在這篇論文中，我們引進兩個加權距離(Wilcoxon rank and Log rank)去分析52張撲克牌必須洗幾次才會均勻的問題。Bayer和Diaconis (1992)使用 variation distance 的方法分析洗牌問題。Lin (2006)為了避免複雜的數學算式，使用 deviation distance 的方法分析洗牌問題。這裡，我們提供兩個距離加權法去探討洗牌問題。

In this paper, we cite two weighted rank distances (Wilcoxon rank and Log rank) to analyze how many times must a deck of 52 cards be shuffled to become
sufficiently randomized. Bayer and Diaconis (1992) used the variation distance as a measure of randomness to analyze the card-shuffling. Lin (2006) used the deviation distance to analyze card-shuffling without complicated mathematics formulas. We provide two new ideas to measure the distance for card-shuffling analysis.

1 Introduction
2 A measure of randomness - distance
2.1 The deviation distance
2.2 The absolute distance
3 The weighted rank distance
3.1 The distance based on Wilcoxon rank
3.2 The distance based on Log rank
4 Simulation study
4.1 The chi-square goodness of fit test
4.2 Simulation results
5 Conclusion
Appendix
References

[1] Aldous, D. (1983). ”Random walks on finite groups and rapidly mixing Markov chains.” Springer Lecture Notes in Mathematics, Vol. 986, pp. 243-297.
[2] Aldous, D. and Diaconis, P. (1986). ”Shuffling cards and stopping times”. American Mathematical Monthly, Vol. 93, No. 5, pp. 333-348.
[3] Bayer, D. and Diaconis, P. (1992). ”Trailing the dovetail shuffle to its lair”. Annals of Applied Probability, Vol. 2, No. 2, pp. 294-313.
[4] Gilbert, E. (1955). ”Theory of shuffling”. Bell Telephone Laboratories memorandum.
[5] Lin, C.H. (2006). ”A study of shuffling cards and stopping times for randomness”. Master of Science Thesis, Department of Applied Mathematics, National Sun Yat-sen University.
[6] Pearson, K. (1900). ”On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be
reasonably supposed to have arisen from random samplig”. Philosophical Magazine , Vol. 50, No. 5, pp. 157-175.
[7] Reeds, J. (1981). Unpublished manuscript. Cited in Aldous and Diaconis, 1986. Not consulted by the present author.