Lab · Group slopes实验 · 分组斜率
Does one line fit every faculty?一条线适合每一个学院吗？
A registrar has 48 students: 12 in Arts, 12 in Social Sciences, 12 in Science, 12 in Engineering. For each student the sheet has hours in a writing workshop and the course grade. The registrar can report one rise for all 48, or one rise in each faculty. The heavy line is the single rise, the pooled line. The thin lines are the four faculties. If Engineering rises fast and Arts is flat, the single number matches none of the four.一位教务主任有 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：写作工作坊的时数，和课程成绩。主任可以报告全部 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。细线是四个学院。如果工程学院升得快、文学院很平，这个单一的数四个学院都不符合。
Why this matters.为何重要。 A program can look helpful on average and still be steep in one faculty and flat in another. The average is a compromise among the groups, not a description of each of them.一个项目可以在平均上看起来有用，同时在一个学院里很陡、在另一个学院里很平。平均是各组之间的折中，不是对每一组的描述。
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。
Read the chart读这张图
If you misread this读错了会怎样
If the registrar tells the provost the heavy line means the writing workshop lifts grades about the same in every faculty, the provost 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.如果教务主任告诉校长，粗线表示写作工作坊在每个学院都差不多同样抬高成绩，校长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。
Enough, and not enough够用，以及不够
Enough to see one line through all 48 students sit between the four faculty lines.够用来看见：穿过全部 48 名学生的一条线，可以落在四条学院的线之间。
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.不够写资助说明。说明仍要说为什么按学院分组是对的，一个小学院的线该向其他学院借多少，以及每条线有多不确定。
Equations, assumptions, and limits方程、假设与限度
Each faculty has a fixed offset: −1, −0.35, 0.35, 1. A student’s slope is 0.30 plus the faculty gap times that offset, plus noise. The pooled line is ordinary least squares on all 48 students, with faculty ignored. The thin lines are least squares inside each faculty. Nothing here shrinks a faculty toward the university mean.每个学院有固定偏移：−1、−0.35、0.35、1。一名学生的斜率是 0.30 加上学院差距乘这个偏移，再加上噪声。汇总线是全部 48 名学生上的普通最小二乘，不管学院。细线是每个学院内部的最小二乘。这里没有把一个学院向全校均值收缩。