Big thank you this week to Annie Logue, a fellow freelance writer and one of my go-to folks for economic data. Check out her books and other writing.

Teacher salaries: what a big debate. Are teachers paid too little? Too much? Haven’t teachers’ unions bumped up teacher pay and benefits so that teachers are given a far greater piece of the pie than similar jobs in the private sector? Are teachers whining too much?

Like many numbers in the news, there are myths surrounding teacher pay. Talk to a weary teacher in an inner city school, and you might hear about working far more than 40 hours per week. Another teacher might tell you that he’s got a second job just to help make ends meet.

But many parents and politicians have a very different view. They say that teachers are well compensated, especially given their generous benefits packages and summers off.

There are a lot of caveats about this particular issue, so bear with me. It is challenging to generalize about teachers’ salaries, because like with many other professions they depend on a variety of factors: time on the job, geographic area, and perhaps most important here, the effectiveness (or even presence) of a teachers’ union. But the biggest caveat is this: comparing public sector and private sector jobs is generally not fair. It makes sense to compare teachers’ salaries from state to state or region to region. But what does it mean to compare teachers’ compensation with those of managers? What does it mean to be a manager? How are these categories developed?

And then there is the whole issue of teachers’ time “on the clock.” I’ll address that concern at the end of this post.

Still, I’m taking the risk to make these comparisons, even knowing that they’re not entirely fair–because they’re the only options we have. I’m going to assume (based on the salary ranges and the descriptions in the tables) that managers are college educated (or the equivalent) and have similar responsibilities as teachers. Teachers manage classrooms, budgets and schedules in similar ways that marketing managers manage staffs, budgets and schedules.

And yes, this is a compromise that is not fair. If you have a better idea, I would love to hear it. Honestly.

To help draw some clumsy conclusions, I’ve turned to the Bureau of Labor Statistics, a federal agency that keeps track of things like salaries and benefits packages. You can find all of my numbers here. Please do check them.

First off, how much do primary, secondary and special education teachers earn?

Total teacher compensation is $56.89 per hour worked. About 69% of that ($39.20) is wages and salaries, while 31% ($17.69) is total benefits.

So close to a third of a teacher’s compensation package is devoted to benefits. This is an important point, because the public teacher is one of the few workers in the country who still earns retirement and savings. Most pensions and retirement plans went away years ago. But let’s break down the teachers’ benefit package a little more.

Of the teacher’s benefit package, an average of $2.49 is spent on paid leave, $0.17 is spent on supplemental pay, $6.34 is spent on insurance, $5.85 is spent on retirement and savings and $2.84 is spent on legally required benefits.

All of those amounts are per hour, remember. So for every hour worked, a teacher earns, on average, $5.85 in retirement benefits.

What about managers in the private sector? Turns out that the differences are not all that stark.

Total management compensation is $58.28 per hour worked (full-time). About 68% of that ($39.59) is wages and salaries, while 32% ($18.70) is total benefits.

So again, the benefits packages for managers in the private sector are about a third of the total compensation packages. The per-hour compensation for private-sector managers and public school teachers are pretty darned close. Now we can dig into the compensation packages.

Of the private-sector manager’s benefit package, an average of $5.11 is spent on paid leave, $2.73 is spent on supplemental pay, $4.32 is spent on insurance, $2.78 is spent on retirement and savings and $3.75 is spent on legally required benefits.

The balance is really different here. Managers in the private sector earn more on paid leave and supplemental pay, but less in retirement and insurance. Still, the cost of the total benefit package is remarkably similar to that of teachers’.

Again a warning: it is not really fair to compare these two industries in this way. One of the big issues around this is whether or not the hourly rate has been figured correctly. Most teachers will probably say that their hourly rate is much less than $56.89, because they work far more hours than the schools tabulate.

However, research conducted by a variety of outlets reports that teachers work, on average, fewer hours than most professionals, including managers. In this round up of the research, NPR and StateImpact (a consortium of Ohio public radio stations) report that teachers work three hours fewer per week than other professionals. This research is in line with BLS foundational data for the numbers above.

So why is there such a different impression among teachers? Some teachers work two jobs. They tutor or coach a sport or put in time at the local diner. And of course, just because some teachers put in the minimum, it doesn’t mean that there aren’t many teachers out there who are putting in far more than a 40-hour work week.

