tangent
Not Supported
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n
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x
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,
Search
Returned 99 matches (100 formulae, 63 docs)
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r
x
y
=
∑
i
=
1
n
(
x
i
-
x
¯
)
(
y
i
-
y
¯
)
(
n
-
1
)
s
x
s
y
=
∑
i
=
1
n
(
x
i
-
x
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)
(
y
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i
=
1
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x
i
-
x
¯
)
2
∑
i
=
1
n
(
y
i
-
y
¯
)
2
,
Doc 1
1.0000, 0.0000, 75.0000, 1.2213
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Correlation_and_dependence.html
∑
k
=
1
n
(
x
k
-
x
¯
)
(
y
k
-
y
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)
∑
k
=
1
n
(
x
k
-
x
¯
)
2
∑
k
=
1
n
(
y
k
-
y
¯
)
2
Doc 2
0.5704, 0.0000, 36.0000, 0.5704
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Dimensionless_quantity.html
r
=
r
x
y
=
∑
i
=
1
n
(
x
i
-
x
¯
)
(
y
i
-
y
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)
∑
i
=
1
n
(
x
i
-
x
¯
)
2
∑
i
=
1
n
(
y
i
-
y
¯
)
2
Doc 3
0.4426, -16.0000, 34.0000, 0.8511
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Pearson_product-moment_correlation_coefficient.html
r
=
∑
i
=
1
N
(
X
i
-
X
¯
)
(
Y
i
-
Y
¯
)
∑
i
=
1
N
(
X
i
-
X
¯
)
2
∑
i
=
1
N
(
Y
i
-
Y
¯
)
2
Doc 4
0.4228, -14.0000, 24.0000, 0.4228
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Fisher_transformation.html
r
(
α
,
β
)
=
∑
i
(
j
i
α
-
j
α
¯
)
(
k
i
β
-
k
β
¯
)
∑
i
(
j
i
α
-
j
α
¯
)
2
∑
i
(
k
i
β
-
k
β
¯
)
2
.
Doc 5
0.3422, -24.0000, 20.0000, 0.3422
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Assortativity.html
simil
(
x
,
y
)
=
∑
i
∈
I
x
y
(
r
x
,
i
-
r
x
¯
)
(
r
y
,
i
-
r
y
¯
)
∑
i
∈
I
x
y
(
r
x
,
i
-
r
x
¯
)
2
∑
i
∈
I
x
y
(
r
y
,
i
-
r
y
¯
)
2
Doc 6
0.3257, -37.0000, 23.0000, 0.3257
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Collaborative_filtering.html
=
(
n
+
1
)
n
(
n
-
1
)
(
n
-
2
)
(
n
-
3
)
∑
i
=
1
n
(
x
i
-
x
¯
)
4
(
∑
i
=
1
n
(
x
i
-
x
¯
)
2
)
2
-
3
(
n
-
1
)
2
(
n
-
2
)
(
n
-
3
)
Doc 7
0.3082, -38.0000, 24.0000, 0.6165
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Kurtosis.html
β
^
=
∑
i
=
1
n
(
x
i
-
x
¯
)
(
y
i
-
y
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)
∑
i
=
1
n
(
x
i
-
x
¯
)
2
=
x
y
¯
-
x
¯
y
¯
x
2
¯
-
x
¯
2
=
Cov
[
x
,
y
]
Var
[
x
]
=
r
x
y
s
y
s
x
,
Doc 8
0.2885, -49.0000, 21.0000, 0.2885
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Regression_toward_the_mean.html
λ
=
E
(
V
)
stdev
(
V
)
=
∑
i
=
1
t
c
i
μ
i
Var
(
∑
i
=
1
t
c
i
G
i
)
=
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i
=
1
t
c
i
μ
i
∑
i
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1
t
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i
2
σ
i
2
+
2
∑
i
=
1
t
∑
j
=
i
c
i
c
j
σ
i
j
Doc 9
0.2679, -52.0000, 18.0000, 0.2679
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Standardized_mean_of_a_contrast_variable.html
b
^
=
∑
i
=
1
n
(
x
i
-
x
¯
)
(
y
i
-
y
¯
)
∑
i
=
1
n
(
x
i
-
x
¯
)
2
,
Doc 10
0.2616, -14.0000, 20.0000, 0.7912
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Explained_sum_of_squares.html
similarity
=
cos
(
θ
)
=
A
⋅
B
∥
A
∥
∥
B
∥
=
∑
i
=
1
n
A
i
×
B
i
∑
i
=
1
n
(
A
i
)
2
×
∑
i
=
1
n
(
B
i
)
2
Doc 11
0.2616, -27.0000, 18.0000, 0.2616
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Cosine_similarity.html
β
^
=
∑
i
=
1
n
(
y
i
-
y
¯
)
(
x
i
-
x
¯
)
∑
i
=
1
n
(
x
i
-
x
¯
)
2
=
C
o
v
(
x
,
y
)
V
a
r
(
x
)
Doc 12
0.2616, -30.0000, 16.0000, 0.7643
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Simple_linear_regression.html
1
=
∑
i
(
Y
i
-
Y
^
i
)
2
∑
i
(
Y
i
-
Y
¯
)
2
+
∑
i
(
Y
^
i
-
Y
¯
)
2
∑
i
(
Y
i
-
Y
¯
)
2
.
