Lab · Group slopes练习 · 分组斜率
Does one line fit every faculty?一条线适合每一个学院吗？
A writing workshop has 48 students: 12 in Arts, 12 in Social Sciences, 12 in Science, and 12 in Engineering. For each student the sheet has hours in the workshop and the course grade. The teacher can report one rise for all 48, or one rise in each faculty. The heavy line is the single rise, the pooled line: one line that ignores which faculty a student is in. The thin lines are the four faculties. If Engineering rises fast and Arts is flat, that single number matches none of the four.写作工作坊有 48 名学生：文学院、社会科学学院、理学院、工程学院各 12 人。每人两列：工作坊时数，和课程成绩。老师可以报告全部 48 人共用的上升，也可以每个学院一条。粗线是那条共用的上升，也就是汇总线：不管学生在哪个学院，硬画的一条。细线是四个学院。如果工程学院升得快、文学院很平，这一个数四个学院都不符合。
Why it matters.为什么要紧。 A workshop can look helpful on average and still be steep in one faculty and flat in another. The average is a compromise among the groups. It is not a description of any one of them.一个工作坊可以在平均上看起来有用，同时在一个学院里升得很陡、在另一个学院里几乎是平的。平均是各组之间的折中，不是对其中任何一组的描述。
Try this试一试 Set the faculty gap near zero and watch the lines sit together. Then raise the gap. The heavy line stays in the middle while the thin lines fan out.把学院差距调到接近零，看线靠在一起。再把差距调高。粗线留在中间，细线散开。
How to read the chart怎么读这张图
If you read this wrong要是读错了
If the teacher tells the faculty office that the heavy line means the writing workshop lifts grades about the same in every faculty, the office may fund one campus-wide program. When the thin lines fan out, Engineering can be steep and Arts flat. The heavy line is neither faculty.如果老师告诉院办，粗线表示写作工作坊在每个学院都差不多同样抬高成绩，院办可能会资助一个全校统一的项目。细线散开时，工程学院可以很陡，文学院很平。粗线不是其中任何一个学院。
What this picture is enough for这张图够说明什么
Enough to see one line through all 48 students sit between the four faculty lines.看得出：穿过全部 48 名学生的一条线，可以落在四条学院的线中间。
Not enough for the funding note. It still has to say why faculty is the right grouping, how much a small faculty’s line should borrow from the others, and how uncertain each line is.还不够拿来写资助说明。说明仍要说为什么按学院来分组，一个小学院的线该向其他学院借多少，以及每条线有多没把握。
The equation, and where it stops式子、假设和边界
Each faculty has a fixed offset: −1, −0.35, 0.35, 1. A student’s slope is 0.30 plus “how far the faculties differ” times that offset, plus a little jumpiness. The heavy line is ordinary least squares on all 48 students, with faculty ignored. The thin lines are the same fit inside each faculty. Nothing here pulls a small faculty toward the university average.每个学院有一个固定的偏移：−1、−0.35、0.35、1。一名学生的斜率是 0.30，加上“学院差多远”乘这个偏移，再加上一点跳动。粗线是全部 48 名学生放在一起的普通最小二乘，不管学院。细线是每个学院内部的同样拟合。这里没有把小学院往全校平均上拉。