And then there is new research from the Gates Foundation, which concluded that teachers put in 10 hours and 40 minutes a day, far more than the BLS estimates. (See page 13 of the linked report for the details.) If true, the hourly wages and salary totals for teachers are much, much too high. Then again, BLS could be underestimating for all industries and professions.

Clearly this debate is not easily defined or settled. We need better data about how much teachers actually work and better ways to compare the information we do have. In my mind, the discussion about teacher pay is far from over.

What are your thoughts on the data presented above? On the teacher salary debate? Do you have personal experience as a teacher or a private-sector manager? If so, please share your ideas in the comments section. 

Photo Credit: smkybear via Compfight cc

The post Numbers in the News: Teacher Salaries appeared first on Math for Grownups.

I grew up in a hunting community. On the first day of deer season, students could take an excused absence, as long as they had a note from home that stated they were out hunting.

My 15-year-old daughter enjoys target shooting. Her favorite gun is a semi-automatic.

But despite my background, I have never fired a gun. I admit freely that I don’t particularly like them. At the same time, I have respect for those who hunt and use guns for sport, like target practice. And I respect the professionals in law enforcement who carry weapons in order to keep us safe.

Those are the disclaimers, as I step carefully into this loaded topic (pardon the pun). As a journalist who writes about math, I believe that numbers can help us understand these complex and controversial topics. With that said, I want to break down some of the numbers that describe our country’s relationship with guns.

This is something that anyone who consumes news reports or reacts to current events should be able to do. It’s a grownup skill that requires a little bit of math and a lot of logic. Because of the controversy involved in gun ownership and shooting incidents, it’s a good idea to turn to the numbers. So here we go.

The Small Arms Survey estimates that there are approximately 270 million guns owned by civilians in the U.S.

No one can say for sure that this number is absolutely correct. That’s because there is no way to definitively count the number of civilian-owned guns. There are several reasons for this. First, gun registrations is managed on a state-by-state basis. In fact, eight states explicitly prohibit the registration of firearms (though many have exceptions that require, for example, registration of a firearm by someone who was convicted of a crime). Without registering guns, it’s difficult to count them. Second, notice of the sale or transfer of guns is not required by all states. And stolen or lost guns? These may or may not be reported either.

If we can’t count the number of guns owned by Americans, we certainly have difficulty counting the types of guns in legal circulation. And never mind the number of illegally possessed firearms.

Still 270 million is generally recognized as the best estimate. And no one can deny that this is a large number. This is when we start comparing ourselves to other countries. India comes in second place with 46 million civilian-owned guns. Then China (40 million), Germany (25 million) and Pakistan (18 million). And while these numbers may be interesting, they don’t tell the whole story. Math helps with that.

Take the U.S. and India for example. The U.S. has a lot of guns, and India has a lot of people. So it’s useful to consider the number of guns per 100 people. (Or it could be per person or per 10 people… whatever.) In the U.S. there are 88 firearms per 100 men, women and children. In India, there are 4. Yes, four. Because India’s population is so much larger than the U.S.’s, the rate of guns per 100 people is much lower.

This brings in another interesting comparison: the U.S. and Yemen. In Yemen, there are approximately 11.5 million civilian-owned guns, but there are 55 guns per person. Small population + lots of guns = high rate of guns per person. In fact, the U.S. has more guns per 100 people than any other country in the world. (Yemen comes in second place.)

Now none of this says that guns are good or bad. These are just numbers. But it does point to differences in how countries (and the people in them) think about guns. This is definitely something worth looking into. But for now, let’s consider another number.

In 2013, the CDC reported 33,636 firearm deaths in the U.S.

Again, that’s a big number. It includes all sorts of deaths, including homicides, suicides and accidental deaths. What is often overlooked is the extremely high rate for suicide by firearm. Nearly 2/3 of all firearms deaths were suicides in 2013–that’s 21,175. There were 11,208 homicides by firearms in 2013, just about half the number of suicides. That leaves 1,253 deaths that were accidental or unclassified firearm deaths.

It might also be helpful to break down this really big number.

In 2013, firearms were responsible for about 92 deaths per day. That’s nearly 4 people per hour.

Suddenly those numbers are a whole lot more striking. There aren’t even four people in my immediate family! It’s important to remember that this total number includes both suicides and homicides.

