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The height of a class of students in a school is said to follow Normal Distribution. We know that the domain of a random variable that follows Normal distribution is said to range from minus infinity to plus infinity. But height can never attain a negative value. So how can we say that height follows a Normal Distribution?

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  • $\begingroup$ What probability does this particular distribution assign to having a negative height? How many people would that translate into? $\endgroup$ – whuber Aug 1 '18 at 19:08
  • $\begingroup$ Using this website our author approximates a $6\sigma$ event (above 7'4", below 4') to occur once in every 1 billion people born. Quite unlikely. $\endgroup$ – ERT Aug 1 '18 at 19:25
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    $\begingroup$ Said by whom? What is your evidence that height follows a normal distribution? $\endgroup$ – Ben Aug 2 '18 at 5:37
  • $\begingroup$ Normality is a model, and we should not confuse that with a fact; such statements are typically a rough (if often useful) approximation. $\endgroup$ – Glen_b Aug 2 '18 at 10:59
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It is just an approximation, a generalization so that a high-school statistics classes may take samples of their friends in an interactive classroom activity. It also serves as a nice introductory lesson for these statistics classes, showing the prevalence of the normality assumption in real-life and how close a Gaussian actually comes to modeling real datasets.

According to Wikipedia...

Height is sexually dimorphic and statistically it is more or less normally distributed, but with heavy tails. It has been shown that a log-normal distribution fits the data equally well, besides guaranteeing a non-negative lower confidence limit, which could otherwise attain a non-physical negative height value for arbitrarily large confidence levels.

Also, according to this site (which also makes sweeping approximations and generalizations like the one you are concerned with)...

Note that this density is not Gaussian at all. Instead, it is very flat on top. You might reason that since the average of normal random variables is normal, adult heights should be normal. But we don’t have an average, we have a mixture. The density for the general adult population is a mixture of the male and female distributions. If you assigned a height to married couples as an average of the husband’s height and the wife’s height, the resulting value would be an average than a mixture and would follow a normal density.

photo corresponding to article

Here is a link to an article based in mathematics criticizing the very assumption you are questioning. Quite a nice article, especially the beginning parts.

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For many statistical analyses the exact distribution is not so important. You could call these analyses robust to the exact shape of the probability distribution. For instance, in regression analysis the distribution of the errors $\varepsilon$ is not so important for estimation of coefficients $\beta$: $$y=X\beta+\varepsilon$$ Even for estimation of uncertainty $\sigma_\beta=var[\beta]$ of coefficients the exact distribution of errors is not important in many cases.

Since the exact distribution is rarely known it is important that the statistical methods are robust to the choice of the approximation of these distributions. Otherwise, your results become unreliable. I suppose for many techniques in your field the approximation of the height as a normal variable is good enough. It is quite unlikely to encounter a high schools student who is 1 inch tall. The standard deviation of the height distribution is tight enough to make such a probability near zero, and the exact probability is not important. It's important to capture the approximate shape of the distribution in the range of values that matter.

Having wrote all of this, I must warn that sometime the shape of the distribution in tails does matter. One such case is the financial risk management. In this field one must be careful to consider the so called fat tails. Again, we usually do not know the exact shape of the tails of the probability density function, but we know that Gaussian distribution is often not fat enough in its tails. Choosing Gaussian approximation can lead to underestimation of certain risks, and potential for big financial losses. In this case we often pick other approximations that have fatter tails, such as Student t.

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