# VACETS Regular Technical Column

The VACETS Technical Column is contributed by various members , especially those of the VACETS Technical Affairs Committe. Articles are posted regulary on [email protected] forum. Please send questions, comments and suggestions to [email protected]

Wed, 2 Apr 1997

# The Standard Deviation and The Normal Distribution: The Long and Short Of It.

By T. V. Nguyen

Many years ago while I was involved in statistical works in Vietnam, one of my senior colleagues remarked that: "given a few statistics, a good manager can assess the performance of a company in minutes". Indeed, as I mentioned in the last article that if one knows the mean and the standard deviation of a data set, one can have a pretty good idea of its distribution. For example, if a scientific paper reports that the mean and standard deviation (SD) of blood pressure (BP) of a sample of 100 elderly persons to be 110 and 15 mmHg, respectively, one can infer that approximately 95% of the subjects have BP ranged between 80 to 140 (=110 +/- 2*15). On the other hand, if another researcher reports that the mean and SD of 75 and 40 mmHg, respectively, then one can be sure that the researcher is either (i) wrong in his calculation or (ii) some of his patients were not normal, since 2*SD=80, which is greater than the mean itself. Another way of checking whether a researcher knows what he/she is doing is by asking him/her to provide the mean, SD and median. If the mean and median are not approximately the same, then the distribution of the data must be skewed and hence normal statistical methods may not applicable. In this note, I will discuss how the standard deviation can be used in assessing some interesting social issues.

Consider the following question: If you are a boss, would height play a role in your selection of a successor for your job? In a FORTUNE magazine column in 1981, a discussion concerning height as a factor in Deng Xiao Ping's choice of Hu Yao Bang for his replacement as Chairman of the Chinese Communist Party. The article notes the fact surrounding the case what is enough to arouse suspicions when examined in the light of statistics.

Deng, as we know, is only five feet tall, a height that is short even by Chinese standard. Therefore, the choice of Hu, who is also short (five-feet tall), raised (or lowered) eyebrows because, as the article notes, "the odds against a 'height-blind' decision producing a chairman as short as Deng are abput 40 to 1". In other words, if we possessed the relative frequency distribution of the heights in Chinese population, only one in 41 (i.e. 2.4%) of them would possesses heights less than or equal to five feet. The calculate these odds, the author of the article made some interesting assumptions concerning the relative frequency distribution of the heights of Chinese male population, mostly notable that the distribution follows the "normal" bell-shaped Gaussian curve (as it does in the USA).

It is generally held that a boy's length at birth represents 28.6% of his final height and that, in pre-revolutionary China, the average length of a Chinese boy at birth was 18.9 inches. From this, it can be deduced that the mean height of all mature male Chinese is (18.9/0.286) = 66.08 inches (or 5 feet, 6.08 inches). Assuming that the distribution of the heights of males in China follows a normal distribution (as it does in the US and in fact all countries in the world) with a mean of 66 inches and a standard deviation of 2.7 inches, a figure that look about right for the mean.

If we are willing to accept the assumption, that the heights of adult males are normally distributed wth mean (M) 66 inches and standard deviation (SD) of 2.7 inches, we are ready to calculate the probability that a single adult Chinese male, chosen at random, will have a height that is less than or equal to 5 feet, or equivalently, 60 inches. After some algebra (you may want to check it yourself), this probability is 0.0132 (i.e. 1.32%) or approximately 1 in 76, which corresponding to the odds of 75 to 1.

This odds certainly agree with our intuition, because it is difficult to believe that the proportion of 5-foot tall adult male Chinese is very large. Nevertheless, the validity of our calculated odds depends on the validity of our assumptions. However, it is possible that there is a flaw in our assumptions. It is not clear that the distribution of the heights of all adult male Chinese is a good model for the distribution of the heights of potential candidates for Deng Xiao Ping's replacement. Presumably, the candidates would be very select group of senior, and elderly, members of the Chinese Communist Party. It is a well-known fact that the heights of human decrease as they get older, particularly as they reach 60 years of age or older. Therefore, we would expect the distribution of the heights of the candidates for Deng's post to possess a mean that is less than the mean for the distribution of all adult male Chinese. We would also expect the odds of randomly selecting a person 5-feet tall or less from among the candidates to be larger than the corresponding odds of selection from among the population of heights of all Chinese adult males.

Did Deng Xiao Ping take height into account in selecting his successor? The answer to this question depends on the assumptions that you are willing to make. Consequently, we leave the answer to you. Perhaps, you will have additional reasons for accepting or not accepting the assumptions.

Tuan Nguyen, Ph.D.
[email protected]

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