Yet when we enter into a discussion about gun control, many of us think about homicides — and specifically mass shootings. According to the FBI, there were 17 “active shooter incidents” in 2013. In those incidents, 44 people died and 42 people were injured. That’s a lot of people and a lot of scary moments. At the same time, compare those numbers with the number of suicides by firearms. That’s a big difference.

Guns were responsible for more than half of all suicides in 2013 in the U.S..

That year, 41,149 people committed suicide, and as stated above, 21,175 of them chose to end their lives using a firearm. So it seems that the best way to reduce suicides and the number of gun deaths is to reduce the number of suicides by guns. Two birds, one stone.

Certainly better mental health care is a good option. But there is something even easier. In Great Britain in the 1950s and 60s, about half of all suicides were by coal gas from ovens. But by the 1970s, coal gas was replaced by natural gas, which has far less carbon monoxide. The suicide rate by gas had dropped to zero (yes, zero) and the overall suicide rate had fallen by a third.

These are striking numbers that suggest that when the means of suicide is eliminated, the suicide rate drops. What would happen in this country, if suicidal folks did not have access to guns?

In my mind, this leads to a logical conclusion: gun control is a matter of public health. I don’t believe that we should rid our country of civilian-owned firearms or outlaw the possibility of owning guns. I do believe that to reduce the incidence of death by firearms, we can develop safety measures. These will likely include a variety of approaches–much like how we ensure that cars are as safe as possible.

How did I come to that conclusion? The numbers told me so.

What are your thoughts on the numbers behind the gun control debate?  Were any of these figures surprising to you? (NOTE: This is a very volatile subject, so I ask that you please keep your comments civil and on point.)

The post Numbers in the News: Guns appeared first on Math for Grownups.

Myth #5: Common Core is Overflowing with Fuzzy Math

First, a definition: fuzzy math is a derogatory term for an educational movement called reform math. Therefore the claim of fuzzy math isn’t so much a myth as an attempt to insult  the way that many math teachers and education researchers advocate teaching mathematics to K-12 students.

Second, some history: in 1989, the National Council of Teachers of Mathematics (disclaimer: I was once a member) published a document called Curriculum and Evaluation Standards for School Mathematics, which recommended a newish philosophy of math education. The group followed with Principles and Standards for School Mathematics in 2000. School officials and curriculum companies responded by implementing many of the approaches offered by the NCTM and as a result, the way we teach mathematics began to change. This change is what advocates call reform math and critics often call fuzzy math.

Before the NCTM’s publications, math teachers focused on the math — in particular series of steps (algorithms) designed to get the right answer to a problem or question. With reform math, educators became more focused on how students best learn mathematics. Suddenly, context and nuance and “why?” were at least as important as the answer. And it is true that Common Core Standards for Mathematics are largely based on the NCTM’s publications.

If this is truly fuzzy math, then we don’t have a myth here. (Although, to be fair, there is a legitimate branch of set theory and logic called “fuzzy mathematics.” But somehow, I don’t think Common Core critics using this term have real math in mind.) I include the fuzzy-math criticism as a myth because it suggests that teaching math in a conceptual way is a bad idea.

Throughout this series, I have asserted that the best way for students to understand and remember mathematical concepts is by returning over and over to the concepts behind the applications. Why is 24 such a flexible number? Because it has eight factors: 1, 2, 3, 4, 6, 8, 12 and 24. Students who really get this will have an easier time adding and subtracting fractions, reducing fractions, simplifying algebraic expressions and eventually solving algebraic equations through factoring.

This is numeracy, folks.

Students will not become numerate (think literate but with math) without a solid, conceptual understanding of mathematical ideas and properties. Numeracy does not typically evolve from memorizing multiplication tables or long division or pages and pages of practice problems. (Disclaimer: some kids will certainly become numerate regardless of how they’re being taught, but many, many others won’t.)

Numeracy is a life-long quest concentrated between the ages of five and 18 years old. Grownups can gain numeracy, but isn’t it better for our kids to enter into adulthood with this great understanding?

If Common Core critics want to call this whole philosophy “fuzzy math,” so be it. Just know that the ideas behind reform mathematics are deeply rooted in research about how kids learn math, not some ridiculous idea that was made up in the board rooms of a curriculum development company or smoke-filled political back rooms.

In short, the problems with Common Core math are not found in the standards themselves. Instead, the application and heated discourse are clouding Common Core’s real value and promise.

