Let’s Summarize

If we have a quantitative data set from a population with mean [latex]µ[/latex] and standard deviation [latex]σ[/latex], the model for the theoretical sampling distribution of means of all random samples of size [latex]n[/latex] has the following properties:

• The mean of the sampling distribution of means is [latex]µ[/latex].
• The standard deviation of the sampling distribution of means is [latex]σ[/latex] divided by the square root of the sample size, [latex]n[/latex]. This is also called the standard error of the mean.
  • Notice that as [latex]n[/latex] grows, the standard error of the sampling distribution of means shrinks. That means that larger samples give more accurate estimates of a population mean.
  • For a large enough sample size, the sampling distribution of means is approximately normal (even if the population is not normal). This is called the central limit theorem.
  • Even if a distribution is non-normal, if the sample size is sufficiently large, a normal distribution can be used to calculate probabilities involving sample means and sample sums. This is even true for exponential distributions and uniform distributions.
  • The general rule is that if [latex]n[/latex] is at least 30, then the sampling distribution of means will be approximately normal. However, if the population is already normal, then any sample size will produce a normal sampling distribution.
• The Central Limit Theorem is not for calculating probabilities involving an individual value.