Sample Size Derivation via CLT

Worked Solution

How to Think About It: This is one of the most fundamental calculations in statistics, and it comes up constantly in practice -- sizing a backtest, deciding how much data you need for a signal study, or figuring out if your track record is long enough to be meaningful. The logic is always the same: the CLT tells you $\bar{X}$ is approximately normal, so you just need the standard deviation of $\bar{X}$ to be small enough relative to your tolerance $\epsilon$. The three parts progressively strip away simplifying assumptions.

Quick Estimate: Suppose you want 95% confidence ($z_{0.025} = 1.96$) and your data has standard deviation $\sigma = 10$, and you want $\epsilon = 1$. Then $n \ge (1.96 \times 10 / 1)^2 = 384.16$, so you need $n \ge 385$. If the data is positively autocorrelated with, say, a design effect (DEFF) of 3, you would need roughly $385 \times 3 = 1{,}155$ observations. That is why autocorrelation is such a killer for precision -- it can triple your data requirements.

Approach: Apply the CLT in each setting, invert the probability statement to solve for $n$.

Formal Solution:

(i) Known variance $\sigma^2$:

By the CLT, $\bar{X} \approx N(\mu, \sigma^2/n)$, so $\frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \approx N(0,1)$. The confidence requirement becomes:

$P\left(\left|\frac{\bar{X} - \mu}{\sigma/\sqrt{n}}\right| \le \frac{\epsilon \sqrt{n}}{\sigma}\right) \ge 1 - \alpha$

This holds when $\frac{\epsilon \sqrt{n}}{\sigma} \ge z_{\alpha/2}$, where $z_{\alpha/2}$ is the upper $\alpha/2$ quantile of the standard normal. Solving for $n$:

$\boxed{n \ge \left(\frac{z_{\alpha/2} \, \sigma}{\epsilon}\right)^2}$

This is the textbook formula. Notice $n$ scales as $\sigma^2 / \epsilon^2$ -- halving your error tolerance quadruples your sample size.

(ii) Unknown variance:

When $\sigma$ is unknown, you replace it with the sample standard deviation $s$ and use the $t$-distribution with $n - 1$ degrees of freedom:

$n \ge \left(\frac{t_{\alpha/2, \, n-1} \cdot s}{\epsilon}\right)^2$

This is an implicit equation because the $t$-quantile depends on $n$. In practice, you handle this in one of two ways:

• Pilot study approach: Collect a small pilot sample of size $n_0$, compute $s$ from it, then use the formula with $z_{\alpha/2}$ instead of $t_{\alpha/2, n-1}$ (valid when $n$ is large enough that $t \approx z$).
• Iterative approach: Start with $n_0 = (z_{\alpha/2} \cdot s / \epsilon)^2$, compute $t_{\alpha/2, n_0 - 1}$, update $n$, and repeat until convergence. This typically converges in 2-3 iterations.

For $n \ge 30$ or so, $t_{\alpha/2, n-1} \approx z_{\alpha/2}$ and the distinction is negligible.

(iii) Autocorrelated data:

When observations are serially correlated, the variance of $\bar{X}$ is no longer $\sigma^2 / n$. Instead:

$\text{Var}(\bar{X}) = \frac{1}{n} \left(\gamma_0 + 2 \sum_{k=1}^{n-1} \left(1 - \frac{k}{n}\right) \gamma_k \right) \approx \frac{\sigma_{LR}^2}{n}$

where $\gamma_k = \text{Cov}(X_t, X_{t+k})$ and the long-run variance is:

$\sigma_{LR}^2 = \gamma_0 + 2 \sum_{k=1}^{\infty} \gamma_k = \sigma^2 \left(1 + 2 \sum_{k=1}^{\infty} \rho_k \right)$

The ratio $\text{DEFF} = \sigma_{LR}^2 / \sigma^2$ is called the design effect. The required sample size is:

$\boxed{n \ge \left(\frac{z_{\alpha/2} \, \sigma_{LR}}{\epsilon}\right)^2 = \text{DEFF} \cdot \left(\frac{z_{\alpha/2} \, \sigma}{\epsilon}\right)^2}$

Equivalently, you can think of it as the i.i.d. sample size divided by the effective sample fraction: $n_{\text{eff}} = n / \text{DEFF}$. In practice, $\sigma_{LR}^2$ is estimated using the Newey-West HAC estimator with a kernel bandwidth $M$ (typically $M \approx n^{1/3}$):

$\hat{\sigma}_{LR}^2 = \hat{\gamma}_0 + 2 \sum_{k=1}^{M} w(k, M) \, \hat{\gamma}_k$

where $w(k, M) = 1 - k/(M+1)$ is the Bartlett kernel weight.

For a stationary AR(1) process with autocorrelation $\rho$, the DEFF simplifies to $(1+\rho)/(1-\rho)$. For $\rho = 0.5$, DEFF $= 3$, so you need 3x as many observations as the i.i.d. formula suggests.

Answer: The required sample sizes are: (i) $n \ge (z_{\alpha/2} \sigma / \epsilon)^2$ for known variance; (ii) the same formula with $s$ replacing $\sigma$ and iterating on the $t$-quantile for unknown variance; (iii) $n \ge \text{DEFF} \cdot (z_{\alpha/2} \sigma / \epsilon)^2$ for autocorrelated data, where $\text{DEFF} = \sigma_{LR}^2 / \sigma^2$ captures the inflation due to serial correlation.