Mean Difference and Standard Deviation
1  Set of Wave Values
{W_{0}, W_{1}, W_{2}, ... W_{n2}, W_{n1}, W_{n}}
2  Approximate Projection Set
First difference calculation.
S_{0} = {y_{0}, y_{1}, y_{2}, y_{3}, y_{4}, y_{5}} = {W_{0}, W_{1}, W_{2}, W_{3}, W_{4}, W_{5}}
Plug values into following equations.
a_{0} = 720y_{0}  1800y_{1} + 2400y_{2}  1800y_{3} + 720y_{4}  120y_{5}
a_{1} = 1044y_{0} + 3510y_{1}  5080y_{2} + 3960y_{3}  1620y_{4} + 274y_{5}
a_{2} = 580y_{0}  2305y_{1} + 3720y_{2}  3070y_{3} + 1300y_{4}  225y_{5}
a_{3} = 155y_{0} + 685y_{1}  1210y_{2} + 1070y_{3}  475y_{4} + 85y_{5}
a_{4} = 20y_{0}  95y_{1} + 180y_{2}  170y_{3} + 80y_{4}  15y_{5}
a_{5} = 1y_{0} + 5y_{1}  10y_{2} + 10y_{3}  5y_{4} + 1y_{5}
y_{6} = (1a_{0} + 7a_{1} + 49a_{2} + 343a_{3} + 2401a_{4} + 16807a_{5}) / 120
Dy_{0} = y_{6}  W_{6} First difference between approximate and actual.
Second difference calculation.
S_{1} = {y_{0}, y_{1}, y_{2}, y_{3}, y_{4}, y_{5}} = {W_{1}, W_{2}, W_{3}, W_{4}, W_{5}, W_{6}}
Plug values into following equations.
a_{0} = 720y_{0}  1800y_{1} + 2400y_{2}  1800y_{3} + 720y_{4}  120y_{5}
a_{1} = 1044y_{0} + 3510y_{1}  5080y_{2} + 3960y_{3}  1620y_{4} + 274y_{5}
a_{2} = 580y_{0}  2305y_{1} + 3720y_{2}  3070y_{3} + 1300y_{4}  225y_{5}
a_{3} = 155y_{0} + 685y_{1}  1210y_{2} + 1070y_{3}  475y_{4} + 85y_{5}
a_{4} = 20y_{0}  95y_{1} + 180y_{2}  170y_{3} + 80y_{4}  15y_{5}
a_{5} = 1y_{0} + 5y_{1}  10y_{2} + 10y_{3}  5y_{4} + 1y_{5}
y_{6} = (1a_{0} + 7a_{1} + 49a_{2} + 343a_{3} + 2401a_{4} + 16807a_{5}) / 120
Dy_{1} = y_{6}  W_{7} Second difference between approximate and actual.
Third difference calculation.
S_{2} = {y_{0}, y_{1}, y_{2}, y_{3}, y_{4}, y_{5}} = {W_{2}, W_{3}, W_{4}, W_{5}, W_{6}, W_{7}}
Plug values into following equations.
a_{0} = 720y_{0}  1800y_{1} + 2400y_{2}  1800y_{3} + 720y_{4}  120y_{5}
a_{1} = 1044y_{0} + 3510y_{1}  5080y_{2} + 3960y_{3}  1620y_{4} + 274y_{5}
a_{2} = 580y_{0}  2305y_{1} + 3720y_{2}  3070y_{3} + 1300y_{4}  225y_{5}
a_{3} = 155y_{0} + 685y_{1}  1210y_{2} + 1070y_{3}  475y_{4} + 85y_{5}
a_{4} = 20y_{0}  95y_{1} + 180y_{2}  170y_{3} + 80y_{4}  15y_{5}
a_{5} = 1y_{0} + 5y_{1}  10y_{2} + 10y_{3}  5y_{4} + 1y_{5}
y_{6} = (1a_{0} + 7a_{1} + 49a_{2} + 343a_{3} + 2401a_{4} + 16807a_{5}) / 120
Dy_{2} = y_{6}  W_{8} Third difference between approximate and actual.
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Continue advancing subset through the set of Wave Values.
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Second to last difference calculation.