Doc 3
0.4426, -16.0000, 34.0000, 0.8511
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Pearson_product-moment_correlation_coefficient.html
n
(
X
¯
-
μ
0
)
∑
i
=
1
n
(
X
i
-
X
¯
)
2
/
(
n
-
1
)
Doc 13
0.2252, -8.0000, 16.0000, 0.3793
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Degrees_of_freedom_(statistics).html
C
n
=
∑
i
=
1
n
(
x
i
-
x
¯
n
)
(
y
i
-
y
¯
n
)
Doc 14
0.2213, -5.0000, 17.0000, 0.7569
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Algorithms_for_calculating_variance.html
s
x
y
=
1
N
∑
n
=
1
N
(
x
n
-
x
¯
)
(
y
n
-
y
¯
)
.
Doc 15
0.2213, -8.0000, 13.0000, 0.2213
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Concordance_correlation_coefficient.html
r
=
1
n
-
1
∑
i
=
1
n
(
X
i
-
X
¯
)
(
Y
i
-
Y
¯
)
s
X
s
Y
Doc 16
0.2213, -12.0000, 13.0000, 0.2213
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Pivotal_quantity.html
∑
i
=
1
n
(
y
i
-
y
¯
)
(
x
i
-
x
¯
)
-
β
^
∑
i
=
1
n
(
x
i
-
x
¯
)
2
=
0
Doc 12
0.2616, -30.0000, 16.0000, 0.7643
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Simple_linear_regression.html
β
^
=
1
T
∑
t
=
1
T
(
x
t
-
x
¯
)
(
y
t
-
y
¯
)
1
T
∑
t
=
1
T
(
x
t
-
x
¯
)
2
,
Doc 17
0.2213, -23.0000, 13.0000, 0.2213
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Errors-in-variables_models.html
r
x
y
=
∑
x
i
y
i
-
n
x
¯
y
¯
(
n
-
1
)
s
x
s
y
=
n
∑
x
i
y
i
-
∑
x
i
∑
y
i
n
∑
x
i
2
-
(
∑
x
i
)
2
n
∑
y
i
2
-
(
∑
y
i
)
2
.
Doc 1
1.0000, 0.0000, 75.0000, 1.2213
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Correlation_and_dependence.html
∑
i
=
1
n
2
(
y
^
i
-
y
¯
)
(
y
i
-
y
^
i
)
=
∑
i
=
1
n
2
b
^
(
(
y
i
-
y
¯
)
(
x
i
-
x
¯
)
-
b
^
(
x
i
-
x
¯
)
2
)
=
2
b
^
(
∑
i
=
1
n
(
y
i
-
y
¯
)
(
x
i
-
x
¯
)
-
b
^
∑
i
=
1
n
(
x
i
-
x
¯
)
2
)
=
2
b
^
∑
i
=
1
n
(
(
y
i
-
y
¯
)
(
x
i
-
x
¯
)
-
(
y
i
-
y
¯
)
(
x
i
-
x
¯
)
)
=
2
b
^
⋅
0
=
0.
