Understanding the t-Distribution: Small Sample Mean Estimation Explained

1 Distribution

The previous article on large‑sample mean estimation relied on the Central Limit Theorem, which breaks down when the sample size is small and the population standard deviation is unknown. In such cases we replace the normal approximation with the t‑distribution, which has ν = n‑1 degrees of freedom.

Derivation of the t‑distribution:

Let Z be a standard normal variable (Z ~ N(0,1)) and let V be a chi‑square variable with ν degrees of freedom (V ~ χ²_ν), independent of Z. Then the random variable T = Z / sqrt(V/ν) follows a t‑distribution with ν degrees of freedom, denoted T ~ t_ν.

To understand the chi‑square distribution, consider ν independent standard normal variables X₁,…,X_ν. Their squared sum Σ X_i² follows a chi‑square distribution with ν degrees of freedom:

V = Σ_{i=1}^{ν} X_i² ∼ χ²_ν.

Thus the normal distribution leads to the chi‑square distribution, which together with the normal distribution yields the t‑distribution.

The probability density function of the t‑distribution involves the gamma function Γ(·), but memorizing the full formula is unnecessary for most applications. The key points are the generation process and its practical use.

As the degrees of freedom increase, the t‑distribution’s shape approaches that of the standard normal distribution. When ν > 20, the difference is negligible.

When ν grows larger, the sample mean remains centered at 0, the standard deviation shrinks, and the distribution becomes increasingly normal.

2 Small Sample Population Mean Estimation

For a small sample of size n, a (1−α)% confidence interval for the population mean μ is

\(\bar{x} \pm t_{\alpha/2,\,\nu}\, \frac{s}{\sqrt{n}}\)

where \(\bar{x}\) is the sample mean, s is the sample standard deviation, ν = n−1 is the degrees of freedom, and \(t_{\alpha/2,\,\nu}\) is the critical value from the t‑distribution.

3 Summary

This article introduced the basic properties of the t‑distribution and presented the formula for constructing confidence intervals for a population mean when the sample size is small.

References

https://zhuanlan.zhihu.com/p/110207817

https://baike.baidu.com/item/t%E5%88%86%E5%B8%83/299142?fr=aladdin

https://baike.baidu.com/item/%E4%BC%BD%E7%8E%9B%E5%87%BD%E6%95%B0/3540177?fromtitle=%E4%BC%BD%E9%A9%AC%E5%87%BD%E6%95%B0&fromid=11217190&fr=aladdin