Got a question about the Common Core Standards for Mathematics? Please ask! Disagree with my assessment above? Share it! And if you missed Myth #1, Myth #2, Myth #3, Myth #4, you can find them here, here, here and here.

The post Common Core Common Sense: Myths About the Standards, Part 5 appeared first on Math for Grownups.

Myth #3: The Standards Introduce Algebra Too Late

One of the reasons for Common Core is to be sure that when students graduate from high school they are ready for college and/or the job market. And these days that means having some advanced math skills under their belts. But if you read the Common Core course headings, algebra is not mentioned until high school.

Up to this point, the math is referred to by the grade level, not subject(s) covered. So at first glance, this looks suspiciously like there is no mention of algebra in middle school. You have to dig a little deeper to learn that tough algebraic concepts are covered in the middle school standards. In fact, algebra is introduced (in an extremely conceptual way, with no mention of the word algebra) in kindergarten!

The Common Core math standards are divided into domains — or mathematical concepts. Here is the full list:

• Counting & Cardinality
• Operations & Algebraic Thinking
• Number & Operations in Base Ten
• Number & Operations — Fractions
• Measurement & Data
• Geometry
• Ratios & Proportional Relationships
• The Number System
• Expressions & Equations
• Functions
• Statistics & Probability

Of this list, you can find algebraic ideas and skills in at least four domains: Operations & Algebraic Thinking, Ratios & Proportional Relationships, Expressions & Equations and Functions. (You can argue that algebra appears in others as well.) In kindergarten, students are introduced to the idea of an equation, like this: 3 + 2 = 5. They also answer questions like this: What number can you add to 9 to get 10? (Algebraically speaking this question is x + 9 = 10, what is x?)

Variables aren’t introduced until much later, in 6th grade, when students are expected to “write, read, and evaluate expressions in which letters stand for numbers.” At this point, students begin to learn the language of algebra, with vocabulary words like coefficient (in the expression 3x, 3 is the coefficient) and term (in the expression 3x – 6, 3x and 6 are terms). Also in 6th grade, they start solving simple equations and inequalities, like 4 + x = 7 and 5x = 15.

In 8th grade, radicals and exponents are introduced, and students learn to solve simple equations with these operations. In addition, they graph lines and put equations into point-slope form and slope-intercept form, and begin solving systems of equations (pairs of equations with two variables). They also make connections between an equation of a line and the graph of a line. Finally, functions are introduced in 8th grade.

All of that happens well before high school, leaving lots of time in high school to delve into polynomials, quadratic equations and conic sections.

But here’s the most important thing: under Common Core, students are given a tremendous amount of context for all of this math, as well as time to develop true numeracy. This can speed along algebraic understanding. For example, students who are fluent with multiples and factors of whole numbers and decimals will have a much easier time learning how to solve equations by factoring. That’s because they will have the foundation of factoring or expanding. They will be able to use the distributive property with ease and focus their attention on the new concepts being presented.

In other words, this slow build develops numeracy.

So don’t let the Common Core headings fool you. Algebraic concepts and skills are meted out throughout the grade levels, allowing students to truly understand foundational concepts and fluently perform basic algebraic skills well before high school begins.

Got a question about the Common Core Standards for Mathematics? Please ask! Disagree with my assessment above? Share it! And if you missed Myth #1 or Myth #2, you can find the here and here.

The post Common Core Common Sense: Myths About the Standards, Part 3 appeared first on Math for Grownups.

Myth #2: The Standards Omit Basic Math Facts

While grabbing a latte at the local Starbucks a few weeks ago, I ran into a friend of mine. She was taking a break from teaching cursive to high school students at a nearby private school’s summer program.

“Kids don’t learn cursive in elementary school anymore, and so they can’t sign their names,” she explained. “Kids aren’t even required to learn their multiplication tables these days!” 

Well, I know for a fact that multiplication facts are covered in math classes across the country, including those in our fair city. But there’s this idea out there that third-graders are using calculators to find 8 x 2. While I don’t doubt that this has happened on at least one occasion, it’s not a trend in education. And math facts are a part of the Common Core.

The Common Core Standards emphasize critical thinking. And without a foundation in basic facts, students will not be able to apply critical thinking skills to problem solving of any kind.

