Central Limit Theorem
The central limit theorem (CLT) is, arguably, the most important theorem in statistics. It is, in many cases, an introductory statistics course.
The central limit theorem (CLT) is the most powerful foundation pilar in statistics.
Just about everything in statistics is based on CLT. Without it, the notion that sample data determines the population’s probability distribution wouldn’t exist.
The CLT is woven into the very fabric of our society that most of us don’t even notice it is there. Think about elections polls; not everyone is contacted and asked who they are voting for. Yet the results tend to be very close to the polls.
What exactly is Central Limit Theorem?
Basic Concepts of CLT
CLT is fairly simple. A large enough, randomly selected sample with a finite variance, the distribution of the sample means approach a normal distribution, regardless of the shape of the original population.
It is important to note that CLT promises the sampling distribution will, at some point, reach normal. The path it takes to get there it is entirely dependent on the original population.
The population creates a “friction” factor. Symmetrical populations have low friction, where a normal curve is produced almost immediately. When the population is skewed, it has friction, demandind a larger population before it behaves like a standard normal distribution.
Why Does a Normal Curve Matter?
Reaching the normal curve means that the data has reached Nirvana state in the statistical realm. It unlocks the most traditional statistical tests.
When you have a normal curve, it is a lot easier to use parametric tests. It is critically important when calculating confidence interval, standard and precise margin of error as it makes much easier to work with a predictable geometry of the normal curve.
The Distributions
Distributions directly impact how fast CLT kicks in and how large of the sample size needs to achieve for sampling distribution.
Taking from Kwak and Kim (2017), if you take an unbiased, 6-sided dice and roll at random 𝑛 times, what is the probability of landing on number 3 times? This is known as binomial distribution (the probability that models the exact number of successes in a fixed number of independent yes/no or binary trials). More here.
The image above shows the outcome of the test. When 𝑛=10, the probability is skewed. As they continued to roll the dice, the graph changed. Notice on panel C, at 𝑛=30, the distribution is close to normal. Increasing to 100 did produce a smoother shape, but the overall pattern remained about the same.
Conclusion
Understanding this concept is critical for future work. Anytime I’m working with a sample, reaching the normal curve help me ensure the data is large enough for statistical calculations.
In fact, the reason I revisited this concept was because I was working with an NHANES dataset and trying to determine whether the sample was sufficient to yield the answers I was seeking with the data available.
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