CLT
What is the Central Limit Theorem and Why Does It Matter?
The central limit theorem (CLT) is one of the most important and useful concepts in statistics. It tells us that under certain conditions, the distribution of sample means approximates a normal distribution, regardless of the shape of the population distribution.
What does the CLT say?
Suppose we have a population with some unknown distribution and we want to estimate its mean. We can do this by taking random samples from the population and calculating their means. For example, if we want to estimate the average height of people in a country, we can randomly select some people and measure their heights.
The CLT states that as the sample size increases, the distribution of sample means becomes more and more normal, even if the population distribution is not normal. This means that we can use the properties of the normal distribution to make inferences about the population mean based on our sample mean.
The CLT also tells us two important facts about the sampling distribution:
- The mean of the sampling distribution is equal to the mean of the population distribution. This means that our sample mean is an unbiased estimator of the population mean.
- The variance (or standard deviation) of the sampling distribution is equal to the variance (or standard deviation) of the population distribution divided by the sample size. This means that as our sample size increases, our sampling error decreases and our estimate becomes more precise.
How large should our sample size be?
There is no definitive answer to this question, as it depends on how close we want our sampling distribution to be to a normal distribution. However, a common rule of thumb is that if our sample size is at least 30, then we can assume that our sampling distribution is approximately normal.
Of course, this rule may not apply in some situations where our population distribution is extremely skewed or has outliers. In those cases, we may need a larger sample size to ensure normality.
Why does it matter?
The CLT matters because it allows us to use powerful statistical tools that are based on normal distributions, such as confidence intervals and hypothesis tests. These tools help us make reliable conclusions about populations based on samples.
For example, suppose we want to test whether there is a difference between two groups in terms of their average scores on some test. We can use a t-test to compare their sample means and see if they are significantly different from each other. The t-test relies on an assumption that both groups have normally distributed scores or at least have large enough samples for their sampling distributions to be approximately normal.
If this assumption holds true, then we can use critical values from a t-distribution table or calculate p-values using software programs like Excel or R to determine whether our results are statistically significant or not.
However, if this assumption does not hold true and our samples are too small or too skewed for their sampling distributions to be approximately normal, then using a t-test may lead us to incorrect conclusions. We may either reject a true null hypothesis (type I error) or fail to reject a false null hypothesis (type II error).
Therefore, it is important to check whether our samples satisfy
the conditions for applying CLT before using any statistical methods
that rely on it.
Summary
The central limit theorem (CLT) states that under certain conditions,
the distribution of sample means approximates a normal
distribution regardless of
the shape of
the population
distribution.
This allows us
to use
normal-based statistical methods
to make inferences about populations based on samples.
We should always check whether our samples meet these conditions before applying CLT-based methods.
References:
: https://en.wikipedia.org/wiki/Central_limit_theorem
: https://www.statology.org/central-limit-theorem/
: https://www.investopedia.com/terms/c/central_limit_theorem.asp