S_{n8} = {y_{0}, y_{1}, y_{2}, y_{3}, y_{4}, y_{5}} = {W_{n8}, W_{n7}, W_{n6}, W_{n5}, W_{n4}, W_{n3}}
Plug values into following equations.
a_{0} = 720y_{0}  1800y_{1} + 2400y_{2}  1800y_{3} + 720y_{4}  120y_{5}
a_{1} = 1044y_{0} + 3510y_{1}  5080y_{2} + 3960y_{3}  1620y_{4} + 274y_{5}
a_{2} = 580y_{0}  2305y_{1} + 3720y_{2}  3070y_{3} + 1300y_{4}  225y_{5}
a_{3} = 155y_{0} + 685y_{1}  1210y_{2} + 1070y_{3}  475y_{4} + 85y_{5}
a_{4} = 20y_{0}  95y_{1} + 180y_{2}  170y_{3} + 80y_{4}  15y_{5}
a_{5} = 1y_{0} + 5y_{1}  10y_{2} + 10y_{3}  5y_{4} + 1y_{5}
y_{6} = (1a_{0} + 7a_{1} + 49a_{2} + 343a_{3} + 2401a_{4} + 16807a_{5}) / 120
Dy_{n8} = y_{6}  W_{n2} Second to last difference between approximate and actual.
Next to last difference calculation.
S_{n7} = {y_{0}, y_{1}, y_{2}, y_{3}, y_{4}, y_{5}} = {W_{n7}, W_{n6}, W_{n5}, W_{n4}, W_{n3}, W_{n2}}
Plug values into following equations.
a_{0} = 720y_{0}  1800y_{1} + 2400y_{2}  1800y_{3} + 720y_{4}  120y_{5}
a_{1} = 1044y_{0} + 3510y_{1}  5080y_{2} + 3960y_{3}  1620y_{4} + 274y_{5}
a_{2} = 580y_{0}  2305y_{1} + 3720y_{2}  3070y_{3} + 1300y_{4}  225y_{5}
a_{3} = 155y_{0} + 685y_{1}  1210y_{2} + 1070y_{3}  475y_{4} + 85y_{5}
a_{4} = 20y_{0}  95y_{1} + 180y_{2}  170y_{3} + 80y_{4}  15y_{5}
a_{5} = 1y_{0} + 5y_{1}  10y_{2} + 10y_{3}  5y_{4} + 1y_{5}
y_{6} = (1a_{0} + 7a_{1} + 49a_{2} + 343a_{3} + 2401a_{4} + 16807a_{5}) / 120
Dy_{n7} = y_{6}  W_{n1} Next to last difference between approximate and actual.
Last difference calculation.
S_{n6} = {y_{0}, y_{1}, y_{2}, y_{3}, y_{4}, y_{5}} = {W_{n6}, W_{n5}, W_{n4}, W_{n3}, W_{n2}, W_{n1}}
Plug values into following equations.
a_{0} = 720y_{0}  1800y_{1} + 2400y_{2}  1800y_{3} + 720y_{4}  120y_{5}
a_{1} = 1044y_{0} + 3510y_{1}  5080y_{2} + 3960y_{3}  1620y_{4} + 274y_{5}
a_{2} = 580y_{0}  2305y_{1} + 3720y_{2}  3070y_{3} + 1300y_{4}  225y_{5}
a_{3} = 155y_{0} + 685y_{1}  1210y_{2} + 1070y_{3}  475y_{4} + 85y_{5}
a_{4} = 20y_{0}  95y_{1} + 180y_{2}  170y_{3} + 80y_{4}  15y_{5}
a_{5} = 1y_{0} + 5y_{1}  10y_{2} + 10y_{3}  5y_{4} + 1y_{5}
y_{6} = (1a_{0} + 7a_{1} + 49a_{2} + 343a_{3} + 2401a_{4} + 16807a_{5}) / 120
Dy_{n6} = y_{6}  W_{n} Next to last difference between approximate and actual.
Dy = {Dy_{0}, Dy_{1}, Dy_{2}, ... Dy_{n8}, Dy_{n7}, Dy_{n6}} Set of differences between approximate and actual
The set of differences will always be 6 less than the number of the Wave Data set count. For that reason, there needs to be at least 10 or more points in the Wave to calculate the mean and standard deviation.
3  Mean Difference
m = ( [0 to n  6] å Dy_{i} ) / (n  6)
m = (Dy_{0} + Dy_{1} + Dy_{2} + ... + Dy_{n8} + Dy_{n7} + Dy_{n6}) / (n  6)
4  Standard Deviation of Difference

s = Ö( [0 to n  6] å (Dy_{i}  m)²) / (n  6) 

s = Ö((Dy_{0}  m)² + (Dy_{1}  m)² + (Dy_{2}  m)² + ... + (Dy_{n8}  m)² + (Dy_{n7}  m)² + (Dy_{n6}  m)²) / (n  6) 
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