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Q: The distribution of sample means is not always a normal distribution Under what circumstances will the distribution of sample means not be normal?

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It need not be if: the number of samples is small; the elements within each sample, and the samples themselves are not selected independently.

Yes. You could have a biased sample. Its distribution would not necessarily match the distribution of the parent population.

It approaches a normal distribution.

No, many sample statistics do not have a normal distribution. In most cases order statistics, such as the minimum or the maximum, are not normally distributed, even when the underlying data themselves have a common normal distribution. The geometric mean (for positive-valued data) almost never has a normal distribution. Practically important statistics, including the chi-square statistic, the F-statistic, and the R-squared statistic of regression, do not have normal distributions. Typically, the normal distribution arises as a good approximation when the sample statistic acts like the independent sum of variables none of whose variances dominates the total variance: this is a loose statement of the Central Limit Theorem. A sample sum and mean, when the elements of the sample are independently obtained, will therefore often be approximately normally distributed provided the sample is large enough.

The central limit theorem basically states that for any distribution, the distribution of the sample means approaches a normal distribution as the sample size gets larger and larger. This allows us to use the normal distribution as an approximation to binomial, as long as the number of trials times the probability of success is greater than or equal to 5 and if you use the normal distribution as an approximation, you apply the continuity correction factor.

Related questions

It need not be if: the number of samples is small; the elements within each sample, and the samples themselves are not selected independently.

The distribution of the sample mean is bell-shaped or is a normal distribution.

Yes. You could have a biased sample. Its distribution would not necessarily match the distribution of the parent population.

It approaches a normal distribution.

Not necessarily. It needs to be a random sample from independent identically distributed variables. Although that requirement can be relaxed, the result will be that the sample means will diverge from the Normal distribution.

The F distribution is used to test whether two population variances are the same. The sampled populations must follow the normal distribution. Therefore, as the sample size increases, the F distribution approaches the normal distribution.

No, many sample statistics do not have a normal distribution. In most cases order statistics, such as the minimum or the maximum, are not normally distributed, even when the underlying data themselves have a common normal distribution. The geometric mean (for positive-valued data) almost never has a normal distribution. Practically important statistics, including the chi-square statistic, the F-statistic, and the R-squared statistic of regression, do not have normal distributions. Typically, the normal distribution arises as a good approximation when the sample statistic acts like the independent sum of variables none of whose variances dominates the total variance: this is a loose statement of the Central Limit Theorem. A sample sum and mean, when the elements of the sample are independently obtained, will therefore often be approximately normally distributed provided the sample is large enough.

The central limit theorem basically states that for any distribution, the distribution of the sample means approaches a normal distribution as the sample size gets larger and larger. This allows us to use the normal distribution as an approximation to binomial, as long as the number of trials times the probability of success is greater than or equal to 5 and if you use the normal distribution as an approximation, you apply the continuity correction factor.

The form of this question incorportates a false premise. The premise is that the data are normally distributed. Actually, is the sample mean which, under certain circumstances, is normally distributed.

Because as the sample size increases the Student's t-distribution approaches the standard normal.

A half.

Frequently it's impossible or impractical to test the entire universe of data to determine probabilities. So we test a small sub-set of the universal database and we call that the sample. Then using that sub-set of data we calculate its distribution, which is called the sample distribution. Normally we find the sample distribution has a bell shape, which we actually call the "normal distribution." When the data reflect the normal distribution of a sample, we call it the Student's t distribution to distinguish it from the normal distribution of a universe of data. The Student's t distribution is useful because with it and the small number of data we test, we can infer the probability distribution of the entire universal data set with some degree of confidence.

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