Stuff like this really honks me off, and this is even worse (Darren, you owe me blood pressure medication).

Teachers at Soquel High School have agreed not to wear “Educators for Obama” buttons in the classroom after a parent complained that educators were attempting to politically influence his daughter and other students.

These teachers must not have much to do in the classroom, if they have all this time to waste on topics that have nothing to do with the curriculum. But I promised myself I wouldn’t rant, so I won’t. Instead, I’ll offer an alternative for those who just can’t keep from bringing the election into the classroom — an alternative that does not push a candidate or a party, and actually has something to do with learning the class material — and critical thinking, in the literal, and not the “think like a slobbering leftist” education school definition. Wow, how about that!

Student interest is a great motivator, particularly when you teach something many students find boring, or even intimidating, like I did. One thing we did that was very successful was build several applications with the tools we were going to cover that grabbed student interest when we said, “At the end of the semester, you’ll be able to do this, too.”

One of these was a simulation model that based on the scores for all of the games that season predicted the winner of the Superbowl (we had one for the NBA finals and another for the World Series, depending on which semester we were in).

So if you absolutely must address the election in class, here is one way you can do it where the students will actually learn something, and contains not a hint of advocacy or indoctrination.

Have students build an application that predicts the results of the election. Remind them that the more variables they incorporate, the more accurate it will likely be, and encourage them to make it as complex as they like.

You’d want to break them into teams to do this, and give them time to talk about what variables they would want to incorporate, and how. You should probably give them a list of sources for data, like realclearpolitics.com, gallup.com, and rasmussenreports.com. In fact, give them a whole class period to do nothing but plan their model, figure out where they’d get the data, and assign people in the team to do various tasks.

I’d give them a week to turn in the models. After going through them, you can pull several up with different results and as a class, pick apart the applications and discuss why they got different results (this is what is known as a learning experience). You can then, again as a class, discuss which of the models is/are most likely to accurately predict the results, and why. You can even give bonus points to the team whose model most accurately predicts the election.

See? You addressed the election, and you didn’t have them sing creepy Hitler Youth songs.

If you think about it, these models incorporate a lot of mathematical knowledge in many different areas, and all through the model. Take collecting the data, say, polls. How are they going to deal with the different levels of statistical error in different polls? How will they deal with different party weights in different polls? What, other than polls, will they use as input variables, and how will they incorporate them into the model? For example, if they’re going to look at the number of voters who went for Hillary in the primaries and turn that into support for McCain, how, exactly, are they going to do it? What algorithm will they use, and what will they base it on? And would they also want to use another variable, say, Democrat respondents who only lean Democrat in the election, or are undecided to calculate their Hillary conversion variable?

And what about actual election day statistics, will they use those? If so, which variables? How will they incorporate them?

You can turn just about anything into a real, learning experience in the classroom if you just think about it. Unfortunately, “thinking” seems to be an alien concept to many teachers these days.

The learning isn’t only in creating the models. The learning — and critical thinking — is also in analyzing the models and comparing them once they’ve been done. What makes a good model? What makes this model more accurate than that one? Would this be a more accurate model if we tweaked the algorithms, and if so, how would we tweak them? You get the idea.

When my students are working in teams, I usually migrate from team to team, playing devil’s advocate, and gently nudging them when they’re completely off track (I call this guided constructivism). With a project like this, I would probably limit my input to making sure they understood, and correcting fundamental errors, like only taking into consideration the popular vote. Oh. And I would only do something like this after the students had all of the necessary knowledge and skills to actually build a working application. Sorry, but if you think turning students loose on their own to do complex projects like this is a good way to introduce them to new skills, you have no business within a hundred miles of a classroom.

(We talked about doing this with one of the sports championships, don’t remember which now, but decided against it because making the data usable would require complex Excel text functions we had not covered in class. This would definitely not be a way to teach them how to build a simulation.)

Cross-posted at Kitchen Table Math

Have you noticed that the only numbers we’re getting is the amount of the bailout? In the last couple of days, I’ve heard 700 billion and 1 trillion. But you know, there are some things we — not to mention Congress — need to know.

First, what will be the estimated economic cost of the bailout?

Second, what will be the estimated economic cost of no bailout?

Surely, economists somewhere are calculating these numbers, right? If the cost of the bailout is greater than the cost of no bailout, then passing a bailout bill would be insane. No informed decision can be made without these two numbers, so I repeat, economists are calculating them somewhere, surely?