Doc 10
0.2616, -14.0000, 20.0000, 0.7912
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Explained_sum_of_squares.html
q
x
y
=
∑
(
x
-
x
¯
)
(
y
-
y
¯
)
I
(
x
,
y
)
∑
I
(
x
,
y
)
Doc 18
0.2140, -10.0000, 15.0000, 0.7075
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Gravitational_lensing_formalism.html
q
x
y
=
∑
(
x
-
x
¯
)
(
y
-
y
¯
)
w
(
x
-
x
¯
,
y
-
y
¯
)
I
(
x
,
y
)
∑
w
(
x
-
x
¯
,
y
-
y
¯
)
I
(
x
,
y
)
Doc 18
0.2140, -10.0000, 15.0000, 0.7075
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Gravitational_lensing_formalism.html
sim
(
d
j
,
q
)
=
𝐝
𝐣
⋅
𝐪
∥
𝐝
𝐣
∥
∥
𝐪
∥
=
∑
i
=
1
N
w
i
,
j
w
i
,
q
∑
i
=
1
N
w
i
,
j
2
∑
i
=
1
N
w
i
,
q
2
Doc 19
0.2140, -37.0000, 14.0000, 0.2140
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Vector_space_model.html
𝐖
=
∑
i
=
1
n
x
(
𝐱
i
-
𝐱
¯
)
(
𝐱
i
-
𝐱
¯
)
′
+
∑
i
=
1
n
y
(
𝐲
i
-
𝐲
¯
)
(
𝐲
i
-
𝐲
¯
)
′
n
x
+
n
y
-
2
Doc 20
0.2116, -31.0000, 15.0000, 0.2116
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Hotelling's_T-squared_distribution.html
β
1
^
=
∑
(
x
i
-
x
¯
)
(
y
i
-
y
¯
)
∑
(
x
i
-
x
¯
)
2
and
β
0
^
=
y
¯
-
β
1
^
x
¯
Doc 21
0.2079, -23.0000, 15.0000, 0.2079
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Regression_analysis.html
r
=
∑
i
=
1
n
i
(
∏
j
=
1
i
n
-
j
+
1
j
)
(
[
L
]
K
d
)
i
1
+
∑
i
=
1
n
(
∏
j
=
1
i
n
-
j
+
1
j
)
(
[
L
]
K
d
)
i
=
∑
i
=
1
n
i
(
n
i
)
(
[
L
]
K
d
)
i
1
+
∑
i
=
1
n
(
n
i
)
(
[
L
]
K
d
)
i
Doc 22
0.2062, -74.0000, 16.0000, 0.2062
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Dissociation_constant.html
R
M
D
(
S
)
=
∑
i
=
1
n
∑
j
=
1
n
|
y
i
-
y
j
|
(
n
-
1
)
∑
i
=
1
n
y
i
Doc 23
0.2005, -18.0000, 15.0000, 0.3524
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Mean_absolute_difference.html
S
E
β
^
=
1
n
-
2
∑
i
=
1
n
(
y
i
-
y
^
i
)
2
∑
i
=
1
n
(
x
i
-
x
¯
)
2
Doc 24
0.1944, -23.0000, 15.0000, 0.3871
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Student's_t-test.html
=
[
∑
i
=
1
N
(
x
i
2
-
x
i
1
)
(
x
i
2
-
x
i
1
)
′
]
-
1
∑
i
=
1
N
(
x
i
2
-
x
i
1
)
(
y
i
2
-
y
i
1
)
=
F
D
T
=
2
Doc 25
0.1944, -38.0000, 14.0000, 0.5004
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Fixed_effects_model.html
Cov
(
X
,
Y
)
=
Cov
(
X
-
k
x
,
Y
-
k
y
)
=
∑
i
=
1
n
(
x
i
-
K
x
)
(
y
i
-
K
y
)
-
(
∑
i
=
1
n
(
x
i
-
K
x
)
)
(
∑
i
=
1
n
(
y
i
-
K
y
)
)
/
n
n
.
Doc 14
0.2213, -5.0000, 17.0000, 0.7569
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Algorithms_for_calculating_variance.html
t
score
=
(
β
^
-
β
0
)
n
-
2
SSR
/
∑
i
=
1
n
(
x
i
-
x
¯
)
2
.
