![]() The figure below illustrates a normally distributed characteristic, X, in a population in which the population mean is 75 with a standard deviation of 8. Central Limit Theorem with a Normal Population ![]() If the population is normal, then the result holds for samples of any size (i.e, the sampling distribution of the sample means will be approximately normal even for samples of size less than 30). In order for the result of the CLT to hold, the sample must be sufficiently large (n > 30). This means that we can use the normal probability model to quantify uncertainty when making inferences about a population mean based on the sample mean.įor the random samples we take from the population, we can compute the mean of the sample means:Īnd the standard deviation of the sample means:īefore illustrating the use of the Central Limit Theorem (CLT) we will first illustrate the result. In fact, this also holds true even if the population is binomial, provided that min(np, n(1-p)) > 5, where n is the sample size and p is the probability of success in the population. If the population is normal, then the theorem holds true even for samples smaller than 30. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). ![]() The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. ![]()
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