Unfortunately, I doubt it. I’m sure this will be decided with no data or reference to cost or the real world. It’s a Democratic Congress, after all.

Oh. I’m not seeing those missing numbers, but these economists don’t like the bailout (h/t Andy Roth).

Rasmussen reports the number of respondents opposed to the bailout is growing.

even Newsweek calls him on it.

Obama’s Social Security Whopper

He tells Social Security recipients their money would now be in the stock market under McCain’s plan. False.

What happened to those hope-y change-y politics? Let’s see, Dukakis tried it, Mondale tried it, Gore tried it, Kerry tried it, and you can see how far it got them. Come on, Obama, get back to the changeyness! Try to come up with an original lie to scare the old folks, at least.

“The Palin Effect,” Noemie Emery, which is as much about Hillary and the Democrats as it is Palin and the Republicans. Sharp analysis.

If Obama wins, she gets to see her party in power, if that is her object. The problem is that the party is no longer hers. Or hers and her husband’s. If Obama wins, the Clintons become history. They also slip down considerably on the great grid of power: She is eclipsed by a president who defeated her, a first lady who hates her, a loquacious vice president with a large, lively family, and a legion of people who early on threw in their lots with Obama, and have prior claims upon him and his loyalty. She becomes in effect a footnote to history, remembered perhaps for her personal dramas, her historic run in the primaries no longer remarkable, but overshadowed by Sarah Palin’s run for vice president. Win or lose, Palin becomes the country’s most visible she-politician, culture phenomenon, as well as the best bet to succeed John McCain at the head of her party. Hillary is yesterday’s news, and has the rest of her life to brood on the mistakes that caused her to lose–very narrowly–the great prize she wanted and pursued, some will tell you, for the past 30 years.

This changes, however, if McCain wins. At once, she becomes the most important Democrat, the shipwreck survivor, the frontrunner for her party’s 2012 nomination; the road not taken; the one that, if followed, would have led to the outcome for which her party has struggled so long. For four long years, she will be saying “I told you so”–to the super-delegates who didn’t flock to her even when she won all those big primaries; to Obama, now back in the Senate, who didn’t name her when he had his big chance. A deflated Messiah, a wünderkind who couldn’t quite hack it, Obama would join Al Gore and John Kerry in the weary line of pitiful losers who tried and failed to match Bill Clinton’s success. Bill Clinton himself becomes the Big Dog again, the one shining light in the overall darkness, the only Democrat to be elected twice since Franklin D. Roosevelt, the most successful Democrat since the mid-1960s, when Lyndon Johnson’s luck, along with his party’s good fortune, ran out. (Granted, this is a fairly low bar to get over. But still.) If you were Hillary Clinton, which prospect would you find more appealing? Let’s guess.

[ . . . ]

The truth is that Hillary’s feminists were never the key to her primary victories. Her triumphs in the big states that were so impressive–Ohio and Texas, Pennsylvania, Kentucky, and West Virginia–were fueled by (Andrew) Jacksonian voters, in less elite venues, who found her the more conservative of the two Democrats; the least urban, the least elitist, the most likely to be strong and assertive in foreign affairs. These are not people for whom Roe v. Wade (either way) is a big voting issue. They are people for whom toughness is. They perceive, correctly, that each is a woman you would want to have on your wagon train if you were crossing the continent, and to them, each has the same gutsy, tough-woman vibe. It is not irrelevant that the places where the McCain people expect Palin to help most are the states in which Clinton managed to mop the floor with Obama, the states Obama offended with his “God and guns” ridicule. Clinton and Palin cannot afford to offend all of each other’s constituents, and perhaps they don’t want to.

And so, Hillary is missing in action from the Palin-hating brigade. She and McCain are said to be friends, and to work well together. In the primaries, she often compared Obama unfavorably to her friend in the Senate. Her comment that she and McCain had credentials in the national security area while Obama had a speech made four years ago has already appeared in McCain’s commercials, and it is hard to believe when she said it that she could not foresee this happening. It is also hard to believe that after she and Bill vote for McCain in the privacy of the voting booth up in Chappaqua, they will not be among the first to make phone calls to Sarah Palin, and then to John McCain.

