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I have spent some time trying to understand the Central Limit Theorem. I have come to the conclusion that the CLT says that the sampling distribution of the sample means gets closer to a normal distribution around the population mean as the sample size increases. (Is this correct?)

However, I am still wondering what understanding the concept of the CLT would help me with in statistics. Maybe my real question might be: "Why do I need to know the CLT?" Without knowing it, what difficulty would I have while studying statistics?

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    $\begingroup$ yes, your conclusion is correct. $\endgroup$ – carlo Sep 3 '20 at 21:12
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    $\begingroup$ The limit theorem will help you to understand the behaviour of sums of variables and why many experimental variables (that are such a sums) can be considered Gaussian distributed, and also why many (that are not such simple sums) are not. $\endgroup$ – Sextus Empiricus Sep 3 '20 at 22:37
  • $\begingroup$ The biggest takeaway to me would be importance in understanding sampling. $\endgroup$ – MaxW Sep 3 '20 at 23:47
  • $\begingroup$ @carlo Which conclusion is correct? The one in the first paragraph? $\endgroup$ – Dilip Sarwate Sep 4 '20 at 12:41
  • $\begingroup$ yes, that's the only one $\endgroup$ – carlo Sep 4 '20 at 23:46
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This is actually a very controversial subject. In my career I've noticed that people who understand the CLT often have worse understanding of what is really important when it comes to real-world data. And too often they don't take the time to do simple simulations that show that the CLT can require far greater sample sizes to work than they thought.

The idea of large sample theory and asymptotics is not appealing once you get comfortable with the Bayesian paradigm, which focuses on exact inference using flexible models. For example, the Bayesian t-test has parameters for two things we don't know: the ratio of the variances in the two populations, and a parameter for the degree of non-normality in the true unknown distribution. Bayesian posterior inference is exact at all sample sizes and will account for unequal variance and non-normality, and in addition will give you the probability of non-normality. This is explained in my BBR course in section 5.9.3 of the course notes.

Another way to get around any need for normality is to use semiparametric models which encompass basic nonparametric tests as special cases. This is also discussed in BBR.

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    $\begingroup$ The argument does not hinge on being or acting Bayesian. What matters generally is performance of methods with the samples in hand, not whether a procedure is good because of how it behaves with arbitrarily large samples. Even people who should know better push dubious methods. Thus one test for a normal distribution hinges on the asymptotic behaviour of sample skewness and kurtosis, but simulations show that even if the parent distribution really is normal, convergence of sampling distributions is extremely slow. (That such a test is often useless on other grounds is a different story.) $\endgroup$ – Nick Cox Sep 3 '20 at 11:56
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    $\begingroup$ On the face of it, misapplying the CLT is evidence that one does not understand it. $\endgroup$ – whuber Sep 3 '20 at 14:20
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    $\begingroup$ I'm not sure this answers the question anywhere. $\endgroup$ – gung - Reinstate Monica Sep 3 '20 at 19:54
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    $\begingroup$ @whuber one could even say that applying the CLT is evidence that one does not understand it, unless one already knows more about the data generating mechanism that I usually do. $\endgroup$ – Frank Harrell Sep 4 '20 at 11:20
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It's important to understand where distributions come from, when a particular distribution is an appropriate model, what conditions are assumed by a particular model, how different models are approximations of others, etc. For instance, with the Central Limit Theorem, the underlying distribution needs to have a finite standard deviation, and it's generally restricted to IID samples. Understanding the CLT helps you understand how the assumption of IID is used, and how it can be relaxed. Understanding how quickly it converges and what affects the convergence will help you understand things like that more skew will increase the time it takes to converge to a normal distribution, and with a highly skewed distribution a Poisson distribution may be a better model. Understanding when to use a normal or Poisson or student-t or $\chi2$, etc. is an important skill.

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  • $\begingroup$ also, CLT helps you understand the very nature of gaussian distribution. CLT is a key piece of understanding why gaussian distribution is so popular $\endgroup$ – carlo Sep 3 '20 at 21:05
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    $\begingroup$ The CLT provides no clue as to the rate of convergence to a normal distribution. Perhaps you meant: en.m.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem $\endgroup$ – Frans Rodenburg Sep 4 '20 at 1:41
  • $\begingroup$ If the CLT is a clue to why the gaussian distribution is so popular as a raw data distribution we need to change how we teach. $\endgroup$ – Frank Harrell Sep 4 '20 at 11:21
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    $\begingroup$ So how else do we arrive at normality, absent CLT? $\endgroup$ – BigBendRegion Sep 4 '20 at 14:19
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    $\begingroup$ That could not be further from the truth. Just look at the log normal distribution and the wrong answer that the CLT provides even for n=50,000. The CLT has absolutely nothing to do with raw data distributions and its use makes for bad statistical practice in typical sample sizes. "Significance and meaning" come from the accuracy of inference for our own small sample sizes. $\endgroup$ – Frank Harrell Sep 5 '20 at 12:45
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Your question runs to the heart of the difference between education and training. Instead of statistics, consider pharmacy and medicine.

A pharmacist has to have extensive coursework in chemistry and biology, yet their primary function or their nearly exclusive function is to count manufactured pills. Very few pharmacists compound drugs anymore. And, while their advising role couldn’t be substituted for by someone else, most of their advice is repetitive.

Likewise, for a general practitioner, in terms of frequency of behaviors, their most-used skills are taking blood pressure, looking in your mouth, and taking your pulse. That is hardly a good use for the calculus, chemistry, biology, and higher-end medical training that they received.

