The standard deviation of the distribution of the sample means,, is called the standard error of the mean. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal n times the population mean. If the size ( n) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distributions regardless of the shape of the population. If the size ( n) of the sample is sufficiently large, then ~ N( μ, ) and ΣΧ ~ N( nμ, ( )( σ)). Central Limit Theorem Given a random variable (RV) with known mean μ and known standard deviation, σ, we are sampling with size n, and we are interested in two new RVs: the sample mean,, and the sample sum, ΣΧ. Glossary Average a number that describes the central tendency of the data there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean. The random variable has a different z-score associated with it from that of the random variable X. The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases. To put it more formally, if you draw random samples of size n, the distribution of the random variable, which consists of sample means, is called the sampling distribution of the mean. The variable n is the number of values that are averaged together, not the number of times the experiment is done. Standard deviation is the square root of variance, so the standard deviation of the sampling distribution is the standard deviation of the original distribution divided by the square root of n. The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by the sample size. The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means form their own normal distribution (the sampling distribution). ![]() ![]() If you draw random samples of size n, then as n increases, the random variable which consists of sample means, tends to be normally distributed and Using a subscript that matches the random variable, suppose: Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution). The Central Limit Theorem for Sample Means (Averages)
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