And “How Liberal Trolls Are Working To Get McCain Elected President,” by DJ Drummond. The main thesis is interesting, but what I find fascinating (naturally) is that he has compared the polls to their crosstabs and noticed that the numbers don’t add up (the Firefox spelling checker doesn’t know crosstabs?)

So, put it all together, and in the past week Obama has stayed steady or lost support in every party identification group, yet Gallup says his overall support went up four points. And McCain stayed steady or went up in every party identification group, yet we are supposed to accept the claim that his overall support went down by four points? Anyone have an answer for how that is even possible?

He has the answer. Go read it. And California Conservative points out a few other incongruities in the polls.

I also am leery of polls this year. I may write about why when and if I can hunt down a believable hypothesis.

These are the questions they could use calculators on.

5. Write these percentages as decimals: 34% 52% 8%

6. Write these decimals as fractions: 0.5 0.03 0.95

7. Betty got 13 of the 20 questions correct in a biology test. What percentage did Betty get?

8. Gary ate 25% of a cake. What fraction of the cake did he eat?

Er no, contrast. ABC. Match the candidate to the action.

In response to an impending hurricane:

1. Postpones the beginning of the Convention and travels to the area where the hurricane will land.

     A. John McCain
     B. Barack Obama

2. Uses the hurricane as a springboard to attack his opponent 1,900 miles from the hurricane.

     A. John McCain
     B. Barack Obama

Do I really need to give you the answers?

Simulations 101

A simulation is a statistical model used to make informed predictions. Since simulations seem to have a mystical aura these days, due to all this climate stuff, it’s in everybody’s best interest to understand that no, they aren’t magic, and yes, they’re actually quite simple.

You need four things. You need data, from which you can create a model. You need simulated input data to feed the model, and that will produce simulated output data. Because the output are simulated, you need to run the simulation model repeatedly (these are called iterations), and the more iterations you run, the more reliable the output data are. Because you have multiple iterations and therefore, multiple output data, they are interpretated statistically to produce a single output.

Sound complicated? Well, simulations can be nightmarishly complex, but the general concept is actually pretty simple.

Let’s say you are a new freshman at Some State University, and your finances are tight. You did not purchase a parking sticker at registration because you are mathematically savvy and you wanted to determine whether it would be cheaper to buy the sticker or pay parking tickets. Let’s say you know that there is a 30% chance that if you park illegally in the lot outside your classroom building, you will get a ticket (we’ll ignore how you’d get that information). A parking sticker would cost $140 per semester, and each parking ticket would set you back $25. There are 15 weeks in the semester, and after looking at your schedule, you have determined that you would have to park in seven different lots every week (that’s 105 times a semester, and each time, you have a 30% chance of being ticketed).

The 30% chance of being ticketed is the probability you have extracted from the data (again, for the purposes of this, we’ll ignore how you got it). Let me show you how simple this is.

Imagine a roulette wheel with 100 pockets, numbered 1 through 100. Get a piece of paper and a pencil, and write Ticket Y and Ticket N on it. Spin the roulette wheel, toss the ball onto it, and wait for it to land in a pocket. If the number of the pocket is 1-30, make a hash mark under Ticket Y; otherwise, mark Ticket N. Now, because you are going to do this 105 times throughout the semester, repeat this process 105 times.

You have just completed one iteration of the simulation. The more iterations you do, the more reliable your results will be, so do 99 more iterations (by the way, do you see why these are known as Monte Carlo simulations?)

When you have finished all 100 iterations, average the Y and N hashes for all of the iterations (we do other things too, like look at the standard error and so forth, but that’s for another time). Now, multiply the average number under Y, multiply it by $25, and compare it to the cost of a parking sticker.

Using the roulette wheel is simulated input data. It isn’t real, because it’s not really parking in those lots. But it produces a random number, and since you know that the probability of getting a ticket is 0.3, you can determine, based on the simulated data, whether you get ticketed or not. So you can create a simulation model to determine whether it will be cheaper to buy a sticker or pay the parking tickets.