Indeed, the reason that physician assistants and pharmacy assistants exist in U.S. medicine is that most things of importance can be trained into a person and do not need higher-end reasoning.

The Central Limit Theorem is that sort of thing. If you completely skipped it, you could still do a t-test, estimate a Bayesian posterior density, find the sample median, or perform the Kolmogorov-Smirnov test. For 95% of the applications out there, you would be skilled enough, and you would be competent enough to provide advice to others as well.

The difficulty would happen when you believed you knew what to do, but you were wrong. For example, there are distributions where the assumptions of the Central Limit Theorem are strongly violated, and the sample mean is without meaning at all. In some areas of knowledge, that is a common problem. In other areas, it is never a problem. The Central Limit Theorem, at its most basic application, lets you know that sampling distributions exist as a concept. At the advanced level, it will keep your work from imploding.

EDIT For The Comments Consider prices set in a double auction, $p_1$ and $p_2$ with quantities $q_1$ and $q_2$. Return is defined as $$r_1=\frac{p_2}{p_1}\times\frac{q_2}{q_1}-1.$$ Let us define $R=r+1$. For brevity, let us ignore dividends and when $q_2=0$ due to bankruptcy and when $q_2^j=kq_1^j$ and for mergers, or this will go on for about forty pages.

In a double auction there is no winner's curse, so the rational action of each actor is to bid their expectation as to its value. Again, for brevity as this is not required if we can go on for forty pages, let us assume there are very many actors. The limit book, which in later operations will be scaled by the variance, should be normally distributed around an equilibrium price $p^*$. Ignoring stock splits and stock dividends, $q_1=q_2$, so $$R=\frac{p_2}{p_1}.$$

Now, noting that $R$ is a slope, we can find the ratio distribution of the slopes. Unfortunately, if you do that in Cartesian coordinates around $(0,0)$ you end up with a messy mixture distribution of a Cauchy distribution and a distribution with finite variance. It isn't useful, at least in economics, because it requires data that could not be reached because the necessary extra data was never recorded.

However, if you integrate around the equilibrium prices $(p^*_1,p^*_2)$ and formally account for the cost of liquidity and the effect of bankruptcy, then you end up with a distribution that looks like real world data. Note that $\Re^2$ is not an ordered set, so the idea of $(0,0)$ is a bit arbitrary. You would then transform the distribution by adding back in the equilibrium return of $\frac{p_2^*}{p_1^*}.$ It is easier if you think about this as a vector in polar coordinates. The distribution of the slopes of the vector of bivariate shocks $(\epsilon_1,\epsilon_2)$ has no mean or variance. The shocks, individually, are normally distributed.

As a visual example, consider the distribution of daily returns for Carnival Cruise Lines below. The process gets complex when you consider annual returns instead because equity returns are not scale-invariant. You can see multi-week long shifts in the location of the supply and demand curves and those long shifts can be observed in annual returns sometimes as multiple peaks or splits in the scale parameter. The red line is the fitted line. enter image description here

Because the distribution lacks a first moment, standard tools such as least-squares will produce spurious results. That is the source of the failure of models like the Capital Asset Pricing Model or Ito models such as Black-Scholes, or time-series tools like GARCH to fail in validation over the population of data.

In fact, when Fama and MacBeth decisively falsified models like the CAPM in 1973, one would have thought they would have gone away. Indeed, the third to the last paragraph in Black and Scholes seminal paper on options pricing states they tested their model and it failed to pass validation. Likewise, the paper introducing GARCH as a concept tested the tool on equity returns and found the assumptions so strongly violated that they stated it shouldn't be used for equities.

However, what every economist learns is that $\hat{\beta}=(X'X)^{-1}(X'Y)$ and it or a cousin, such as FGLS, fills the literature.

The Central Limit Theorem doesn't apply to a range of real data types, other than equity securities. If you do not know that, your field can produce 3800 papers on one small anomaly in options pricing as finance has.

Just a final note on the picture above, it is possible to improve the fit. The solution I used was a bit crude but vastly superior to assuming normality. Hundreds of thousands of hours have been spent in research in finance, financial economics, and macroeconomics by ignoring the fact that returns are not data. Prices are data. Volumes are data. Returns are a statistic and a function of prices, volumes, and dividends.

It is no more proper to assume a statistic's distribution than it would be proper to assume the sampling distribution of the difference of two means is the $\chi^2$ distribution because you didn't check to see if that was correct.

You can find examples of this type of phenomenon in physics, hydrology, biology and medicine.

The Central Limit Theorem not only says what happens when it works, but it also sets the conditions of when it does not work. It is both a blessing and a warning.

You are correct, there are practical limitations on the CLT, but technicians never know that. Personally, I have yet to be given an infinitely large data set. My guess is that my laptop is happier with that state of affairs anyway.

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  • $\begingroup$ It would be very nice to illustrate what you say in the last paragraph with one or two examples. Which distribution? Which area? $\endgroup$ – ttnphns Sep 5 '20 at 3:52
  • $\begingroup$ I would like the same demonstration because I'm dubious of this final claim. And in the text before that, the sample mean can have meaning even if the sample standard deviation doesn't. CLT requires meaningful SDs and requires the SD to be statistically independent of the mean, which doesn't happen with skewed raw data distributions. $\endgroup$ – Frank Harrell Sep 5 '20 at 12:47

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