Index

A statistic which assigns a single number to several individual statistics in order to quantify trends. The best-known index in the United States is the consumer price index, which gives a sort of ``average'' value for inflation based on the price changes for a group of selected products.

Let $p_n$ be the price per unit in period $n$, $q_n$ be the quantity produced in period $n$, and $v_n\equiv p_nq_n$ be the value of the $n$ units. Let $q_a$ be the estimated relative importance of a product. There are several types of indices defined, among them those listed in the following table.

Index	Abbr.	Formula
Bowley Index	$P_B$	${\textstyle{1\over 2}}(P_L+P_P)$
Fisher Index	$P_F$	$\sqrt{P_LP_P}$
Geometric Mean Index	$P_G$	$\left[{\prod \left({p_n\over p_0}\right)^{v_0}}\right]^{1/\sum v_0}$
Harmonic Mean Index	$P_H$
Laspeyres' Index	$P_L$	${\sum p_nq_0 \over \sum p_0q_0}$
Marshall-Edgeworth Index	$P_{ME}$	${\sum p_n(q_0+q_n)\over \sum (v_0+v_n)}$
Mitchell Index	$P_M$	${\sum p_nq_a\over \sum p_0q_a}$
Paasche's Index	$P_P$	${\sum p_nq_n\over \sum p_0q_n}$
Walsh Index	$P_W$	${\sum \sqrt{q_0q_n}\,p_n\over \sum \sqrt{q_0q_n}\ p_n}$

See also Bowley Index, Fisher Index, Geometric Mean Index, Harmonic Mean Index, Laspeyres' Index, Marshall-Edgeworth Index, Mitchell Index, Paasche's Index, Residue Index, Walsh Index

References

Fisher, I. The Making of Index Numbers: A Study of Their Varieties, Tests and Reliability, 3rd ed. New York: Augustus M. Kelly, 1967.

Kenney, J. F. and Keeping, E. S. ``Index Numbers.'' Ch. 5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 64-74, 1962.

Mudgett, B. D. Index Numbers. New York: Wiley, 1951.