Sure, there is no Common Core Standard that says students must be able to recite the multiplication tables 1 through 12 by heart. Instead, Common Core focuses on the concept of multiplication — which is pretty darned complex — encouraging teachers to illustrate multiplication with arrays (the picture below is an array), equal-sized groups, and area. The difference boils down to this: We grownups probably memorized that 8 x 2 = 16, while today’s students might figure it out on their own with a drawing like this:

• • • • • • • •

• • • • • • • •

The array above gives context to multiplication. Students can see for themselves that there are two rows of eight dots and 16 dots in all. The simple illustration even offers students a way to discover (or remember) the math fact themselves before memorization naturally occurs. In short, it’s much more meaningful than flash cards.

And while the example above is very visual, the idea behind it is flexible, allowing students with different learning styles to understand multiplication. A more kinetic (tactile) student can arrange 16 pennies in an array. A student with an aural learning style can count the dots out loud — in rows, in columns and in total. And so on.

There are plenty of other math facts included in the Common Core Standards, from the properties of number systems to formulas for area and volume. But I admit, you won’t find anything like, “Students will recite the value of π to the ninth decimal place.”

And this is a great change from more traditional approaches. Because, nothing sucks the life out of learning like memorization. Besides, can you remember the formula for the surface area of a cube? If not, could you figure it out or find it online? In my opinion, we want students to kick ass in the figuring-out option — to know that a cube has six sides that are exactly alike, and that surface area is figured when you add the area of each of the sides. Knowing those little details means that a formula isn’t necessary.

So yeah, Common Core hasn’t eliminated math facts. They’re just not front and center, leaving much more room for critical thinking. And that’s a good thing.

Got a question about the Common Core Standards for Mathematics? Please ask! Disagree with my assessment above? Share it! And if you missed Myth #1, you can find it here.

The post Common Core Common Sense: Myths About the Standards, Part 2 appeared first on Math for Grownups.

Myth #1: Common Core is a Curriculum

This is perhaps the most pervasive misunderstanding. In fact, the Common Core Standards are simply that: standards. In education-speak, this means they are statements of what students should know, upon completing a course or grade. Common Core does something a bit more than other sets of standards, giving a clear expectation of the depth of this understanding. Compare these fifth-grade math standards, one from Virginia’s Standards of Learning (SOL) and it’s corresponding objective from Common Core:

SOL: The student will describe the relationship found in a number pattern and express the relationship.

Common Core: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

The Common Core Standard isn’t just longer — it expresses much more depth. Students begin to pay attention to the relationships between numerical expressions, algebraic expressions and graphing. The goal is for students to know that these number patterns can be shown in a variety of different ways. And that’s a pretty big deal when students get into more complex algebra.

But here’s the thing: How students are taught is left completely to school districts and/or states. Some select ready-made curriculum, like Everyday Mathematics. Others opt to develop their own curriculum, which is exactly what my daughter’s middle school did.

Certainly, curriculum development companies have leapt on the opportunity to create new lessons, textbooks, activities and online components that correspond with Common Core. That’s capitalism at work in our country. (And it’s fed my bottom line quite well over the last three years. I’ve turned away more work this summer than I was able to accept.) There is nothing in the Common Core that dictates which curriculum must adopt, however. Localities still have control over that decision and process.

This is not to say that the Common Core hasn’t forced some pretty major changes in how mathematics is taught. Under these standards, students are encouraged to discover mathematical concepts, rather than be told how math works or should be understood. For traditionalists this could be a bad change. Yet, I believe that a discovery-based approach helps students conceptualize mathematics, which gives them a much better chance at developing strong numeracy than those who learn merely by rote. More on that in a later myth.

But regardless of what you think of the standards themselves, it’s important to know that they are merely a guideline for teachers and schools. Just like state educational standards — and each state has them — Common Core is merely outlining what the students should know, once they’ve mastered the material. Now how states and districts choose to measure students’ understanding of the standards is a different story — and a completely separate discussion of the standards themselves.

Got a question about the Common Core Standards for Mathematics? Please ask! Disagree with my assessment above? Share it!

The post Common Core Common Sense: Myths About the Standards, Part 1 appeared first on Math for Grownups.

When I was a boy and I would see scary things in the news, my mother would say to me, ‘Look for the helpers. You will always find people who are helping.’ – Fred Rogers

This is an amazing idea in the midst of the mayhem and terror that followed the explosions. There were dozens and dozens and dozens of people who ran toward the bomb sites, because that’s what they do – help those in need.

