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    the central limit theorem states that sampling distributions are always the same shape as the population distribution from whence the data came.


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    The Central Limit Theorem

    Within probability and statistics are amazing applications with profound or unexpected results. This page explores the amazing application of the central limit theorem.

    The Central Limit Theorem – How to Tame Wild Populations

    People come in a variety of shapes and sizes. Get a few million people together in one place, say in Rhode Island or South Carolina, and it would be impossible to predict what a single person selected from either state would be like. Try to compare all Rhode Islanders to all South Carolinians and the task gets even more complex. Obviously, something is needed to simplify the process, and that’s why we have statistics.

    The first step is to decide on a measurement, such as weight. Yes, this ignores that Mary Jane in SC has freckles and John in RI has a tattoo but we have to focus on something or we can’t make a comparison. Unfortunately, even after focusing on a single measurement, we still have two different populations of data points consisting of millions of wildly different numbers. These populations include everything from two pound preemies to 400+ pound bubbas.

    Again, we must simplify and so we’ll focus on a parameter that can characterize the weights of all individuals in a population. A parameter is a number which summarizes a specific characteristic generated from measurements of every member of a population. Using a parameter it’s possible to represent a property of an entire population with a single number instead of millions of individual data points.

    There are many possible parameters to choose from such as the median, mode, or interquartile range. Each is calculated in a different manner and illuminates the data from a different point of view. We’ll use the mean, because it’s one of the most useful and widely used. In spite of its harsh sounding name it has a very nice effect on helping us understand populations.

    The mean, or average, summarizes something called central tendency. This is a fancy way of saying what’s typical or expected in a population. If all the data points were plotted on a line segment the most typical values would usually be found somewhere near the center of the line segment, hence, the term central tendency.

    Of course central tendency isn’t the only issue. There’s also the issue of variability or spread. We like to call this although, admittedly, wildness is not a standard term. Still, seems appropriate, since, the characteristics of individuals drawn from populations with lots of variability or spread tend to be unpredictable.

    While the mean gives us a single number to describe a complex population, it’s, unfortunately, a parameter. We have to use every single data point in a population to calculate a parameter. With millions of data points to collect, this could be a real problem. By the time we got all the measurements, the two pound preemies might have turned into 400+ pound adults.

    The solution is to use a randomly chosen sample of the population and calculate a statistic. Statistics are always based on samples. We would carefully collect some data points chosen at random and calculate a sample mean. We’ll call this statistic x-bar. Clearly, x-bar is not a parameter since it’s not calculated from the entire population. It’s only an estimate of the parameter.

    If we selected another sample and calculated a second x-bar we might find that it differs considerably from the first. By their nature, random samples can sometimes give unexpected results even when flawlessly collected. For example, the first could be made up mostly of preemies while the second could be made up of sumo wrestlers. While such extremes are unlikely, selecting large sized samples would prevent them. After all, there are just so many preemies and sumo wrestlers available. A really large sized sample could never be made up entirely of either preemies or sumo wrestlers.

    Ultimately, if one sample mean or x-bar is wildly different from another we would not be any better off than trying to look at the entire population. Obviously, we need to understand just how wild or variable sample means or x-bars are likely to be.

    Central Limit Theorem Applet

    The attached applet simulates a population by generating 16,000 floating point random numbers between 0 and 10. Each time the "New Population" button is pressed it generates a new set of random numbers. The plot labeled Population Distribution shows a histogram of the 16,000 data points.

    The applet uses two different pseudo random number generators (PRNG) . The "Uniform Distr" option uses Java's standard PRNG in which every value has an equal probability. The "Normal Distr" option uses a PRNG from Java in which the probability of generating a particular value is determined by a normal distribution. The skewed distribution uses the same PRNG, however, the left side is truncated. The binomial distribution is simply two skewed distributions that are mirror images and scaled appropriately.

    Both PRNGs are referred to as pseudo random

    number generators since they generate numbers from equations which over a long period of time repeat. However, the numbers they generate are very similar to ideal random numbers.

    Below the population histogram is a histogram representing the sampling distribution of x-bar. Each time the resample button is pressed a new set of samples is obtained from the population and the sampling distribution histogram is re-plotted.

    The slider called "Sample Size" helps illustrate the central limit theorem. When the sample size is increased, the sampling distribution becomes narrower as predicted by equation (1).

    Source : www.intuitor.com

    The Central Limit Theorem

    In all cases, by the time n = 30, the distribution in very symmetric and the variance continually decreases as we noticed for the home run data in the

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    The Central Limit Theorem

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    Chapter: Biostatistics for the Health Sciences: Sampling Distributions for Means

    In all cases, by the time n = 30, the distribution in very symmetric and the variance continually decreases as we noticed for the home run data in the previous section.


    Section 7.1 illustrated that as we average sample values (regardless of the shape of the distribution for the observations for the parent population), the sample average has a distribution that becomes more and more like the shape of a normal distribution (i.e., symmetric and unimodal) as the sample size increases. Figure 7.4, taken from Kuzma (1998), shows how the distribution of the sample mean changes as the sample size increases from 1 to 2 to 5 and finally to 30 for a uni-form distribution, a bimodal distribution, a skewed distribution, and a symmetric distribution.

    In all cases, by the time = 30, the distribution in very symmetric and the variance continually decreases as we noticed for the home run data in the previous section. So, the figure gives you an idea of how the convergence depends on both the sample size and the shape of the population distribution function.

    What we see from the figure is remarkable. Regardless of the shape of the popu-lation distribution, the sample averages will have a nearly symmetric distribution approximating the normal distribution in shape as the sample size gets large! This is a surprising result from probability that is called the central limit theorem. Let us now state the results of the central limit theorem formally.

