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Distribution of the Sample Mean Sampling ‘n’ items from a population with mean μ and standard deviation σ Each sample of ‘n’ items has a mean These sample means have a central tendency and form a distribution If ‘n’ is large enough (>30) the sample means form a normal distribution. The Central Limit Theorem What is the mean of a group of these samples? What is the standard deviation of a group of these samples? This distribution has: mean approximately equal to the population mean μ standard deviation SD n The Central Limit Theorem Do different population distributions change the result? A normal distribution is produced regardless of the nature of the original distribution The Central Limit Theorem What effect does the changing the sample size ‘n’ have? The larger the sample size the more accurate the sample mean is. The distribution of sample means then has a smaller standard deviation (there is less error). The Central Limit Theorem What is the standard error? SD n is also called the standard error of the sample means. Example Samples of size 20 are taken from a population of bats with a mean weight of 90 g and standard deviation 10 g. Calculate the mean and s.d. of the means of samples of bats taken from this population. E( ) = 90 g SD ( )