Okay, sure, you can look at the probability and the rest of the data and figure out that it’s going to be cheaper to buy the sticker. But that was merely a very simple model meant only to explain exactly what a simulation is. A simulation can be as complex as we need it to be. For example, weather affects the chance of being ticketed (meter maids don’t like being out in the rain and snow any more than you do). So if the weather is bad, the probability of being ticketed decreases. Again, as long as we know the probabilities, we can easily create a simulation. Also, lots are policed more at the beginnings of semesters (to catch the new students) and in the final two weeks (studying for and taking those final exams). If you have the probabilities, you can create the simulation. Staffing is tight, so lots are policed in shifts throughout the week, so the probability of being ticketed in a particular lot depends on the day of the week and the time. But again, as long as you have the data and can extract the probabilities, you can create the simulation.

This is what we call a manual simulation, where we use a raw probability to calculate the outcome, and for all but the simplest problems, isn’t very sophisticated. But we can use other software packages (the @Risk add-in for Excel, for example) which uses the distribution of past data instead of probabilites extracted from it to create highly sophisticated simulation models.

A simulation is only as good as the input data and the model. If you got, say, the probability of being ticketed wrong, your simulation output would give you an incorrect prediction. Likewise, if you set up your model wrong and got one of the calculations incorrect, you would get an incorrect prediction. Keep that in mind as you read about what this or that simulation predicts.

Just kidding. But other than pundits and blog readers, the US is just starting to discover who Sarah “Barracuda” Palin is. If you’re one, this is for you. If you know all about her, you’ll appreciate the links.

Note: These are just from yesterday. There are lots more already posted on the blogs I read since then, and the dextrosphere will continue to be all Sarah all the time for a while now.

Here they are, in no particular order (Firefox bookmarks capabilites are surprisingly — and annoyingly — limited):

• The Corner on National Review Online–steyn
• Pajamas Media » Why Palin Is a Fantastic Choice
• Pajamas Media » More to Sarah Palin Than Meets the Eye
• McCAIN CHOOSES | Daily Telegraph Tim Blair Blog
• The Campaign Spot on National Review Online
• Riehl World View: Wow
• The Corner on National Review Online
• The Corner on National Review Online
• 08/29 04:53 PM
• Campaign2008VictoryA: Palin-Clinton: The Battle Begins
• Bush: Palin ‘exciting,’ Boxer: ‘Dangerous’: The Swamp
• Bold and ballsy - Maggie’s Farm
• Palin Pick Causes ‘Elation’ among Evangelical Leaders - The Brody File: David Brody Blog - CBN News
• Jules Crittenden » Palin Comparison …
• The Club For Growth - http://www.clubforgrowth.org
• The Jawa Report: VPILF: Sarah Palin
• The Corner on National Review Online–levin
• Palin pick already yielding rhetorical dividends?
• Roger L. Simon » Palin: Obama, the stodgy; McCain, the maverick
• The Countertop Chronicles » McCain Palin 2008
• SayUncle » More Palin on guns
• What some Christian bloggers are (already) saying about Sarah Palin | blogs4God
• Hot Air » Blog Archive » Palin: Let’s shatter that glass ceiling that Hillary cracked; Update: Full video added
• Hot Air » Blog Archive » Pessimistic blogger feeling pessimistic about pick
• Hot Air » Blog Archive » Ferraro: Palin a “historic” choice for McCain
• Hot Air » Blog Archive » What Palin does for McCain, and to Obama
• Dirty Harry’s Place… » Want Proof Sarah Palin Is A Great Choice…?
• Centre Daily Local News - State College and Centre County | Centre Daily
• TOP 10 REASONS McCAIN’S VP SHOULD BE ALASKA GOVERNOR SARAH PALIN : Democracy Project
• Blue Crab Boulevard » Obama Campaign Slams Small Towns Again
• BLACKFIVE: Palin! Palin! Palin! -McCain’s Veep rises up.
• Reformers in Washington Applaud the Choice
• Palin Will Be Welcomed by Social and Economic Conservatives - Michael Barone (usnews.com)
• The Weekly Standard–fred
• Ace of Spades HQ–the children!
• From the Maas
• Ace of Spades HQ
• Sarah Palin - Wikipedia, the free encyclopedia
• Ace of Spades HQ
• Ace of Spades HQ–experience?
• Ace of Spades HQ–obama stealing palin
• Ace of Spades HQ–palin
• John McCain picks Alaska’s Gov. Sarah Palin as his VP | Top of the Ticket | Los Angeles Times
• Obama Shows Sexist, Overblown Reaction to Palin Pick
• Townhall.com::Blog
• Palin is GOP’s beacon - Pittsburgh Tribune-Review
• BeldarBlog: Would Alaska Gov. Sarah Palin be a home-run hit as McCain’s Veep?
• Draft Sarah Palin For Vice President: A weekend to remember.
• Daily Kos: State of the Nation
• Chris Kelly: Governor Palin Has What It Takes To Be The Next Dick Cheney - Politics on The Huffington Post
• The Corner on National Review Online
• VP Choice of the Day: Sarah Palin - McCain Now
• Alaska Governor Sarah Palin
• adn.com | politics : To slim down budget, Palin dumps chef

