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~Q
]|Expr|[#b @`bb#_b#_b#_})'# b'4" *|: ;bP:&c0!)Weighted Total |
  |Least Squares}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~V?m0 (P)~p0
1 ~V?f0 ('x)~p0
1 ~V?v0 ('g)~p0
1 ~V?v0 ('b)~p0
1 ~V?v0 ('a)~p0
1 ~V?v0 ('s)~p0
1 ~V?v0 ('l)~p0
1 ~V?v0 (n)~p0
1 ~V?m0 ('S)~p0
1 ~V?m0 (S)~p0
1 ~V?m0 (X)~p0
1 ~V?m0 (Y)~p0
1 ~V?f0 (Mean)~p0
1 ~V?f144 (RowsOf)~p0
1 ~V?f161 (min)~p0
1 ~V?v0 (YY)~p0
1 ~V?v0 (XX)~p0
1 ~V?v0 (XY)~p0
1 ~V?v0 (C)~p0
1 ~V?v0 (k)~p0
1 ~V?v0 (m)~p0
1 ~V?v0 (b)~p0
1 ~V?v0 (y)~p0
1 ~V?v0 (x)~p0
1 ~V?c1 ('p)~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b'4["! ) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})!# b'4" *|: ;bP8&c0!*Introduction|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b'F[#! ) # b'F|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})b""# b'F" *|: ;bP8&c0!*When both|
  | the X and Y data are subject to errors that are independently|
  | and identically distributed with zero mean and common variance|
  |,L an orthogonal residual is required,L and unweighted TLS yields|
  | a ,Bbest fit,B in the maximum likelihood sense,N  When the data|
  | do not have common variance,L a weighted TLS analysis is required|
  |,L and the residual will not be orthogonal,N|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}) # b'F}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#L})!# b'4" *|: ;bP8&c0!*Data|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(X=(100;200;600))~p0
1 ~d~A(Y=(2;6;1))~p0
255 ~d~Q
]|Expr|[#b @`bb#_b#_b#_})0# b'4" Symbol^: ;bP8&c0  S"!*|:!&c0!* |
  |is a known positive definite covariance matrix,N|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~A('S=('s_XX,'s_XY;'s_XY,'s_YY))~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b'4["! )0# b'4" *|: ;bP8&c0!*Let |
  |$^"!Symbol^:!&c0  s^: &c0!*X_ ,] standard deviation of X data|
  |,L}& b!( b"0 b#8 b$@ b%H b&P!WW})3# b'4and $^:!&c0  s^: &c0!*Y|
  |_ ,] standard deviation of Y data,N  Then|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b%H[#! );# b%H$^" Symbol^: ;bP8&c0  s|
  |(!"!*|:!&c0!*XX}_ ,] $^$^: &c0  s^:!&c0!*X__^2,L the variance|
  | of X or covariance of X and X,N}& b!( b"0 b#8 b$@ b%H b&P!WW}|
  |);# b%H$^: &c0  s(!:!&c0!*YY}_ ,] $^$^: &c0  s^:!&c0!*Y__^2,L|
  | the variance of Y or covariance of Y and Y,N|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})b @# b%H$^: &c0  s(!:!&c0!*XY}|
  |_,L the covariance of X and Y,L is zero when the data errors |
  |are uncorrelated,N}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b'4["! )5# b'4" *|: ;bP8&c0!*Consider|
  | uncorrelated X and Y data with $^"!Symbol^:!&c0  s^: &c0!*X_|
  | ,] 100 and, }& b!( b"0 b#8 b$@ b%H b&P!WW})b @# b'4$^:!&c0  s|
  |^: &c0!*Y_ ,] 1,N  Then $^:!&c0  s(!: &c0!*XX}_ ,] 10000,L $^|
  |:!&c0  s(!: &c0!*XY}_ ,] 0,L and $^:!&c0  s(!: &c0!*YY}_ ,] 1|
  |,N}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~A('s_XX=10000)~p0
1 ~d~A('s_XY=0)~p0
255 ~d~A('s_YY=1)~p0
255 ~d~Q
]|Expr|[#b @`bb#_b#_b#_})!# b'4["! ) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})!# b'4" *|: ;bP8&c0!*Statistics|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~A(n=RowsOf(X))~p0
1 ~d~A(Mean(?u)=1/n*Summation(k):(1):(n)*?u_k)~p0
1 ~d~Ht(C*(?u,?v)):(1/n*Summation(k):(1):(n)*(?u_~
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1 ~A(C_XY=C*(X,Y))~p1
1 ~A(C_XY=-233.33333333333331)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"H$_c/__c/__}
^ _~A(C_~
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2 ~sb/_!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(C_XY=~
