Means, Medians, Logs, Ratios. Consider the following tuitions and their natural logs for five colleges:
|
College
|
In-State
|
Out-of-State
|
Out/In Ratio
|
Log(In-State)
|
Log(Out-of-State)
|
|
A
|
$1,000
|
$ 3,000
|
3
|
6.9078
|
8.0064
|
|
B
|
$4,000
|
$ 8,000
|
2
|
8.2941
|
8.9872
|
|
C
|
$5,000
|
$ 30,000
|
6
|
8.5172
|
10.3090
|
|
D
|
$8,000
|
$ 32,000
|
4
|
8.9872
|
10.3735
|
|
E
|
$40,000
|
$ 40,000
|
1
|
10.5966
|
10.5966
|
(a) Find the average In-State tuition. Find the average log(In-State). Confirm that the log of the average isnot the same as the average of the logs. (b) Find the median In-State tuition and the median of the logs of In-State tuitions. Verify that the log of the median is the same as the median of the logs. (c) Compute the median of the ratios. Compute the differences of logged tuitions-log(Out- of-State) minus log(In-State) and compute the median of these differences. Verify that the median of the differences (of log tuitions) is equal to the natural log of the median of ratios (aside from some minor rounding error).