Final answer:
Different groups of numbers can have the same summary statistics like mean and variance, but consist of distinct data points, illustrating how summary statistics can sometimes hide the true nature of the data distribution. The central limit theorem explains how sample means form their own normal distribution, echoing the mean of the original population and variance adjusted by sample size. This is key for hypothesis testing, which often assumes equal population means and variances.
Step-by-step explanation:
When different groups of numbers have exactly the same summary statistics, such as mean and variance, it means that these groups, while consisting of distinct values, share certain statistical properties. These properties do not capture the nature of the individual data points, which can be very different from one another. For example, two datasets can have the same mean and variance, yet one could be a collection of values tightly clustered around the mean, while the other could be widely spread out with more extremes.
The central limit theorem plays a crucial role in understanding why sample means form their own normal distribution, with a mean identical to the original distribution and a variance that is equal to the original variance divided by the sample size. This phenomenon occurs regardless of the original population's distribution. It explains why the standard deviation of the sampling distribution of the means decreases, becoming more consistent, as the sample size increases.
Therefore, even with a difference in individual data points or entire datasets, the mean and variance may remain consistent if the overall distribution does not change. This is especially important in hypothesis testing, such as testing the null hypothesis, where it is assumed that all group population means are equal with equal variances.