Final answer:
A die follows a uniform distribution, where each number on the die has an equal chance of appearing. As the number of dice rolled increases, the mean of the sample means remains approximately the same, but the spread of the sample means gets smaller.
Step-by-step explanation:
A die is a cube with six faces, each numbered from 1 to 6. When you roll a die, each number on the die has an equal chance of appearing. This means that the probability of rolling any specific number is 1/6. This uniform distribution of probabilities means that each number on the die is equally likely to occur.
When you roll multiple dice and calculate the mean of the numbers rolled, you are calculating the average value. As you increase the number of dice rolled, the mean of the sample means remains approximately the same. However, the spread of the sample means, measured by the standard deviation, gets smaller. This means that the sample means become more tightly clustered around the mean. Additionally, the graph of the sample means becomes steeper and thinner as the number of dice rolled increases.
This pattern is described by the Central Limit Theorem, which states that as the number of dice rolled increases, the sample means tend toward a normal distribution. In other words, the sampling distribution of the means becomes more bell-shaped as the number of dice increases.