What is the Central Limit Theorem and why is it important?

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The Central Limit Theorem (CLT) is a fundamental concept in statistics that explains how the distribution of sample means behaves.

✅ Definition:

The Central Limit Theorem states that:

When you take sufficiently large random samples from any population (regardless of its original distribution), the distribution of the sample means will approach a normal distribution (bell curve), provided the sample size is large enough.

📘 Key Points:

• Applies even if the population is not normally distributed.

• Works best when the sample size is ≥ 30.

• The mean of the sampling distribution equals the population mean.

• The standard deviation of the sampling distribution (called the standard error) is:

  $$\text{SE} = \frac{\sigma}{\sqrt{n}}$$

  where σ = population standard deviation, n = sample size.

🎯 Why It's Important:

1. Enables Inferential Statistics

   • Lets us make predictions about a population using sample data.

   • Supports confidence intervals and hypothesis testing.

2. Simplifies Analysis

   • Many statistical methods assume normality. CLT justifies using those methods even if the original data isn’t normal.

3. Real-World Applications

   • Used in quality control, polling, medical trials, A/B testing, etc.

🧠 Example:

Imagine you measure the average height of 50 people across many random samples. Even if height isn’t perfectly normally distributed, the distribution of those sample averages will be close to normal.

✅ Summary:

The Central Limit Theorem is powerful because it allows us to use normal distribution techniques to analyze and interpret data, even when the original data isn’t normal—as long as the sample size is large enough.

Read More

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What is the difference between population and sample?

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