Final answer:
The Central Limit Theorem states that as sample size increases, the distribution of sample means will approach a normal distribution. A frequency distribution chart clustered near the mean suggests the influence of the Central Limit Theorem, and this is especially observable when sample sizes are large, typically at or over 30.
Step-by-step explanation:
If the measurements in a frequency distribution chart are grouped near the mean in a normal distribution, it suggests that these measurements are indeed the result of several different random effects. This phenomenon is explained by the Central Limit Theorem (CLT), which is a fundamental concept in statistics. The CLT states that when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.
The key to the CLT is the size of the sample. As you take more samples, or the sample size increases, you begin to see the sample means form their own normal distribution, which is called the sampling distribution. Importantly, this sampling distribution will have the same mean as the original distribution, and its variance is the original variance divided by the sample size (n).
For practical application, statisticians have found that a sample size of 30 is often sufficient for the CLT to hold true. This means that the distribution of sample means will approximate a normal distribution, irrespective of the population's original distribution. This normal distribution is characterized by its bell shape and is reflected in histograms of sample means.
An example of this is rolling dice. If you roll one die, the distribution of the means is simply the distribution of the die's faces. As you increase the number of dice rolled, the distribution of these sample means will more closely resemble a normal distribution. Therefore, the graph for the sample means will be steeper and thinner as the sample size increases.