that this is wasteful, since it yields much more information than just the median (., the upper and lower quartile points, the deciles, etc.). In fact, we saw in § that the element x(N+1)/2 can be located in of order N operations. | Do Two Distributions Have the Same Means or Variances 615 that this is wasteful since it yields much more information than just the median . the upper and lower quartile points the deciles etc. . In fact we saw in that the element x N 1 2 can be located in of order N operations. Consult that section for routines. The mode of a probability distribution function p x is the value of x where it takes on a maximum value. The mode is useful primarily when there is a single sharp maximum in which case it estimates the central value. Occasionally a distribution will be bimodal with two relative maxima then one may wish to know the two modes individually. Note that in such cases the mean and median are not very useful since they will give only a compromise value between the two peaks. CITED REFERENCES AND FURTHER READING Bevington . 1969 Data Reduction and Error Analysis for the Physical Sciences New York McGraw-Hill Chapter 2. Stuart A. and Ord . 1987 Kendal s Advanced Theory of Statistics 5th ed. London Griffin and Co. previous eds. published as Kendall M. and Stuart A. The Advanced Theory of Statistics vol. 1 Norusis . 1982 SPSS Introductory Guide Basic Statistics and Operations and 1985 SPSS-X Advanced Statistics Guide New York McGraw-Hill . Chan . Golub . and LeVeque . 1983 American Statistician vol. 37 pp. 242-247. 1 Cramer H. 1946 Mathematical Methods of Statistics Princeton Princeton University Press . 2 Do Two Distributions Have the Same Means or Variances Not uncommonly we want to know whether two distributions have the same mean. For example a first set of measured values may have been gathered before some event a second set after it. We want to know whether the event a treatment or a change in a control parameter made a difference. Our first thought is to ask how many standard deviations one sample mean is from the other. That number may in fact be a useful thing to know. It does relate to the strength or importance
