1、What is the T Distribution?
T分布(也叫Student 's T分布)是一組與正態分布曲線幾乎相同的分布，只是更短更胖一點。當有小樣本時，使用t分布而不是正態分布。樣本容量越大，t分布越接近正態分布。事實上，對於大於20的樣本容量(如自由度)，其分布與正態分布幾乎完全相同。
The T distribution (also called Student’s T Distribution) is a family of distributions that look almost identical to the normal distribution curve, only a bit shorter and fatter. The t distribution is used instead of the normal distribution when you have small samples (for more on this, see: t-score vs. z-score). The larger the sample size, the more the t distribution looks like the normal distribution. In fact, for sample sizes larger than 20 (e.g. more degrees of freedom), the distribution is almost exactly like the normal distribution.
2、How to Calculate the Score for a T Distribution
當你看t分布表時，你會發現你需要知道df。這意味着“自由度”，也就是樣本容量減去1。
When you look at the t-distribution tables, you’ll see that you need to know the “df.” This means “degrees of freedom” and is just the sample size minus one.
方法：
Step 1: Subtract one from your sample size. This will be your degrees of freedom. Step 2: Look up the df in the left hand side of the t-distribution table. Locate the column under your alpha level(the alpha level is usually given to you in the question).
3、Uses
T分布(以及相關的T得分)用於假設檢驗，當您想知道您應該接受或拒絕原假設時
The T Distribution (and the associated t scores), are used in hypothesis testing when you want to figure out if you should accept or reject the null hypothesis.
圖中的中心區域是接收區域，尾部是拒絕區域。在這個雙尾測試的特殊圖表中，拒絕區域是藍色的。尾部區域可以用z分數或t分數來描述。例如，左邊的圖像顯示了5%(每邊2.5%)尾部的區域。z分數是1.96(來自z表格)，也就是離 平均值1.96個標准差。如果z小於-1.96或大於1.96，零假設將被拒絕。
The central region on this graph is the acceptance area and the tail is the rejection region, or regions. In this particular graph of a two tailed test, the rejection region is shaded blue. The area in the tail can be described with z-scores or t-scores. For example, the image to the left shows an area in the tails of 5% (2.5% each side). The z-score would be 1.96 (from the z-table), which represents 1.96 standard deviations from the mean. The null hypothesis will be rejected if z is less than -1.96 or greater than 1.96
一般而言，當樣本量較小(小於30)或不知道總體標准差時，就會使用該分布。出於實際目的(即在現實世界中)，情況幾乎總是如此。不像初級統計課，現實生活中會用到它而不是正態分布。如果樣本容量足夠大，這兩個分布實際上是相同的。
In general, this distribution is used when you have a small sample size (under 30) or you don’t know the population standard deviation. For practical purposes (i.e. in the real world), this is nearly always the case. So, unlike in your elementary statistics class, you’ll likely be using it in real life situations more than the normal distribution. If the size of your sample is large enough, the two distributions are practically the same.
4、參考文獻
https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/t-distribution/