Doc 24
0.1944, -23.0000, 15.0000, 0.3871
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Student's_t-test.html
∑
i
=
1
n
(
x
i
-
x
¯
)
(
θ
-
x
¯
)
=
0
Doc 26
0.1871, -3.0000, 14.0000, 0.1871
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Sufficient_statistic.html
variance
=
s
2
=
∑
i
=
1
n
(
x
i
-
x
¯
)
2
n
-
1
Doc 14
0.2213, -5.0000, 17.0000, 0.7569
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Algorithms_for_calculating_variance.html
σ
^
i
=
∑
k
=
1
n
(
x
k
-
x
¯
i
)
2
n
-
1
σ
^
i
,
j
=
∑
k
=
1
n
(
x
k
-
x
¯
i
)
(
x
k
-
x
¯
j
)
n
-
1
Doc 27
0.1843, -36.0000, 13.0000, 0.1843
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Experimental_uncertainty_analysis.html
r
=
r
x
y
=
∑
x
i
y
i
-
n
x
¯
y
¯
(
n
-
1
)
s
x
s
y
Doc 3
0.4426, -16.0000, 34.0000, 0.8511
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Pearson_product-moment_correlation_coefficient.html
F
=
∑
i
=
1
n
(
y
i
^
-
y
¯
)
2
/
k
∑
j
=
1
k
∑
i
=
1
n
j
(
y
i
j
-
y
i
^
)
2
/
(
n
-
k
-
1
)
Doc 28
0.1810, -31.0000, 11.0000, 0.3601
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Omnibus_test.html
σ
2
=
∑
i
=
1
N
(
x
i
-
x
¯
)
2
N
and
s
2
=
N
N
-
1
⋅
σ
2
=
N
N
2
-
N
⋅
∑
i
=
1
N
(
x
i
-
x
¯
)
2
.
Doc 29
0.1810, -36.0000, 12.0000, 0.6531
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Mean_square_weighted_deviation.html
σ
2
=
∑
i
=
1
N
w
i
(
x
i
-
x
¯
*
)
2
∑
i
=
1
N
w
i
Doc 29
0.1810, -36.0000, 12.0000, 0.6531
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Mean_square_weighted_deviation.html
F
=
∑
j
=
1
k
n
j
(
y
¯
j
-
y
¯
)
2
/
(
k
-
1
)
∑
j
=
1
k
∑
i
=
1
n
j
(
y
i
j
-
y
¯
j
)
2
/
(
n
-
k
)
Doc 28
0.1810, -31.0000, 11.0000, 0.3601
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Omnibus_test.html
S
x
y
=
1
2
N
2
∑
i
=
1
N
∑
j
=
1
N
(
x
i
-
x
j
)
(
y
i
-
y
j
)
Doc 30
0.1736, -17.0000, 13.0000, 0.1736
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Gyration_tensor.html
𝐐
𝐧
=
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
(
x
i
-
x
¯
)
T
.
Doc 31
0.1705, -10.0000, 15.0000, 0.6822
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Estimation_of_covariance_matrices.html
S
=
∑
i
=
1
n
(
x
i
-
x
¯
)
(
x
i
-
x
¯
)
T
∈
𝐑
p
×
p
.
Doc 31
0.1705, -10.0000, 15.0000, 0.6822
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Estimation_of_covariance_matrices.html
𝐐
=
1
n
-
1
∑
i
=
1
n
(
x
i
-
x
¯
)
(
x
i
-
x
¯
)
T
,
Doc 31
0.1705, -10.0000, 15.0000, 0.6822
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Estimation_of_covariance_matrices.html
q
j
k
=
1
N
∑
i
=
1
N
(
x
i
j
-
x
¯
j
)
(
x
i
k
-
x
¯
k
)
Doc 32
0.1705, -13.0000, 14.0000, 0.4876
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Sample_mean_and_sample_covariance.html
q
j
k
=
1
N
-
1
∑
i
=
1
N
(
x
i
j
-
x
¯
j
)
(
x
i
k
-
x
¯
k
)
,
Doc 32
0.1705, -13.0000, 14.0000, 0.4876
