Consider the dot plots for populations with a normal and uniform distribution. The distribution of sample means for different sample sizes shows that it approaches a normal distribution as the sample size increases – this is the core principle of the central limit theorem. Although the mean of the sample means is the same as the population mean, its standard deviation is smaller than the population standard deviation. However, this rule does not apply to populations that are not normal and with a sample size of less than or equal to 30. By knowing that the sample means are normally distributed, one can make better statistical analysis using the properties of normal distribution. For example, the empirical rule that applies to the normal distribution helps to determine the probability of a group of people having mean weights within one, two, or three standard deviations away from the mean of the sample means. These values can also be standardized into z scores. So, one could determine the probability of a group of randomly selected people with a mean weight of less than 80 kg.