Define the notation
![[n]f_0=f_(-(n-1)/2)+...+f_0+...+f_((n-1)/2)](/images/equations/SpencersFormula/NumberedEquation1.svg) |
(1)
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and let 
be the central difference, so
 |
(2)
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Spencer's 21-term moving average formula is then given by
![f_0^'=([5][5][7])/(5·5·7)(1-4delta^2)f_0,](/images/equations/SpencersFormula/NumberedEquation3.svg) |
(3)
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which, written explicitly, gives
![f_0^'=1/(350)[60f_0+57(f_(-1)+f_1)+47(f_(-2)+f_2)+33(f_(-3)+f_3)+18(f_(-4)+f_4)+6(f_(-5)+f_5)-2(f_(-6)+f_6)-5(f_(-7)+f_7)-5(f_(-8)+f_8)-3(f_(-9)+f_9)-(f_(-10)+f_(10))]](/images/equations/SpencersFormula/NumberedEquation4.svg) |
(4)
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Spencer's Formula
See also
Moving Average, SmoothingExplore with Wolfram|Alpha
References
Spencer, J. J. I. A. 38, 334, 1904.Spencer, J. J. I. A. 38, 339, 1904.Spencer, J. J. I. A. 41, 361, 1907.Whittaker, E. T. and Robinson, G. "Spencer's Formula." §144 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 290-294, 1967.Referenced on Wolfram|Alpha
Spencer's FormulaCite this as:
Weisstein, Eric W. "Spencer's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpencersFormula.html