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Sample_mean_and_sample_covariance.html
∑
i
=
1
n
(
x
i
-
μ
)
(
x
i
-
μ
)
T
=
∑
i
=
1
n
(
x
i
-
x
¯
)
(
x
i
-
x
¯
)
T
=
S
Doc 31
0.1705, -10.0000, 15.0000, 0.6822
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Estimation_of_covariance_matrices.html
s
M
=
∑
i
=
1
m
x
i
;
s
Σ
=
∑
i
=
1
m
(
x
i
-
x
¯
)
2
Doc 33
0.1676, -14.0000, 12.0000, 0.1676
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Well-behaved_statistic.html
λ
1
=
(
y
2
-
y
3
)
(
x
-
x
3
)
+
(
x
3
-
x
2
)
(
y
-
y
3
)
det
(
T
)
=
(
y
2
-
y
3
)
(
x
-
x
3
)
+
(
x
3
-
x
2
)
(
y
-
y
3
)
(
y
2
-
y
3
)
(
x
1
-
x
3
)
+
(
x
3
-
x
2
)
(
y
1
-
y
3
)
,
Doc 34
0.1655, -67.0000, 13.0000, 0.3310
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Barycentric_coordinate_system.html
λ
2
=
(
y
3
-
y
1
)
(
x
-
x
3
)
+
(
x
1
-
x
3
)
(
y
-
y
3
)
det
(
T
)
=
(
y
3
-
y
1
)
(
x
-
x
3
)
+
(
x
1
-
x
3
)
(
y
-
y
3
)
(
y
2
-
y
3
)
(
x
1
-
x
3
)
+
(
x
3
-
x
2
)
(
y
1
-
y
3
)
,
Doc 34
0.1655, -67.0000, 13.0000, 0.3310
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Barycentric_coordinate_system.html
σ
y
2
=
1
n
∑
i
=
1
n
(
y
i
-
y
¯
)
2
=
(
1
n
∑
i
=
1
n
y
i
2
)
-
y
¯
2
Doc 35
0.1600, -23.0000, 11.0000, 0.1600
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Variance.html
K
=
(
N
-
1
)
∑
i
=
1
g
n
i
(
r
¯
i
⋅
-
r
¯
)
2
∑
i
=
1
g
∑
j
=
1
n
i
(
r
i
j
-
r
¯
)
2
,
Doc 36
0.1600, -30.0000, 10.0000, 0.1600
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Kruskal–Wallis_one-way_analysis_of_variance.html
s
1
=
(
x
2
-
x
2
)
2
+
(
y
2
-
y
1
)
2
(
x
2
-
x
1
)
2
+
(
y
2
-
y
1
)
2
=
(
y
2
-
y
1
)
2
(
x
2
-
x
1
)
2
+
(
y
2
-
y
1
)
2
,
Doc 37
0.1600, -46.0000, 11.0000, 0.3066
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Rational_trigonometry.html
Γ
=
∑
i
,
j
=
1
n
a
i
j
b
i
j
∑
i
,
j
=
1
n
a
i
j
2
∑
i
,
j
=
1
n
b
i
j
2
Doc 38
0.1568, -25.0000, 14.0000, 0.1568
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Rank_correlation.html
s
=
∑
i
=
1
n
(
x
i
-
x
¯
)
2
n
-
1
Doc 39
0.1541, -7.0000, 12.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Xbar_and_s_chart.html
∑
i
=
1
n
(
x
i
-
μ
)
2
=
∑
i
=
1
n
(
x
i
-
x
¯
)
2
+
n
(
x
¯
-
μ
)
2
Doc 40
0.1541, -20.0000, 12.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Normal_distribution.html
s
2
=
∑
i
=
1
n
(
x
i
-
K
)
2
-
(
∑
i
=
1
n
(
x
i
-
K
)
)
2
/
n
n
-
1
.
Doc 14
0.2213, -5.0000, 17.0000, 0.7569
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Algorithms_for_calculating_variance.html
SSE
=
∑
i
=
1
n
(
X
i
-
X
¯
)
2
+
∑
i
=
1
n
(
Y
i
-
Y
¯
)
2
+
∑
i
=
1
n
(
Z
i
-
Z
¯
)
2
Doc 13
0.2252, -8.0000, 16.0000, 0.3793
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Degrees_of_freedom_(statistics).html
∑
i
=
1
n
(
y
i
-
y
¯
)
2
=
∑
i
=
1
n
(
y
i
-
y
^
i
)
2
+
∑
i
=
1
n
(
y
^
i
-
y
¯
)
2
.
