by the degrees of freedom (calculated as n – 1). With a smaller number of observations, the t-distribution is flatter at the peak and has “thicker” tails compared to the normal distribution (figure 1). As the sample size increases, especially when n exceeds 30, the t-distribution starts to resemble the normal distribution. Perhaps contrary to Gosset’s expectations, as reflected in his letter to Fisher, “you’re the only man that’s ever likely to use them!” the t-distribution has become one of the most famous statistical distributions. It is widely applied in both everyday life and academic research, and of course, a staple in statistics courses. So, the next time you encounter the Student’s t-distribution (or find yourself grappling with it in class), take a moment to appreciate the “Student” behind it, William Sealy Gosset, and the fascinating story of its creation. 專題研習是否能提升學生的學業表現？哪位候選人更有 可能在選舉中勝出？某種新藥對治療特定疾病是否有效？ 雖然這些情境看似毫無關聯，但它們都指向一個核 心概念：我們需要從樣本收集資訊，然後針對某個總體 （population）作出推論，總體可以是全體學生、全 體選民，也可以是全體病人。這過程被稱為「統計推論」 （statistical inference）。假設抽樣過程是隨機且無偏 的，從樣本中計算出的統計數值，如樣本平均值和樣本方 差等，在每次取樣皆會有所不同。 因此，從多次抽樣中得到的樣本平均值會遵循一個特 定的分佈。撇開嚴謹的數學證明不說，根據中心極限定理 （central limit theorem），當樣本量足夠大時，即使總 體並非正態分佈（normally distributed），樣本平均值 的抽樣分佈（sampling distribution of the sample mean）仍會近似正態分佈。 但如果樣本量很小，且我們對總體的標準差沒有頭緒 時，又該怎麼辦呢？今天我們早已習以為常：在這種情況 下，只要總體為正態分佈，樣本平均值的抽樣分佈就會遵 循學生t 分佈（註一）。這一重要發現要歸功於一位名叫 William Sealy Gosset（1876–1937）的釀酒師 [1–3]。 Gosset出生於英格蘭的坎特伯雷，1899 年於牛津 大學取得化學一級榮譽學位。當時，位於都柏林的健力士 （Guinness）啤酒廠意識到在釀酒過程中進行嚴格品質 控制的重要性，因而開始從牛津和劍橋大學招募畢業生， Gosset 便是其中之一。 1. Karl Pearson (1857–1936) was a British statistician and a key figure in the development of modern statistics [7]. His work laid the foundation for many statistical methods and concepts still in use today, including the Pearson correlation coefficient and the chi-squared distribution. Notably, Pearson founded the first university statistics department in the world at University College London in 1911. 2. Ronald Aylmer Fisher (1890–1962) was a British statistician and geneticist [8]. Hailed as “a genius who almost single-handedly created the foundations of modern statistical science,” Fisher’s contributions to statistics include the significance test, analysis of variance (ANOVA), and maximum likelihood estimation, among many others. In genetics, he is regarded as one of the three founding fathers of population genetics, a key component of the modern synthesis that combines Mendelian genetics with Darwin’s theory of evolution.
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