True or False: only the sums of normal distributions are also normal distributions.

Question
Normal distributions
asked 2021-02-27
True or False: only the sums of normal distributions are also normal distributions.

Answers (1)

2021-02-28
In statistics, in a population whose distribution may be known or unknown, if the size (n) of samples is sufficiently large, the distribution of the sample means will be approximately normal.
The mean of the sample means will equal the population mean.
The Standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size (n).
The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution even if the original population is not normally distributed. Additionally, if the original population has a mean of x and a standard deviation of ox, the mean of the sums is nx and the standard deviation is \sqrt{n\sigma_{x}} where n is the sample size.
Therefore, the given statement “only the sums of normal distributions are also normal distribution” is false because the sums of any distribution approach a normal distribution as the sample size increase.
Conclusion:
False, the sums of any distribution approach a normal distribution as the sample size increases.
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