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]
For discussion on
this column, join [email protected]
Copyright ©
1997 by VACETS and Tuan Nguyen
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