    Figure 7.4. The effect of shape of population distribution and sample size on the distribution of meansof random samples. (Source: Kuzma, J. W. Mountain View, California: Mayfield Publishing Company, 1984, Figure 7.3, p. 82.)

    Suppose we have taken a random sample of size from a population (generally, needs to be at least 25 for the approximation to be accurate, but sometimes largersamples sizes are needed and occasionally, for symmetric populations, you can do fine with only 5 to 10 samples). We assume the population has a mean and a standard deviation . We then can assert the following:

    1. The distribution of sample means  is approximately a normal distribution regardless of the population distribution. If the population distribution is nor-mal, then the distribution for  is exactly normal.

    2. The mean for the distribution of sample means is equal to the mean of the population distribution (i.e.,  = where  denotes the mean of the distri-bution of the sample means). This statement signifies that the sample mean is an unbiased estimate of the population mean.

    3. The standard deviation of the distribution of sample means is equal to the standard deviation of the population divided by the square root of the sample size [i.e.,  = (/), where  is the standard deviation of the distribution of sample means based on observations]. We call  the standard error of the mean.

    Property 1 is actually the central limit theorem. Properties 2 and 3 hold for any sam-ple size when the population has a finite mean and variance.

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    Source : www.pharmacy180.com

    central limit theorem

    Let's say I want to test if two independent samples have different means. I know the underlying distribution is not normal. If I understand correctly, my test statistic is the mean, and for large ...

    Independent samples t-test: Do data really need to be normally distributed for large sample sizes?

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    Asked 5 years, 11 months ago

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    Let's say I want to test if two independent samples have different means. I know the underlying distribution is not normal.

    If I understand correctly, my test statistic is the mean, and for large enough sample sizes, the mean should become normally distributed even if the samples are not. So a parametric significance test should be valid in this case, right? I have read conflicting and confusing information about this so I would appreciate some confirmation (or explanation why I'm wrong).

    Also, I've read that for large sample sizes, I should use the z-statistic instead of the t-statistic. But in practice, the t-distribution will just converge to the normal distribution and the two statistics should be the same, no?

    Edit: Below are some sources describing the z-test. They both state that the populations must be normally distributed:

    Here, it says "Irrespective of the type of Z-test used it is assumed that the populations from which the samples are drawn are normal." And here, the requirements for the z-test are listed as "Two normally distributed but independent populations, σ is known".



    z-test Share

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    edited Mar 30, 2016 at 17:33

    asked Mar 30, 2016 at 17:17

    Lisa 5376 6 silver badges 15 15 bronze badges

    What you are saying makes sense. You are using the central limit theorem to assume normality in the distribution of the sample means. Also, you are using the t-test because you don't have the population variance, and you are estimating it based on the sample variance. But can you link or post any of these conflicting sources? –

    Antoni Parellada

    Mar 30, 2016 at 17:25

    Thanks for your reply! Here for example, the requirements for the z-test are listed as "Two normally distributed but independent populations, σ is known", so they are talking about the distribution of the population, not the mean - is that wrong? –


    Mar 30, 2016 at 17:29

    @AntoniParellada I incorporated some sources in the original post! –


    Mar 30, 2016 at 17:34

    Check on Wikipedia –

    Antoni Parellada

    Mar 30, 2016 at 17:34

    @AntoniParellada Sweet, thanks again! Do you want to add your comments as an answer? –


    Mar 30, 2016 at 17:41

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    I think this is a common misunderstanding of the CLT. Not only does the CLT have nothing to do with preserving type II error (which no one has mentioned here) but it is often not applicable when you must estimate the population variance. The sample variance can be very far from a scaled chi-squared distribution when the data are non-Gaussian, so the CLT may not apply even when the sample size exceeds tens of thousands. For many distributions the SD is not even a good measure of dispersion.

    To really use the CLT, one of two things must be true: (1) the sample standard deviation works as a measure of dispersion for the true unknown distribution or (2) the true population standard deviation is known. That is very often not the case. And an example of n=20,000 being far too small for the CLT to "work" comes from drawing samples from the lognormal distribution as discussed elsewhere on this site.

    The sample standard deviation "works" as a dispersion measure if for example the distribution is symmetric and does not have tails that are heavier than the Gaussian distribution.

    I do not want to rely on the CLT for any of my analyses.

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    edited Apr 1, 2016 at 22:33

    answered Mar 31, 2016 at 12:55

    Frank Harrell 74.3k5 5 gold badges 148 148 silver badges 322 322 bronze badges 4

    The CLT may be a bit of a red herring. It often can happen that the sample mean has a decidedly non-normal distribution and the sample SD is decidely non-chi in shape, but nevertheless the t-statistic is usefully approximated by a Student t distribution (in part due to dependence between the two statistics). Whether this is the case ought to be evaluated in any given situation. However, because the CLT asserts little about samples (and says absolutely nothing about them), its invocation in support of distributional assumptions is usually invalid. –

    whuber ♦

    Mar 31, 2016 at 14:30

    Would it be fair to say that we are discussing (and learning in my case) a procedure (comparing two sample means from unknown distributions with a t-test) that is performed routinely (and possibly mindlessly) on a daily basis everywhere, although its justification can be weak? And, are there any uses of the CLT in practice, that would be tolerable/acceptable, even if not ideal? –

    Antoni Parellada

    Mar 31, 2016 at 14:41

    The t t

    -statistic very often has a distribution that is very far from the

    t t

    distribution when the data come from a non-Gaussian distribution. And yes I would say that the justification for using the

    t t

    -test is weaker than most practitioners think. That's why I prefer semi- and non-parametric methods. –

    Source : stats.stackexchange.com

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