SD - Central Limit Theorem (CLT) Lesson

The Central Limit Theorem states that the means of LARGE samples are always normally distributed. But then the question becomes "How large is large?" This is a difficult question to answer: the larger the sample size, the closer a distribution is to being normal. There is no exact point where we can say that the sample size is large enough to warrant an assumption that the sampling distribution is normal. In reality, we tend to use n = 30 as a cutoff point concluding that for samples of size n >30 we can assume that a sampling distribution is approximately normal, otherwise we do not.

THE CENTRAL LIMIT THEOREM ALLOWS US TO USE NORMAL PROBABILITY CALCULATIONS TO ANSWER QUESTIONS ABOUT SAMPLE MEANS OR PROPORTIONS FROM MANY OBSERVATIONS EVEN WHEN THE POPULATION DISTRIBUTION IS NOT KNOWN TO BE NORMAL.

More specifically, the Central Limit Theorem tells us that as the sample size, n, increases, the mean of those n independent values has a sampling distribution that tends toward a normal model with mean equal to the population mean μ , and standard deviation = σ /sqrt (n). When we estimate the standard deviation of a sampling distribution we call it a standard error. Proportions are a special case of means and are also covered by the CLT.

DO NOT CONFUSE the Central Limit Theorem (CLT) with the Law of Large Numbers (LLN). The Central Limit Theorem says "the sampling distribution of ANY mean (or proportion) becomes Normal as the sample size grows larger" regardless of the SHAPE of the population from which the sample was drawn. YAY...this allows us to use normal calculations!!! OKAY so a sampling distribution must be a distribution of SAMPLES...MANY, MANY THEORETICAL SAMPLES.

On the other hand, the Law of Large Numbers says that as the sample size gets larger, the sample AVERAGE is more likely to be closer to the ACTUAL population mean...a BETTER ESTIMATOR! (No mention of sampling distributions.) And remember, these estimators are NEVER 100% accurate. The use of the Central Limit Theorem depends on some basic assumptions being made along with verifying a few conditions. The conditions involve assumptions that we have been making all along but now become a more formal part of our analysis.

1. Random Sampling Condition: The values must be the result of a RANDOM sample or a sampling distribution makes no sense.
2. Independence Assumption: The values must be INDEPENDENT. Remember, one way to verify this is that if the sample is draw without replacement, the trials are generally independent. Often, there is no good way to check for independence so this is considered an assumption.

3. 10% Condition: The sample size must be less than 10% of the total population.

All of these conditions must be met. If we do not have direct knowledge about any particular one, then we must ASSUME the condition has been met.

More detail and examples are contained in the lessons below.

Central Limit Theorem Links to an external site.

Sampling Distribution of a Sample Mean Links to an external site.

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