{\displaystyle n}
sample size
n
1
{\displaystyle n_{1}}
n
2
{\displaystyle n_{2}}
x
¯
{\displaystyle {\overline {x}}}
sample mean
μ
0
{\displaystyle \mu _{0}}
population mean
μ
1
{\displaystyle \mu _{1}}
μ
2
{\displaystyle \mu _{2}}
σ
{\displaystyle \sigma }
population standard deviation
σ
2
{\displaystyle \sigma ^{2}}
population variance
s
{\displaystyle s}
sample standard deviation
s
2
{\displaystyle s^{2}}
sample variance
s
1
{\displaystyle s_{1}}
s
2
{\displaystyle s_{2}}
t
{\displaystyle t}
t statistic
d
f
{\displaystyle df}
degrees of freedom
d
¯
{\displaystyle {\overline {d}}}
d
0
{\displaystyle d_{0}}
s
d
{\displaystyle s_{d}}
p
^
{\displaystyle {\hat {p}}}
x/n = sample proportion , unless specified otherwise
p
0
{\displaystyle p_{0}}
p
1
{\displaystyle p_{1}}
p
2
{\displaystyle p_{2}}
min
{
n
1
,
n
2
}
{\displaystyle \min\{n_{1},n_{2}\}}
n 1 and n 2
x
1
=
n
1
p
1
{\displaystyle x_{1}=n_{1}p_{1}}
x
2
=
n
2
p
2
{\displaystyle x_{2}=n_{2}p_{2}}
χ
2
{\displaystyle \chi ^{2}}
Chi-squared statistic
F
{\displaystyle F}
F statistic
In general, the subscript 0 indicates a value taken from the null hypothesis , H0 , which should be used as much as possible in constructing its test statistic.
^ NIST handbook: F-Test for Equality of Two Standard Deviations (should say "Variances")