Doc 10
0.2616, -14.0000, 20.0000, 0.7912
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Explained_sum_of_squares.html
C
=
μ
^
4
σ
^
4
=
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
4
(
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
2
)
2
,
Doc 41
0.1541, -32.0000, 12.0000, 0.3082
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Jarque–Bera_test.html
s
n
2
=
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
2
=
∑
i
=
1
n
(
x
i
2
)
n
-
(
∑
i
=
1
n
x
i
)
2
n
2
Doc 42
0.1541, -32.0000, 12.0000, 0.3082
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/Bessel's_correction.html
V
2
,
n
=
1
n
2
∑
i
=
1
n
∑
j
=
1
n
1
2
(
x
i
-
x
j
)
2
=
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
2
,
Doc 43
0.1541, -34.0000, 12.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/V-statistic.html
S
=
μ
^
3
σ
^
3
=
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
3
(
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
2
)
3
/
2
,
Doc 41
0.1541, -32.0000, 12.0000, 0.3082
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Jarque–Bera_test.html
b
1
=
m
3
s
3
=
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
3
[
1
n
-
1
∑
i
=
1
n
(
x
i
-
x
¯
)
2
]
3
/
2
,
Doc 44
0.1541, -35.0000, 12.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Skewness.html
g
2
=
m
4
m
2
2
-
3
=
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
4
(
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
2
)
2
-
3
Doc 7
0.3082, -38.0000, 24.0000, 0.6165
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Kurtosis.html
g
1
=
m
3
m
2
3
/
2
=
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
3
(
1
n
∑
i
=
1
n
(
x
i
-
x
¯
)
2
)
3
/
2
,
Doc 45
0.1541, -36.0000, 12.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/D'Agostino's_K-squared_test.html
=
(
n
+
1
)
n
(
n
-
1
)
(
n
-
2
)
(
n
-
3
)
∑
i
=
1
n
(
x
i
-
x
¯
)
4
k
2
2
-
3
(
n
-
1
)
2
(
n
-
2
)
(
n
-
3
)
Doc 7
0.3082, -38.0000, 24.0000, 0.6165
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Kurtosis.html
s
2
=
1
3
N
{
∑
n
=
1
N
(
x
n
,
1
-
x
¯
)
2
+
∑
n
=
1
N
(
x
n
,
2
-
x
¯
)
2
+
∑
n
=
1
N
(
x
n
,
3
-
x
¯
)
2
}
Doc 47
0.1541, -40.0000, 9.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Intraclass_correlation.html
R
2
=
∑
(
y
^
i
-
y
¯
)
2
∑
(
y
i
-
y
¯
)
2
=
y
T
P
T
L
P
y
y
T
L
y
=
1
-
y
T
M
y
y
T
L
y
=
1
-
SSR
TSS
Doc 46
0.1541, -40.0000, 9.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Ordinary_least_squares.html
s
α
^
=
s
β
^
1
n
∑
i
=
1
n
x
i
2
=
1
n
(
n
-
2
)
(
∑
j
=
1
n
ε
^
j
2
)
∑
i
=
1
n
x
i
2
∑
i
=
1
n
(
x
i
-
x
¯
)
2
Doc 12
0.2616, -30.0000, 16.0000, 0.7643
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Simple_linear_regression.html
s
2
=
1
n
-
1
∑
i
=
1
n
(
x
i
-
x
¯
)
2
=
∑
i
=
1
n
(
x
i
2
)
n
-
1
-
(
∑
i
=
1
n
x
i
)
2
(
n
-
1
)
n
=
(
n
n
-
1
)
s
n
2
.
Doc 42
0.1541, -32.0000, 12.0000, 0.3082
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/Bessel's_correction.html
∑
i
=
1
n
(
y
i
-
y
¯
)
2
=
∑
i
=
1
n
(
y
i
-
y
^
i
)
2
+
∑
i
=
1
n
(
y
^
i
-
y
¯
)
2
+
∑
i
=
1
n
2
(
y
^
i
-
y
¯
)
(
y
i
-
y
^
i
)
.
Doc 10
0.2616, -14.0000, 20.0000, 0.7912
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Explained_sum_of_squares.html
1
N
∑
i
=
1
N
(
x
i
-
x
¯
)
2
=
1
N
(
∑
i
=
1
N
x
i
2
)
-
x
¯
2
=
(
1
N
∑
i
=
1
N
x
i
2
)
-
(
1
N
∑
i
=
1
N
x
i
)
2
.
Doc 48
0.1541, -50.0000, 11.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Standard_deviation.html
S
S
err
=
∑
i
=
1
N
(
y
i
-
y
i
^
)
2
S
S
tot
=
∑
i
=
1
N
(
y
i
-
y
¯
)
2
S
S
reg
=
∑
i
=
1
N
(
y
i
^
-
y
¯
)
2
and
y
¯
=
1
N
∑
y
i
i
=
1
N
.
