Central Limit Theorem

The central limit theorem states that when you repeatedly draw samples and calculate their means, the distribution of those means approaches a normal distribution as sample size increases. This holds true even if the original population is not normally distributed. The mean of the sample means equals the population mean, but with smaller standard deviation, making statistical predictions more reliable.

As sample size increases, the sampling distribution of means becomes increasingly bell-shaped and approaches a normal distribution more closely. For populations that are not normal, a sample size of at least 30 is generally required for this approximation to hold. Smaller samples from non-normal populations may not produce normally distributed means.

The standard deviation of the sampling distribution equals the population standard deviation divided by the square root of the sample size. This reduction occurs because averaging multiple values reduces variability. As sample size increases, this standard error becomes even smaller, creating a tighter distribution of sample means around the population mean.

The empirical rule for normal distributions states that approximately 68%, 95%, and 99.7% of data fall within one, two, and three standard deviations from the mean, respectively. Since sample means follow a normal distribution, you can apply this rule to determine the probability that a group's mean weight falls within specific ranges around the sample mean.

Sample means can be standardized into z scores, which measure how many standard deviations a value lies from the mean. Using z scores and area under the curve, you can determine precise probabilities for sample means. For example, you can calculate the probability that randomly selected people have a mean weight less than 80 kg by converting to a z score and finding the corresponding probability.

The central limit theorem applies to non-normal populations when the sample size is sufficiently large, typically at least 30 observations. If the original population is far from normal, larger sample sizes may be needed for the sample means to follow a normal distribution. For populations that are already normal, the theorem applies regardless of sample size.

The central limit theorem is powerful because it allows statisticians to make reliable predictions about sample means even when the original population distribution is unknown or non-normal. By knowing that sample means behave predictably and follow a normal distribution, analysts can use the properties of normal distribution for inference and hypothesis testing across diverse data types.