 1/n*Summation(k):(1):(n)*((100;200;600)_k-Mean(~
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3 ~sb/_!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~A(C_~
 XY=0.33333333333333331*Summation(k):(1):(3)*((~
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4 ~sb/_!!!  }
$&!! c#T"!c"H$_c/__c/__}
^ _~A(C_XX=~
 C*(X,X))~p1
1 ~A(C_XX=46666.666666666664)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"H#_c/__c/__}
^ _~A(C_~
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2 ~sb/_!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(C_XX=~
 1/n*Summation(k):(1):(n)*((100;200;600)_k-Mean(~
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3 ~sb/_!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~A(C_~
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1 ~A(C_YY=4.6666666666666661)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"H#_c/__c/__}
^ _~A(C_~
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2 ~sb/_!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(C_YY=~
 1/n*Summation(k):(1):(n)*((2;6;1)_k-Mean(2;6;1))^~
 2)~p0
3 ~sb/_!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#@})!# b'4`f }["! `fb#@}) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}),# b'4" *|: ;bP8&c0!*Ordinary |
  |least squares,L X independent fit}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(y_1=m_1*x+b_1)~p1
1 ~d~A(m_1=-0.0050000000000000001)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"\"_c/__c/__}
^ _~A(b_~
 1=4.5)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"L"_c/__c/__}
^ _~A(m_~
 1=C_XY/C_XX)~p0
2 ~A(b_1=Mean(Y)-m_1*Mean(X))~p0
2 ~Q
]|Expr|[#b @`bb#_b#_b#_}`f0})!# b'4`f }["! `f0}) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}),# b'4" *|: ;bP8&c0!*Ordinary |
  |least squares,L Y independent fit}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(x_2=m_2*y+b_2)~p1
1 ~d~A(m_2=-50)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"\"_c/__c/__}
^ _~A(b_~
 2=450)~p0
255 ~d~sb/_!!!  }
$&!! c#T"!c"L"_c/__c/__}
^ _~A(m_~
 2=C_XY/C_YY)~p0
2 ~A(b_2=Mean(X)-m_2*Mean(Y))~p0
2 ~G1 1 268 496
0 1 0 4 10
(70...620):(-3...9):(~
 ?=0...2*'p):('p/5):(10)~Q
]|Expr|[#b @`bb#_b#_b#_})!# b#@" *|: ;bP8&c0!*Declarations|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~R8405120 ? (x,y):(y=bottom...top):(x=left...right):(~
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1 ~gc-1 30 ? 0 -32768 -32768 -32768
2541 -32768 -32768 -32768
4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
8757 -32768 -32768 -32768
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22316 -32768 -32768 -32768
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35875 -32768 -32768 -32768
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58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
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4519 -32768 -32768 -32768
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58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
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4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
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56777 -32768 -32768 -32768
58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
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6779 -32768 -32768 -32768
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56777 -32768 -32768 -32768
58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
2541 -32768 -32768 -32768
4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
8757 -32768 -32768 -32768
11299 -32768 -32768 -32768
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49434 -32768 -32768 -32768
