- Return Section 7.6 Mindscapes: graded
- #22
Don't miss the point here: we're talking about the technical definition of the mean -- the sum of all the data, divided by the number of data values -- and not our vague sense of what average is.
In that sense, it's easy to come up with situations where more than half the people are below average -- think of the Lakeside school situation.
Or we could turn it upside down, and we'd have more people above the mean.
Don't confuse the mean and the median.
- #23:
Don't be afraid of decimals! The mean is 84.125.
Don't forget to sort the data before finding the median: remember that when we did the heights of the class, the first thing we did was get sorted.
- I saw some other nice things:
- Some great graphing!
- Nice work on the data sets where the median was higher than the mean. Some good examples.
- #22
- Reminder: Assignment: pp. 581-: #1, 3, 4, 8, 12. These will be
attached to your upcoming exam, coming up 11/3, and constitute 10% of
the exam.
Your exam will cover the material since the previous exam:
- infinity,
- dimension,
- geometry, and
- statistics.
- You might want to check out this information I worked up about the sampling methods using coins that we have looked at (and that you'll look at on Friday).
- A blast from the past: The uncola (7-up) versus the colas
- Sixth grade vice-presidential vote
6 boys prefer 4 boys prefer 8 girls prefer 4 girls prefer First Choice Chad Chad Gwyn Courtney Second Choice Courtney Gwyn Courtney Gwyn Third Choice Gwyn Courtney Chad Chad Three voting schemes, three different winners.
- Plurality voting -- Chad wins
- Vote for Two -- Courtney wins
- Borda Count Method -- Gwyn wins
- Here's my favorite system: Instant run-off, until we have a majority. Who wins?
Look, it seems like this should be easy. We want a system that has the following properties, given each voter's list of ordered rankings for the candidates:
- Better is better: If a voter changes his or her vote in favor of Candidate X, then that shouldn't hurt Candidate X.
- Ignore the irrelevant: throwing out a loser will not change the outcome.
- Go along with the consensus: if everyone agrees, then we have a winner.
Here's the main, astonishing result:
The only voting scheme that satisfies these three very reasonable requirements is a dictatorship! That is, one person makes the decision.
It was not a "fluke" that different methods gave different results. As mentioned in the text, mathematicians can "rig" a seemingly benign system that will hand the election over to a preferred candidate.
- "Transitivity" means that, if we prefer A to B, and B to
C, then we'll prefer A to C. Condorcet's Paradox says there's
no promise of transitivity!
Example:
5 voters' ranking 5 voters' ranking 5 voters' ranking First Place A B C Second Place B C A Third Place C A B Look at the beautiful symmetry in this table.... It's responsible for the problem, in many ways.
- I want to share a story from
my days in the Peace Corps in Togo.
- Our Electoral
College -- the following information is taken primarily from Wikipedia
- We elect our president by an "indirect election".
- Each state has a number of electors equal to the number of its Senators and Representatives in the United States Congress.
- Washington, DC receives as many electors as the smallest state (3).
- The total number of electors is hence 435+100+3=538.
- Most states allow voters to choose between slates of electors pledged to vote for tickets of the parties, and most states use a "winner take all" strategy, giving all the state's weight to the winning candidate's slate.
- Result: "U.S. presidential campaigns concentrate on winning the popular vote in a combination of states that choose a majority of the electors, rather than campaigning to win the most votes nationally."
- A majority would be more than half, or at least 270 votes.
- Each elector casts one vote for President and one vote for Vice President. Electors are technically free to vote for anyone eligible for those offices.
- Here's the map:
- Gore received the most votes in 2000, but Bush won the
2000 election. How could that be? Should that be?