Class today, so while I’m gone, here are some “test your knowledge” quiz questions I have given my students. Each is easily answered with no more than simple descriptive statistics, but tests the student’s knowledge of the concepts (as opposed to whether the student can calculate an arithmetic mean or standard deviation). A student sitting in front of Excel, SPSS, or SAS should be able to answer these three questions in three minutes.

Answers when I get back this afternoon.

1. A tire manufacturer produces a particular model tire whose tread wear life is normally distributed with a mean of 39,000 miles and a standard deviation of 5,300 miles. The manufacturer wishes to provide a guaranteed tread life for this model which would be exceeded by 98% of all tires. What tread life would meet this requirement?
2. The mechanical process which fills 10-lb bags of dog food is subject to random fluctuations in the amount placed in each bag. The amount placed in each bag is approximately normally distributed with a mean of 170 ounces and a standard deviation of 4.3 ounces. Determine an interval centered on the mean such that the weight of the contents of 99% of the bags will fall within that interval.
3. The scores on an exam are approximately normally distributed with a mean of 75 and a standard deviation of 10. If the professor wants 10% of the class to receive As, then what is the minimum score a student can get and receive an A on the exam?

Answers.

All of these are critical value problems, where we calculate either the probability of a value being less than (or greater than) or equal to a critical value, or calculate the critical value based on the probability. The only difference between the first and second problem is that the first problem is one-tailed (we need to find the critical value for which all values will be greater than or equal to the top 98% tail) and the second problem is two-tailed (we need to find the critical value for which all all values will fall between the two tails).

1. A tire manufacturer produces a particular model tire whose tread wear life is normally distributed with a mean of 39,000 miles and a standard deviation of 5,300 miles. The manufacturer wishes to provide a guaranteed tread life for this model which would be exceeded by 98% of all tires. What tread life would meet this requirement?

The first step is always the same for every problem: Enter the data given, then the data we can deduce. We are given the mean, the standard deviation, and the probability of the area of the curve less than the critical value (98%).

Because we know that P(right area) is 98%, we can subtract it from 1 to get the P(left area).

We can use Excel’s NORMSINV function to take the probability and return a z-score, the critical value in standard deviations.

So the critical value is 2.05375 standard deviations below the mean. Now, let’s convert that to a number.

First, we multiply the z(left area) by the standard deviation. That converts the value to a number. Then, all we have to do is add the critical value to the mean to get the answer (we add because the number is negative).

Answer: A tread life of 28,115.14 miles would be exceeded by 98% of all tires.

This is really long, so it’s below the fold.

Ann Althouse has an update about Google shutting down Democrats’ anti-Obama blogs. According to her cited article:

On Monday, Google would not explicitly rebut the idea that it had been tricked but said that the cause of the temporary blockage appeared to be elsewhere. “It appears that our anti-spam filters caused some Blogger accounts to be blocked from creating new posts,” Google spokesman Adam Kovacevich said in a statement. “While we are still investigating, we believe this may have been caused by mass spam e-mails mentioning the ‘Just Say No Deal’ network of blogs, which in turn caused our system to classify the blog addresses mentioned in the e-mails as spam. We have restored posting rights to the affected blogs, and it is very important to us that Blogger remain a tool for political debate and free expression.”

That may or may not be BS (I suspect it is, but that’s not really where I’m going). But Ann adds:

But Kovacevich — unless he’s lying — revealed something about the technique. Google monitors email. (Sidenote: You might want to worry about how Google monitors email.)

Gmail has a spam filter, and in order for it to work, it has to scan incoming messages. I suppose you could call that “monitoring email,” but It seems a bit paranoid to me, especially given the huge volume of messages that Gmail handles. Sure, Google is sleazy, but there’s a difference between that and unworkable.

