# On Some Useful “Inefficient” Statistics

• Frederick Mosteller
Chapter
Part of the Springer Series in Statistics book series (SSS)

## Summary

Several statistical techniques are proposed for economically analyzing large masses of data by means of punched-card equipment; most of these techniques require only a counting sorter. The methods proposed are designed especially for situations where data are inexpensive compared to the cost of analysis by means of statistically “efficient”or “most powerful” procedures. The principal technique is the use of functions of order statistics, which we call systematic statistics.

It is demonstrated that certain order statistics are asymptotically jointly distributed according to the normal multivariate law.

For large samples drawn from normally distributed variables we describe and give the efficiencies of rapid methods:
1. i)

for estimating the mean by using 1, 2,⋯, 10 suitably chosen order statistics; (cf. p. 86)

2. ii)

for estimating the standard deviation by using 2, 4, or 8 suitably chosen order statistics; (cf. p. 89)

3. iii)

for estimating the correlation coefficient whether other parameters of the normal bivariate distribution are known or not (three sorting and three counting operations are involved) (cf. p. 94).

The efficiencies of procedures ii) and iii) are compared with the efficiencies of other estimates which do not involve sums of squares or products.

## Preview

### References

1. [1]
G. W. Brown and J. W. Tukey, “Some distributions of sample means,” Annals of Math. Stat., Vol. 17 (1946), p. 1.
2. [2]
J. H. Curtiss, “A note on the theory of moment generating functions,” Annals of Math. Stat., Vol. 13 (1942), p. 430.
3. [3]
R. A. Fisher and L. H. C. Tippett, “Limiting forms of the frequency distribution of the largest or smallest member of a sample,” Proc. Camb. Phil. Soc., Vol. 24 (1928), p. 180.Google Scholar
4. [4]
R. A. Fisher and F. Yates, Statistical Tables, Oliver and Boyd, London, 1943.
5. [5]
E. J. Gumbel, “Les valeurs extrêmes des distribution statistiques,” Annales de l’Institute Henri Poincaré, Vol. 4 (1935), p. 115.
6. [6]
E. J. Gumbel, “Statische Theorie der grossten Werte,” Zeitschrift für schweizerische Statistik und Volkswirtschaft, Vol. 75, part 2 (1939), p. 250.Google Scholar
7. [7]
H. O. Hartley, “The probability integral of the range in samples of n observations from a normal population,” Biometrika, Vol. 32 (1942), p. 301.
8. [8]
D. Jackson, “Note on the median of a set of numbers,” Bull. Amer. Math. Soc., Vol. 27 (1921), p. 160.Google Scholar
9. [9]
T. Kelley, “The selection of upper and lower groups for the validation of test items,” Jour. Educ. Psych., Vol. 30 (1939), p. 17.
10. [10]
M. G. Kendall, The Advanced Theory of Statistics, J. B. Lippincott Co., 1943.Google Scholar
11. [11]
J. F. Kenney, Mathematics of Statistics, Part II, D. Van Nostrand Co., Inc., 1939, Chap. VII.Google Scholar
12. [12]
K. R. Nair and M. P. Shrivastava, “On a simple method of curve fitting,” Sankhya, Vol. 6, part 2 (1942), p. 121.
13. [13]
K. Pearson, “On the probable errors of frequency constants, Part III,” Biometrika, Vol. 13 (1920), p. 113.Google Scholar
14. [14]
K. Pearson (Editor), Tables for Statisticians and Biometricians, Part II, 1931, p. 78, Table VIII.Google Scholar
15. [15]
N. Smirnoff, “Über die Verteilung des allgemeinen Gliedes in der Variationsreihe,” Metron, Vol. 12 (1935), p. 59.Google Scholar
16. [16]
N. Smirnoff, “Sur la dependance des membres d’un series de variations,” Bull. Univ. État Moscou, Series Int., Sect. A., Math. et Mécan., Vol. 1, fasc.4, p. 1.Google Scholar
17. [17]
L. H. C. Tippett, “On the extreme individuals and the range of samples taken from a normal population,” Biometrika, Vol. 17 (1925), p. 364.Google Scholar
18. [18]
S. S. Wilks, Mathematical Statistics, Princeton University Press, Princeton, N. J., 1943.