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58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
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4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
8757 -32768 -32768 -32768
11299 -32768 -32768 -32768
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29659 -32768 -32768 -32768
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43218 -32768 -32768 -32768
45196 -32768 -32768 -32768
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49434 -32768 -32768 -32768
51976 -32768 -32768 -32768
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~
 ~R8405120 ? (x,y):(x=left...right):(y=bottom...~
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1 ~gc-1 30 ? 0 -32768 -32768 -32768
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4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
8757 -32768 -32768 -32768
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33897 -32768 -32768 -32768
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38417 -32768 -32768 -32768
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43218 -32768 -32768 -32768
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56777 -32768 -32768 -32768
58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
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4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
8757 -32768 -32768 -32768
11299 -32768 -32768 -32768
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27117 -32768 -32768 -32768
29659 -32768 -32768 -32768
31637 -32768 -32768 -32768
33897 -32768 -32768 -32768
35875 -32768 -32768 -32768
38417 -32768 -32768 -32768
40676 -32768 -32768 -32768
43218 -32768 -32768 -32768
45196 -32768 -32768 -32768
47456 -32768 -32768 -32768
49434 -32768 -32768 -32768
51976 -32768 -32768 -32768
54235 -32768 -32768 -32768
56777 -32768 -32768 -32768
58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
2541 -32768 -32768 -32768
4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
8757 -32768 -32768 -32768
11299 -32768 -32768 -32768
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16100 -32768 -32768 -32768
18078 -32768 -32768 -32768
20338 -32768 -32768 -32768
22316 -32768 -32768 -32768
24858 -32768 -32768 -32768
27117 -32768 -32768 -32768
29659 -32768 -32768 -32768
31637 -32768 -32768 -32768
33897 -32768 -32768 -32768
35875 -32768 -32768 -32768
38417 -32768 -32768 -32768
40676 -32768 -32768 -32768
43218 -32768 -32768 -32768
45196 -32768 -32768 -32768
47456 -32768 -32768 -32768
49434 -32768 -32768 -32768
51976 -32768 -32768 -32768
54235 -32768 -32768 -32768
56777 -32768 -32768 -32768
58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
2541 -32768 -32768 -32768
4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
8757 -32768 -32768 -32768
11299 -32768 -32768 -32768
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33897 -32768 -32768 -32768
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58755 -32768 -32768 -32768
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~gc-1 30 ? 0 -32768 -32768 -32768
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4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
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43218 -32768 -32768 -32768
45196 -32768 -32768 -32768
47456 -32768 -32768 -32768
49434 -32768 -32768 -32768
51976 -32768 -32768 -32768
54235 -32768 -32768 -32768
56777 -32768 -32768 -32768
58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~gc-1 30 ? 0 -32768 -32768 -32768
2541 -32768 -32768 -32768
4519 -32768 -32768 -32768
6779 -32768 -32768 -32768
8757 -32768 -32768 -32768