Then, Matt Johnston stumbles here:

Reports that Obama’s female staffers earn, on average, less than his male staffers is complete RUBBISH and should be viewed as nothing more than conservative rabble rousing, and that is hard for me to say as a conservative.

Essentially, what these reporters are doing is comparing female salaries to male salaries, without taking into account the job that is being done.

Uh, Matt, that’s the point. Obama is spouting the “women make 75 cents to every dollar men make!” myth in order to push his latest idiotic idea of “wage fairness.” That pseudo-statistic is exactly what you describe: take the total income females make, without respect to job, and compare it to the total income males make. See how that works? The bloggers are merely applying his own math-challenged methods to what he pays. Fair is fair.

rather, another reason why learning them (as opposed to mindlessly punching a calculator) is a good idea: On one of those house flipping shows, this woman was told that fifty gallons would be enough, so she bought fifty five-gallon buckets — thinking she was buying fifty gallons.

This is funny as hell. Roger Pielke (former director of the University of Colorado’s Center for Science and Technology Policy Research and an associate professor of environmental studies) enlists the help of an undergraduate to help him “understand” a climate change lunatic. A very grateful hat tip to Rich Horton for the link — I’m still laughing!

Over at Wm Briggs. And aren’t all police good Bayesians?

“The Democrats seem to have a living, breathing definition of the word filibuster.” And the pictures are, well, priceless.

Some nitwit academic at, IIRC, U Penn declared that nobody needs to know fractions. I beg to differ. Thursday, I had a conversation with two other people that illustrates real-world need to know fractions. The other two participants were a woman who has a full-time job downtown, and a volunteer who is in the office downtown two hours a week; the woman is more or less my age, and the volunteer is a young’un, in his late 20s, and (this makes it even more horrifying) a PhD student. The relevant portion of the conversation is at the end, and I really didn’t contribute (instead, I ducked out, truly frightened).

Parking is expensive. It costs me $33.75 every week just to park so I can work here.

Oh, I know! It costs me $6 a week to volunteer here!

Parking is 25 cents for 20 minutes. That’s 75 cents an hour. The PhD student parks for two hours a week downtown.

Are you coming in more than once a week now?

God, no, I don’t have the time! I’m trying to write my dissertation.

Where do you park that costs six bucks for two hours?

I have to put six quarters in the parking meter . . .

That was the point at which she — the secretary who is my age — and I looked at each other with terror in our eyes. Her glance was also a plea to jump in and say something, because she was speechless, so I did.

So are you putting dollar coins in the meter?

They make dollar coins?

Yes. It costs a dollar and a half to park for two hours in the garage.

No, it costs six dollars. I know.

It costs six quarters, not six dollars. Six quarters is a dollar and a half.

This stunned him, to judge from the look on his face. You could see the gears turning in his head. But he wasn’t done, oh no. It got even scarier.

So there are three quarters in a dollar?

No, there are four quarters in a dollar.

Then it costs six dollars to park for two hours.

That was the point at which I decided to bow out, and leave this idiot PhD student to the secretary (I really like her, by the way). So I said something about not wanting a parking ticket and left. I figure she could give him his lesson in first-grade mathematics.

Oh, I forgot. He’s working on a PhD in education. Surprised, anyone?

I finally solved the problem (without the help of MS customer support), and got MS Office 2007 installed on my new desktop (I installed Photoshop this morning, and I’ll install SPSS later today). Playing around with Excel reminded me of the reason I love Excel 2007, which in turn reminded me of, well, keep reading.

Months ago, I saw several dishonest statements on edublogs (sorry, it’s been months ago, and I have neither the time nor the inclination to find links at the moment), stating that zeroes are not valid scores. This is not dishonest in itself, but the threads upon which I commented were those in which the author claimed that zeroes were not statistically valid scores.

That is false.

Zeroes are often statistically invalid for calculating descriptive statistics for the class, such as means or standard deviations. But that does not imply that a zero is an invalid measure of a student’s performance. Note that I pointed this out in comments on these blogs, and got no reply. I assume, therefore, that the statements were made not out of ignorance, but dishonesty.

How does a student get a zero on an exam or assignment? Theoretically, a student might put his name in the exam, then get a zero because he didn’t know any of the answers, although the probability of this decreases as the number of questions increases (that is, it’s almost impossible on, say, a 100-question exam, but entirely possible on a 10-question quiz or assignment).