Doc 49
0.1541, -54.0000, 11.0000, 0.1541
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Fraction_of_variance_unexplained.html
F
E
T
=
2
=
[
(
x
i
1
-
x
¯
i
)
(
x
i
1
-
x
¯
i
)
′
+
(
x
i
2
-
x
¯
i
)
(
x
i
2
-
x
¯
i
)
′
]
-
1
[
(
x
i
1
-
x
¯
i
)
(
y
i
1
-
y
¯
i
)
+
(
x
i
2
-
x
¯
i
)
(
y
i
2
-
y
¯
i
)
]
Doc 25
0.1944, -38.0000, 14.0000, 0.5004
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Fixed_effects_model.html
M
D
(
S
)
=
∑
i
=
1
n
∑
j
=
1
n
|
y
i
-
y
j
|
n
(
n
-
1
)
Doc 23
0.2005, -18.0000, 15.0000, 0.3524
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Mean_absolute_difference.html
m
^
h
(
x
)
=
∑
i
=
1
n
K
h
(
x
-
x
i
)
y
i
∑
i
=
1
n
K
h
(
x
-
x
i
)
Doc 50
0.1519, -20.0000, 9.0000, 0.1519
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Kernel_regression.html
k
^
-
1
=
∑
i
=
1
N
(
x
i
k
ln
x
i
-
x
N
k
ln
x
N
)
∑
i
=
1
N
(
x
i
k
-
x
N
k
)
-
1
N
∑
i
=
1
N
ln
x
i
Doc 51
0.1519, -37.0000, 11.0000, 0.1519
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Weibull_distribution.html
=
2
[
∑
i
=
1
N
(
x
i
2
-
x
i
1
)
(
x
i
2
-
x
i
1
)
′
]
-
1
[
∑
i
=
1
N
1
2
(
x
i
2
-
x
i
1
)
(
y
i
2
-
y
i
1
)
]
Doc 25
0.1944, -38.0000, 14.0000, 0.5004
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Fixed_effects_model.html
var
(
r
)
=
N
-
n
N
1
m
x
2
∑
i
=
1
n
(
y
i
-
r
x
i
)
n
-
1
Doc 52
0.1465, -18.0000, 11.0000, 0.2737
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/Ratio_estimator.html
s
2
=
∑
i
=
1
N
w
i
(
∑
i
=
1
N
w
i
)
2
-
∑
i
=
1
N
w
i
2
.
∑
i
=
1
N
w
i
(
x
i
-
x
¯
*
)
2
Doc 29
0.1810, -36.0000, 12.0000, 0.6531
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Mean_square_weighted_deviation.html
q
y
y
=
∑
(
y
-
y
¯
)
2
w
(
x
-
x
¯
,
y
-
y
¯
)
I
(
x
,
y
)
∑
w
(
x
-
x
¯
,
y
-
y
¯
)
I
(
x
,
y
)
Doc 18
0.2140, -10.0000, 15.0000, 0.7075
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Gravitational_lensing_formalism.html
Σ
=
∑
i
=
1
N
w
i
(
∑
i
=
1
N
w
i
)
2
-
∑
i
=
1
N
w
i
2
∑
i
=
1
N
w
i
(
𝐱
i
-
μ
*
)
T
(
𝐱
i
-
μ
*
)
.
Doc 53
0.1465, -37.0000, 10.0000, 0.2872
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Weighted_arithmetic_mean.html
=
w
∑
i
=
1
N
w
i
(
∑
i
=
1
N
w
i
)
2
-
∑
i
=
1
N
w
i
2
.
∑
i
=
1
N
w
i
.
(
x
i
-
x
¯
*
)
2
(
σ
x
i
)
2
Doc 29
0.1810, -36.0000, 12.0000, 0.6531
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Mean_square_weighted_deviation.html
q
j
k
=
∑
i
=
1
N
w
i
(
∑
i
=
1
N
w
i
)
2
-
∑
i
=
1
N
w
i
2
∑
i
=
1
N
w
i
(
x
i
j
-
x
¯
j
)
(
x
i
k
-
x
¯
k
)
.
Doc 32
0.1705, -13.0000, 14.0000, 0.4876
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Sample_mean_and_sample_covariance.html
s
2
=
(
x
2
-
x
1
)
2
+
(
y
2
-
y
2
)
2
(
x
2
-
x
1
)
2
+
(
y
2
-
y
1
)
2
=
(
x
2
-
x
1
)
2
(
x
2
-
x
1
)
2
+
(
y
2
-
y
1
)
2
.
