Question

One of the most important theorems that allows us to do inferential statistics includes the central limit theorem (CLT). What does the CLT say? Your answer should discuss the shape of the sampling distribution, it’s central tendency and dispersion.

Answer 1

The central limit theorem states that given a distribution with a mean μ and variance σ², the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases.

The interesting thing about the central limit theorem is that no matter what the shape of the original distribution, the sampling distribution of the mean approaches a normal distribution.

Two things to be noted about the effect of increasing N: (a) The distribution becomes more and more normal (b) The spread of the distribution decreases.

In non-mathematical terms, the Central Limit Theorem says that when we put together a lot of random events, the aggregate will tend to follow a bell-curve. That's how we get from something distributed linearly (say, the roll of a die, where each number is equally likely) to a curve where most events are near the average and the farther an event is from the average, the less likely it is. Most occurrences in nature may appear to be random (mostly because of the sheer size and the diverse factors in play) but when statistically analyzed, they are seen to fit the “bell-shaped” normal distribution. For example, how tall a person will be is the sum of a number of random variables (what genes the person has, what kind of food she eats, general state of health etc), and so people's heights distributes like a bell curve. The same thing applies to almost every physical property of living things. Political polling tells us that if we sum up a group of randomly-polled people, we will get a pretty good approximation of what would happen if we polled everybody. Thus, many events of life share the same characteristics as the central limit theorem. This is what makes the CLT such an important tool.

A simple demonstration of the CLT can be a numerical example such as - If samples of size 25 are drawn from a population of standard deviation σ , the mean of the sampling distribution will be close to the population mean (m) with the standard deviation, s = Population standard deviation,s/Ö25 = σ/5.