Assuming that each question has four distractors, and therefore that the probability of randomly getting any question correct is 0.25, the probability of getting a zero on a 10-question quiz is 0.056, or 5.6%; the probability of doing the same on a 100-question exam is 3.2072*10^-13, or 0.00000000000032, or 0.000000000032%.

A student could put his name on an exam, quiz, or assignment and answer nothing, that is, turn in a blank. But how frequently does this happen? How stupid can a student be to ensure a zero, when he could randomly answer, and get a somewhat higher score?

Or a student could not show up to take the exam or quiz, or not turn in the assignment. We’ll return to this scenario in a moment.

If you are calculating descriptive stats on a 100-question exam, and if you have zeroes from students who took the exam, but against all odds, managed to get zeroes (as I said, the probability of this is microscopically small), the zeroes are valid, and should be included in the calculation. Why? Because the students who got zeroes took the exam. Therefore, when calculating descriptive stats for the class, that is, answering the question, “How did the class do on the exam?” requires that you include the zeroes.

This, by the way, has never happened to me, in many years of teaching, grading, and calculating stats. The odds are far, far too small.

The same is true if instead of a 100-question exam, you are calculating class stats for a 10-point quiz or assignment (I have had this happen, quite often, because the probability of getting a zero is much, much higher).

To sum up: If you are calculating performance, and the measure of performance for some students is zero, those zeroes are statistically valid. Leaving them out will artificially inflate your class means.

But we have that other scenario, the one I said I’d address, where Johnny got a zero because he didn’t take the exam (or turn in the assignment). What about that?

Are you going to let Johnny make up the exam? If you are not, then his zero should be excluded from the scores when you calculate class stats, because he did not take the exam. Including his zero will artificially lower your class mean since he did not participate in taking the exam.

However, if you are going to let Johnny make up the exam, the question becomes when you let him make it up. If he takes the exam the day after the class took it, say, then include his score (whatever it may be) when you calculate class stats. But if you let him go a week or two, or worse, longer, before he makes up the exam, do not include his score when calculating class stats. By giving him all of that additional time, you make his score a different measure than those of the rest of the class. You cannot compare his performance on the exam to the performance of the rest of the class.

That leads us, of course, to the question of making up exams or accepting late assignments. This, I suspect, was the agenda of those edubloggers who falsely claimed that zeroes are not statistically valid scores, particularly since all were proponents of laissez-fâire grading policies.

If you work in the primary or secondary schools, your grading policy may very well be dictated from above, and you have no choice. But ignoring that, I hold that, at least in the secondary schools, such mushy gooey laissez-fâire grading policies are destructive.

Note that there are very good reasons for not showing up to take an exam, or not handing in an assignment on the due date. Grandmothers really do pass away. Students really do have religious holy days (well, at least some). It is only reasonable to allow students with valid reasons to make up exams or turn in late assignments. I refer here specifically to students who do not have valid reasons for showing up to take the exam (and “my alarm clock didn’t go off” is not a valid reason).

You teach students bad lessons that must be unlearned, with a great deal of pain for those students, when you let Johnny make up the exam. You teach Johnny that scheduling means nothing, that he may come and go as he likes, and do his work or not as he likes, without consequence. Johnny will not remember you kindly later in life when he fails his classes at the university, or is fired from his job because of the lesson you taught him.

Just as bad, perhaps worse, is that you teach the students who are responsible enough to have shown up for the exam that you have no regard or respect for them. You do not care that they are responsible and take education seriously, while Johnny does not. And if you’re sending that message, then you have no right to complain about students not taking education seriously, do you. You do not take it seriously, so why should they?

If you set your grading policies, and if you teach in the secondary schools or above, then there is no excuse for laissez-fâire grading policies, where you allow any student to turn in any assignment at any time he likes, unless none of your assignments is due on a specific date. You have no right to hold Johnny and the rest of the class to two different standards. It’s called fairness.

Of course, we always got a list of every possible religious holy day from every imaginable religion on the planet every semester, far too many to avoid scheduling exams or due dates on holy days. So we set a policy: If you cannot take an exam or turn in an assignment because of religious observance, tell your professor and make alternative arrangements before the date of the exam or due date, and we will happily accomodate you. Come afterwards and claim you couldn’t take the exam because you were at Good Friday services, and you get a zero. For “acts of God,” we only required documentation of some kind.

