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 when you take a sufficiently large sample size (typically n ≥ 30) from a population with a finite mean (μ) and a finite standard deviation (σ), the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution. In practical terms for Coinbase, this means that if we want to estimate the average transaction amount of a large number of users, we can rely on the normal distribution to make inferences about the population mean from sample means.

1. **Define the Population**: Assume the transaction amounts of all users follow an unknown distribution with mean μ and standard deviation σ.
2. **Take a Sample**: Randomly select a sample of n transactions. 
3. **Calculate the Sample Mean**: Compute the mean of the sample transactions. 
4. **Repeat Sampling**: Repeat this process many times (e.g., thousands of samples) to create a distribution of sample means. 
5. **Apply CLT**: According to the CLT, this distribution of sample means will be approximately normal with mean μ and standard deviation σ/√n.

Let's say the average transaction amount for Coinbase users is unknown, but we suspect it might be around $50 with a standard deviation of $20. We take a sample of 36 transactions:

- Sample Transaction Amounts: $30, $45, $55, $70, $60, $40, $50, $65, $80, $75, $35, $55, $45, $50, $60, $70, $80, $40, $30, $45, $60, $75, $55, $65, $70, $80, $50, $40, $60, $55, $45, $70, $80, $30, $50, $60.

- Sample Mean (X̄) = (Sum of all transactions) / n = $52.5.
- Standard Error (SE) = σ/√n = 20/√36 = 20/6 = $3.33.

According to CLT, the distribution of sample means will be approximately normal with mean $50 and standard deviation $3.33.

In practical terms, the result of $52.5 as the sample mean suggests that, based on our sample, we estimate that the average transaction amount for all Coinbase users is around $50. Given the standard error of $3.33, we can create a confidence interval to estimate the range of the true population mean. For instance, a 95% confidence interval would be $52.5 ± 1.96 * $3.33, which results in a range of approximately $46.5 to $58.5. This allows Coinbase to make informed decisions about transaction fees, promotions, and user engagement strategies.

The Central Limit Theorem connects to various statistical concepts:
- **Confidence Intervals**: As demonstrated, CLT allows us to construct confidence intervals for the population mean based on sample data.
- **Hypothesis Testing**: The normal distribution derived from CLT is often used in hypothesis testing to determine if there is enough evidence to reject a null hypothesis regarding the population mean.
- **Regression Analysis**: In regression, the assumptions of normality of residuals often rely on the CLT, especially when dealing with large datasets.
- **Assumptions**: Key assumptions of CLT include that samples are independent and identically distributed (i.i.d), and the population must have a finite mean and variance. If the sample size is small and the population is not normally distributed, the approximation may not hold.

Explain Central Limit Theorem with an example

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