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Distribution of xbar11/7/2023 ![]() However, since we do not have many means at all but only one, produced by one sample, we are dealing with a probability distribution. Thus, what we see is that the majority of means would fall in the centre, fewer to the sides, and fewer still in the tail ends. You can think of a sample mean as one of these balls (all other balls are the means of other samples of the same size). The video above uses a Galton board to demonstrate the connection between randomness and normal curves by showing that balls falling randomly end up distributed approximately into a bell-shaped curve - with the majority in the centre, fewer to the sides, and fewer yet in the “tails”. ![]() If you are wondering about the connection between random sampling and the normal distribution, the following video might help: The latter is noteworthy because it’s true regardless of the shape of the original variable’s distribution (in the population) : a variable might not be normally distributed but its mean (and other statistics) always is. I say estimator and statistic, not mean, because CLT (or a version thereof) applies to all statistical estimators, as they all have a normal distribution with increasing sample size. The CLT provides a description of the sampling distribution: by giving us information about an estimator (in hypothetical repeated sampling), it decreases the uncertainty of the estimation since now we can calculate how close the statistic is to the parameter. To summarize, the sampling distribution provides us with a bridge between sample statistics (i.e., estimators) and population parameters (i.e., the estimated). ![]() Finally, according to the formula for the sampling distribution’s standard deviation (a.k.a the standard error), as the sample size N grows, the standard error becomes smaller - so the distribution will be less variable/spread out, and thus the estimates will be closer to the parameters. Hence, the sampling distribution is centered on the population mean. The next paragraph clarifies each of the CLT’s points in turn.Īs brief as it is, the CLT is conveniently packed with all sorts of useful information: The sampling distribution is normal in shape - so we can apply all we know about the normal distribution to it (for example, that it’s bisected by its mean). This may seem like a lot to take in (what with all the jargon, notation, and all) but it really is simply a description of a distribution. The standard deviation of the sampling distribution (denoted as ) is called the standard error, and is related to the population standard deviation, σ, by the formula.The mean of the sampling distribution (denoted as ) will become the population mean.(That is, the sampling distribution is a bell-shaped curve.) ![]() The distribution of will approach normal distribution in shape.Specifically, the CLT states that with random sampling, as N increases (i.e., for large N), the shape, central tendency, and the dispersion (of the sampling distribution) of the mean,, will be the following: The sample size plays an important role: the CLT applies to “large N”, and is stated for “as the sample size grows”, bringing us back to the point that the larger the N, the better for inference it is. In short, the CLT describes the sampling distribution of the mean. What the CLT does then is provide information about all three of these elements (shape, central tendency, dispersion) but about the distribution of mean. In the previous section I also asked you to imagine the (entirely theoretical, i.e., probability) distribution of the mean (again, in theory, o ver infinitely repeated samples). Recall what we use to describe a variable’s frequency distribution: 1) a graph to visually display the distribution’s shape 2) measures of central tendency and 3) measures of dispersion. Chapter 6 Sampling, the Basis of Inferenceĭespite it’s scary-sounding name, the Central Limit Theorem (CLT) simply describes the sampling distribution - and simultaneously explains why, and how, we can use sample statistics (like the mean of a variable,, obtained through sample data) to estimate population parameters (like the true population mean of that variable, μ). ![]()
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