Doc 37
0.1600, -46.0000, 11.0000, 0.3066
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Rational_trigonometry.html
μ
p
q
=
∑
x
∑
y
(
x
-
x
¯
)
p
(
y
-
y
¯
)
q
f
(
x
,
y
)
Doc 54
0.1407, -14.0000, 11.0000, 0.1407
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Image_moment.html
x
¯
=
∑
i
=
1
n
(
x
i
σ
i
-
2
)
∑
i
=
1
n
σ
i
-
2
,
Doc 53
0.1465, -37.0000, 10.0000, 0.2872
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Weighted_arithmetic_mean.html
s
p
2
=
∑
i
=
1
k
(
n
i
-
1
)
s
i
2
∑
i
=
1
k
(
n
i
-
1
)
Doc 55
0.1407, -17.0000, 9.0000, 0.1407
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Pooled_variance.html
η
2
=
∑
x
n
x
(
y
¯
x
-
y
¯
)
2
∑
x
,
i
(
y
x
i
-
y
¯
)
2
Doc 56
0.1330, -17.0000, 8.0000, 0.1330
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Correlation_ratio.html
S
3
=
N
-
1
∑
i
(
x
i
-
x
¯
)
4
(
N
-
1
∑
i
(
x
i
-
x
¯
)
2
)
2
Doc 57
0.1330, -19.0000, 7.0000, 0.1330
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Moran's_I.html
q
x
x
=
∑
(
x
-
x
¯
)
2
w
(
x
-
x
¯
,
y
-
y
¯
)
I
(
x
,
y
)
∑
w
(
x
-
x
¯
,
y
-
y
¯
)
I
(
x
,
y
)
Doc 18
0.2140, -10.0000, 15.0000, 0.7075
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Gravitational_lensing_formalism.html
1
2
∑
i
=
1
n
∑
j
=
1
n
(
x
i
y
j
-
x
j
y
i
)
2
=
∑
i
=
1
n
x
i
2
∑
i
=
1
n
y
i
2
-
(
∑
i
=
1
n
x
i
y
i
)
2
.
Doc 58
0.1290, -42.0000, 12.0000, 0.1290
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Cauchy–Schwarz_inequality.html
r
=
y
¯
x
¯
=
∑
i
=
1
n
y
∑
i
=
1
n
x
Doc 52
0.1465, -18.0000, 11.0000, 0.2737
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/Ratio_estimator.html
r
x
y
=
x
y
¯
-
x
¯
y
¯
(
x
2
¯
-
x
¯
2
)
(
y
2
¯
-
y
¯
2
)
Doc 12
0.2616, -30.0000, 16.0000, 0.7643
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Simple_linear_regression.html
E
{
(
x
^
-
x
)
(
y
-
y
¯
)
T
}
=
E
{
(
W
(
y
-
y
¯
)
-
(
x
-
x
¯
)
)
(
y
-
y
¯
)
T
}
=
W
E
{
(
y
-
y
¯
)
(
y
-
y
¯
)
T
}
-
E
{
(
x
-
x
¯
)
(
y
-
y
¯
)
T
}
=
W
C
Y
-
C
X
Y
.
Doc 59
0.1272, -66.0000, 10.0000, 0.1272
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Minimum_mean_square_error.html
μ
sig
=
∑
i
=
1
n
(
X
i
-
f
i
)
n
μ
bkg
=
∑
i
=
1
n
(
X
i
-
f
i
)
n
Doc 60
0.1245, -21.0000, 10.0000, 0.1245
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Signal-to-noise_ratio_(imaging).html
R
(
M
,
x
)
=
x
*
M
x
x
*
x
=
∑
i
=
1
n
λ
i
y
i
2
∑
i
=
1
n
y
i
2
Doc 61
0.1194, -24.0000, 9.0000, 0.1194
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Rayleigh_quotient.html
∏
i
=
1
n
x
i
γ
i
∏
i
=
1
n
(
1
-
x
i
)
γ
i
≤
∑
i
=
1
n
γ
i
x
i
∑
i
=
1
n
γ
i
(
1
-
x
i
)
Doc 62
0.1150, -34.0000, 11.0000, 0.1150
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Ky_Fan_inequality.html
R
n
(
ξ
,
x
)
=
r
0
x
∏
i
=
1
n
-
1
(
x
-
x
i
)
∏
i
=
1
n
-
1
(
x
-
x
p
i
)
Doc 63
0.0897, -26.0000, 10.0000, 0.0897
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Elliptic_rational_functions.html