Let’s Summarize

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

• The mean of the sampling distribution of means is $µ$.
• The standard deviation of the sampling distribution of means is $σ$ divided by the square root of the sample size, $n$. This is also called the standard error of the mean.
  • Notice that as $n$ 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 $n$ 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.