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According to the central limit theorem (CLT), which assumes that all samples are the same size and are truly independent of the population distribution shape, the distribution of sample means approaches a normal distribution (often referred to as a "bell curve") as the sample size increases. Based on sample data and conclusions regarding population parameters, the CLT is helpful.
Any random variable having a finite mean and variance, including the sum, difference, product, or ratio of independent random variables, can be subjected to the CLT. Under some circumstances, the CLT can also be used for dependent processes like Markov chains, martingales, and mixing processes.
The CLT also states that the sampling distribution of the mean will have the following properties:
- The mean of the sampling distribution will be equal to the mean of the population distribution: x̄ = μ
- The variance of the sampling distribution will be equal to the variance of the population distribution divided by the sample size: s^2 = σ^2 / n
- The standard deviation of the sampling distribution will be equal to the standard deviation of the population distribution divided by the square root of the sample size: s = σ / √n
Consider a group of retirees who are all 65 years old on average, with a standard deviation of 10. Depending on the randomness of the sample, you might obtain a figure that is far from 65 if you pick a small sample of 5 retirees and calculate their mean age. For this sample size, the mean sampling distribution will not be normal but rather skewed and irregular.
The law of large numbers, though, dictates that if you take a sizable sample of 50 retirees and figure out their mean age, you'll probably end up with a number that's close to 65. Given this sample size, the mean will have a normal sampling distribution with a mean of 65 and a standard deviation of 10/50 = 1.41.
