Spencer's Formula

Define the notation

(1)

and let  be the central difference, so

(2)

Spencer's 21-term moving average formula is then given by

(3)

which, written explicitly, gives

(4)

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.