11299 -32768 -32768 -32768
13558 -32768 -32768 -32768
16100 -32768 -32768 -32768
18078 -32768 -32768 -32768
20338 -32768 -32768 -32768
22316 -32768 -32768 -32768
24858 -32768 -32768 -32768
27117 -32768 -32768 -32768
29659 -32768 -32768 -32768
31637 -32768 -32768 -32768
33897 -32768 -32768 -32768
35875 -32768 -32768 -32768
38417 -32768 -32768 -32768
40676 -32768 -32768 -32768
43218 -32768 -32768 -32768
45196 -32768 -32768 -32768
47456 -32768 -32768 -32768
49434 -32768 -32768 -32768
51976 -32768 -32768 -32768
54235 -32768 -32768 -32768
56777 -32768 -32768 -32768
58755 -32768 -32768 -32768
61015 -32768 -32768 -32768
62993 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~X2 8405120 (left,y):(y=bottom...top):(y)~p0
1 ~gc-1 2 ? 0 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~X1 8405120 (x,bottom):(x=left...right):(x)~p0
1 ~gc-1 2 ? 0 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~V?c64 (left)~p0
1 ~V?c65 (right)~p0
1 ~V?c66 (bottom)~p0
1 ~V?c67 (top)~p0
1 ~A('x(k)=X_k+((-'s_XY+'s_XX*m)*(Y_k-X_k*m-b))/~
 ('s_YY+'s_XX*m^2-2*'s_XY*m))~p1
0 ~A(k=?k)~p0
1 ~A('x(?k)=X_?k+(-'s_XY+'s_XX*m)/('s_YY+'s_XX*m^~
 2-2*'s_XY*m)*(Y_?k-X_?k*m-b))~p0
1 ~d~sb/_!!   }
b00&!  c#T"_c/__c/__!"}
^ _~A(P=~
 (X_1,Y_1;'x(1),'x(1)*m+b;X_2,Y_2;'x(2),'x(2)*m+~
 b;X_3,Y_3;'x(3),'x(3)*m+b))~p0
0 ~d~L1 16711680 ? (x,y_1):(x=left...right)~p0
0 ~gc-1 2 ? 0 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~L1 32768 ? (x_2,y):(y=top...bottom)~p0
0 ~gc-1 2 ? 0 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~L1 255 ? (x,y):(x=left...right)~p0
0 ~gc-1 2 ? 0 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~L1 16744448 ? (P_k):(k=1...2)~p0
0 ~gc-1 2 ? 0 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~L1 16744448 ? (P_k):(k=3...4)~p0
0 ~gc-1 2 ? 0 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~L1 16744448 ? (P_k):(k=5...6)~p0
0 ~gc-1 2 ? 0 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~S17 ? 16744448 ? ? (P_k):(k=1...RowsOf(P)):(6)~p0
0 ~gc-1 6 ? 0 -32768 -32768 -32768
14745 -32768 -32768 -32768
26214 -32768 -32768 -32768
39321 -32768 -32768 -32768
50790 -32768 -32768 -32768
65535 -32768 -32768 -32768
~
 ~S17 ? 255 ? ? (Mean(X),Mean(Y)):(?):(6)~p0
0 ~gc-1 1 ? 0 -32768 -32768 -32768
~t~p0
0 ~Q
]|Expr|[#b @`bb#_b#_b#_}`f#})!# b'4`f }["! `f#}) # b'4|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}))# b'4" *|: ;bP8&c0!*Weighted |
  |total least squares fit}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(y=m*x+b)~p0
1 ~d~A(m=-0.01)~p0
255 ~d~sb/_!!   }
$&!  c#T"!c'L!_c/__c/__}
^ _~A(b=~
 6)~p0
255 ~d~sb/_!!   }
$&!  c#T"!c"L"_c/__c/__}
^ _~A('a=~
 's_XX*C_XY-'s_XY*C_XX)~p0
2 ~d~A('b='s_YY*C_XX-'s_XX*C_YY)~p0
2 ~d~A('g='s_XY*C_YY-'s_YY*C_XY)~p0
2 ~d~A(m=Conditional(1/2*(-sqrt(('b/'a)^2-4*'g/'a)-~
 'b/'a),('a<0);1/2*(sqrt(('b/'a)^2-4*'g/'a)-'b/~
 'a),('a>0);-'g/'b,('a=0)))~p0
2 ~A(b=Mean(Y)-m*Mean(X))~p0
2 ~c1
44 -1 47 -1 ~c7
45 -1 43 -1 29 -1 11 -1 30 -1 12 -1 41 -1 13 -1 ~c2
46 -1 45 -1 42 -1 ~c5
47 -1 46 -1 40 -1 8 -1 41 -1 13 -1 ~c5
49 -1 51 -1 40 -1 8 -1 41 -1 13 -1 ~c5
50 -1 48 -1 29 -1 11 -1 41 -1 13 -1 ~c2
51 -1 50 -1 42 -1 ~c5
53 -1 55 -1 40 -1 8 -1 41 -1 13 -1 ~c5
54 -1 52 -1 30 -1 12 -1 41 -1 13 -1 ~c2
55 -1 54 -1 42 -1 ~c4
58 -1 60 -1 44 -1 19 -1 49 -1 ~c11
59 -1 61 -1 30 -1 12 -1 40 -1 8 -1 41 -1 13 -1 29 -1 11 -1 58 -1 21 -1 ~c4
64 -1 66 -1 44 -1 19 -1 53 -1 ~c11
65 -1 67 -1 29 -1 11 -1 40 -1 8 -1 41 -1 13 -1 30 -1 12 -1 64 -1 21 -1 ~c2
11 68 -1 9 68 -1 10 68 -1 ~c15
71 -1 76 -1 49 -1 19 -1 37 -1 6 -1 44 -1 36 -1 73 -1 5 -1 38 -1 53 -1 74 -1 4 -1 75 -1 3 -1 ~c11
72 -1 77 -1 30 -1 12 -1 40 -1 8 -1 41 -1 13 -1 29 -1 11 -1 71 -1 21 -1 ~e