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Probability is a branch of mathematics that deals with the likelihood of different outcomes. Think of it as a measure of how likely something is to happen. When flipping a coin, there are two possible outcomes: heads or tails. The probability of each outcome is 50%. Probability ranges from 0 to 1, where 0 means an event will not happen and 1 means it definitely will happen.

In the context of the Central Limit Theorem, probability helps us understand the chance of various sample means. As we take more samples, we can predict how these sample means will distribute around the population mean. This predictive power is crucial in statistics.

The normal distribution is a common type of continuous probability distribution. It is symmetric around its mean, with data near the mean being more frequent in occurrence than data far from the mean. This creates a bell-shaped curve when graphed. A key feature of the normal distribution is that it is described by its mean \(\mu\) and standard deviation \(\sigma\).

The Central Limit Theorem states that as the sample size increases, the distribution of the sample means becomes approximately normal, regardless of the shape of the population distribution. This is significant because it allows us to apply normal distribution techniques even when we don’t know the original population distribution.

Sample size refers to the number of observations in a sample. It's a crucial concept in the Central Limit Theorem because a larger sample size tends to result in a more accurate estimate of the population mean. Usually, a sample size \(n > 30\) is considered large enough for the Central Limit Theorem to hold true.

With a larger sample size, individual peculiarities average out, making the sample mean more reliable. Statisticians say that a larger sample size leads to a smaller standard error, which means the sample mean is closer to the true population mean.

The sample mean is the average of the observations in a sample. It is denoted as \(\bar{X}\). Calculating the sample mean involves summing all the observation values and then dividing by the number of observations. The sample mean is used as an estimate of the population mean \(\mu\).

According to the Central Limit Theorem, no matter what the shape of the population distribution is, the distribution of the sample mean \(\bar{X}\) will tend to follow a normal distribution as the sample size increases. This property of the sample mean is incredibly powerful for making inferences about the population.