Explain Central Limit Theorem with an example

Question

Explain Central Limit Theorem in simple terms and provide a concrete example.

Codemia · Accepted Answer

The Central Limit Theorem (CLT) states that if we take a sufficiently large number of samples from a population with a finite level of variance, the distribution of the sample means will approximate a normal distribution, regardless of the shape of the population distribution. Formally, let $ X_1, X_2, \ldots, X_n $ be independent random variables drawn from a population with mean $ \mu $ and finite variance $ \sigma^2 $. The sample mean $ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i $ will have the following properties as $ n \to \infty $:

1. The expected value of the sample mean $ E(\bar{X}) = \mu $
2. The variance of the sample mean $ Var(\bar{X}) = \frac{\sigma^2}{n} $
3. As $ n $ increases, the distribution of $ \bar{X} $ approaches a normal distribution with mean $ \mu $ and variance $ \frac{\sigma^2}{n} $.

To illustrate the Central Limit Theorem, we will conduct an experiment using a dataset of employee salaries at Palantir. Let's say the population of salaries is not normally distributed. We will:

1. **Define the population**: Assume the average salary is $ \mu = 100,000 $ with a standard deviation $ \sigma = 15,000 $.
2. **Sample from the population**: Draw random samples of size $ n = 30 $ from this population.
3. **Calculate the sample mean**: For each sample, compute the sample mean $ \bar{X} $.
4. **Assess distribution of sample means**: Repeat the sampling process 1000 times to observe the distribution of the sample means and compare it to a normal distribution.

Let's assume we draw 1000 samples of size 30 from the population of salaries. After calculating the sample means for each of the 1000 samples, we find:
- Sample Mean Distribution: The means range from approximately 85,000 to 115,000.
- We compute the overall average of these sample means, which should be close to $ \mu = 100,000 $.
- The standard deviation of the sample means (standard error) is calculated as $ \frac{\sigma}{\sqrt{n}} = \frac{15,000}{\sqrt{30}} \approx 2738.61 $.
- The distribution of these sample means will appear normally distributed due to the CLT, even if the original salary distribution was skewed.

The result from our sampling illustrates that the sample means converge towards a normal distribution around the population mean of 100,000. This means if we were to conduct hypothesis tests (e.g., testing if the average salary differs from a benchmark), we can safely use the normal distribution to derive p-values, even if the underlying salary data is not normally distributed. This allows us to make inferences about the population based on sample data, which is vital for business decisions at Palantir.

Understanding the CLT connects to various statistical concepts such as:
- **Hypothesis Testing**: We can formulate null and alternative hypotheses based on sample means.
- **Multiple Testing Correction**: When conducting multiple hypotheses tests, the normal approximation allows for methods like Bonferroni correction or False Discovery Rate (FDR) adjustments.
- **Regression Analysis**: In regression scenarios, CLT ensures the distribution of coefficients estimates approaches normality, making inference valid.
- **Bayesian Inference**: CLT underpins the justification for normal approximations in posterior distributions, especially in large samples.

Explain Central Limit Theorem with an example

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