Still, I always got a few students who didn’t take the exam, and one or two who just disappeared, usually early in the semester, and didn’t drop the class. Those zeroes are invalid, and cannot be included in calculating class stats.

That leads us to Excel 2007. Because I always had zeroes that had to be exlcluded, I could never use the AVERAGE() function, and instead had to use SUM(range)/COUNTIF(range,”>0″). But Excel 2007 now has the AVERAGEIF() function, more than enough reason to upgrade. (Unfortunately, I don’t believe there is a STDEVIF() function.)

But back to zero scores. Yes, they are in many cases, valid scores. A zero is certainly a valid measure of how a student performed if he couldn’t be bothered to take the exam or do the assignment. In other words, a zero is a valid score for assessing that student’s performance. That student chose the zero when he didn’t take the exam. That it may not accurately reflect his knowledge is irrelevant, since by choosing not to take the exam, he made his knowledge irrelevant. Pandering to such irresponsibility undermines the educational mission, both with the irresponsible dolts and with the responsible students, and it undermines your creditiblity as an instructor.

By the way, there’s a rather entertaining article about multiple choice questions and probability here, if that sort of thing turns your crank. And if you’re curious, no, I have never used guessing penalties (you know, where you subtract a value from the score for each incorrect answer), but I did have an otherwise abominable professor in grad school who dealt with random guessing on tests by using paired T-F questions of the following format:

Statement A.
Statement B.

A. Both statements are true
B. Both statements are false
C. The first statement is true and the second statement is false
D. The first statement is false and the second statement is true

I thought it ingenious.

Jay Cost has an interesting mathematical model predicting the Pennsylvania primary.

I don’t expect most people to understand basic probability, not because I think most people are stupid, but because they haven’t been exposed to it. But every once in a while, I run across something that is so breathtakingly, well, stupid, that I really need to address it.

Enter one of those forensics reality shows, you know, where they interview the cops, judges, attorneys, and sometimes jury members, and walk you through the case. We have this woman who was married five times. Each of her husbands died in his early 30s of an apparent heart condition, with no medical history of heart trouble.

“Yeah, right,” you’re saying. That’s because you were born with the common sense gene. Apparently, the cops and townspeople in this case were not. Multiple people are on the camera, saying, “We just thought she was the world’s unluckiest person,” or some variation thereof.

So how “unlucky” would this woman be, to have had five perfectly healthy husbands in their early 30s suddenly drop dead of heart ailments?

First, if these deaths were, indeed, due to happenstance, then they are independent of one another (this will be important in a minute). Second, I went in search of what the probability one’s perfectly healthy spouse would suddenly drop dead of a heart-related condition, but I was not successful. So I’ll supply that.

Let’s say that the probability of your perfectly healthy spouse dropping dead of a heart ailment in his early 30s (p₁) is 1:100, or 0.01. I’m quite sure it’s far smaller — I doubt that one out of one hundred people have had their perfectly healthy spouse drop dead of a heart ailment in his early 30s — but 0.01 is a good, round number.

If these five deaths are independent of one another, as they must be, if they are accidental deaths, then the probability that all five deaths were accidental (p₂), and that she was, in fact, the “unluckiest person in the world” would be the probability of her perfectly healthy spouse dropping dead on the sidewalk of a heart ailment in his early 30s raised to the fifth power, that is:

p₁ = 0.01
p₂ = 0.01^5

or

0.0000000001

or

1:10,000,000,000

or

One out of ten trillion. “Microscopic” doesn’t adquately describe how low the odds are. And if the first probability (p₁) is lower, as I suspect it must be, say 1:1000, or 0.001 (surely no more than one out of a thousand people have their perfectly healthy spouse suddenly drop dead of a heart ailment in his early 30s), then the probability that she was the “unluckiest person in the world” drops exponentially to 0.000000000000001, or 1:1,000,000,000,000,000, or one out of one quintilliion.

These people should have been suspicious when her second husband dropped like a fly, but they weren’t, so why should she have stopped, given that she lived among such stupid people? And I should add that the only reason they started getting suspicious after her fifth husband dies for no apparent reason was that she had taken out a life insurance policy on him just two days before he dropped dead.

This is an edited repost, and it’s long. It’s a detailed discussion of complexity as it relates to problems, students, and teaching (not to mention writing those problems). The post is below the fold (I figured I’d spare the non-math geeks).