In the days that have followed, the FBI and others have been investigating the explosions, gathering information that will likely lead to an arrest and hopefully a conviction. Our natural question in these situations is, “Why?” Catching the person or people who did this will help us find that answer.

It shouldn’t surprise you to know that these investigators will depend on mathematics to help them solve this crime. From measuring the trajectory of the shrapnel to piecing together a timeline of events, math is a critical component in investigation.

A while back, I had the pleasure of interviewing Mary Ellen O’Toole, a former FBI profiler and author of Dangerous Instincts: How Gut Feelings Betray Us. She answered my questions about how she used math as a profiler. And I’m betting that this holds true for the investigation in Boston, as well.

Math at Work Monday: Mary Ellen the FBI profiler

Can you explain what you do for a living?

For half of my career, I worked in Quantico, at the FBI’s Behavioral Analysis Unit, the very unit that is the focus of the television show Criminal Minds. While there I tracked down, studied, and interviewed some of the world’s most infamous criminals, and I analyzed their crime scenes, too. These criminals included Gary Ridgeway (the Green River Killer), Ted Kaczynski (the Unabomber) and Derrick Todd Lee (the serial killer of Baton Rouge.) I worked everything from white-collar crime to work place and school violence to kidnappings to serial murder.

Since my retirement in 2009, I’ve worked as a consultant to law enforcement, corporate security, administrators, and many other professionals. I also teach at the Smithsonian, FBI Academy and many other locations.

When do you use basic math in your job?

As I and other profilers worked to solve a crime, we used every type of math from basic addition to geometry and pattern analysis to statistics and probability to reasoning and logic.

Read the rest of the interview.

If you’d like to share your wishes for the victims of the Boston Marathon bombing, please feel free to do so in the comments section. 

The post Boston Marathon: How FBI profilers use math appeared first on Math for Grownups.

Last Friday, I wrote about how the national polls really don’t matter. That’s because our presidential elections depend on the Electoral College. We certainly don’t want to see one candidate win the popular vote, while the other wins the Electoral College, but it’s those electoral votes that really matter.

Still, polls matter too. I know, I know. Statistics can be created to support *any* cause or person. And that’s true. (Mark Twain popularized the saying, “There are lies, damned lies, and statistics.”) But good statistics are good statistics. These results are only as reliable as the process that created them.

But what is that process? If it’s been a while since you took a stats course, here’s a quick refresher. You can put it to use tomorrow when the media uses exit polls to predict election and referendum results before the polls close.

Random Sampling

If I wanted to know how my neighbors were voting in this year’s election, I could simply ask each of them. But surveying the population of an entire state — or all of the more than 200 million eligible voters in the U.S. — is downright impossible. So political pollsters depend on a tried-and-true method of gathering reliable information: random sampling.

A random sample does give a good snapshot of a population — but it may seem a bit mysterious. There are two obvious parts: random and sample.

The amazing thing about a sample is this: when it’s done properly (and I’ll get to that in a minute) the sample does accurately represent the entire population. The most common analogy is the basic blood draw. I’ve got a wonky thyroid, so several times a year, I need to check to see that my medication is keeping me healthy, which is determined by a quick look at my blood. Does the phlebotomist take all of my blood? Nope. Just a sample is enough to make the diagnosis.

The same thing is true with population samples. And in fact, there’s a magic number that works well enough for most situations: 1,000. (This is probably the hardest thing to believe, but it’s true!) For the most part, researchers are happy with a 95% confidence interval and a ±3% margin of error. This means that the results can be trusted with 95% accuracy, but only outside ±3% of the results. (More on that later.) According to the math, to reach this confidence level, only 1,000 respondents are necessary.

So we’re looking at surveying at least 1,000 people, right? But it’s not good enough to go door-to-door in one neighborhood to find these people. The next important feature is randomness.

If you put your hand in a jar full of marbles and pull one marble out, you’ve randomly selected that marble. That’s the task that pollsters have when choosing people to respond to their questions. And it’s not as hard as you might think.

Let’s take exit polls on Election Day. These are short surveys conducted at the voting polls themselves. As people exit the polling place, pollsters stop certain voters to ask a series of questions. The answers to these questions can predict how the election will end up and what influenced voters to vote a certain way.

The enemy of good polling is homogeneity. If only senior citizens who live in wealthy areas of a state are polled, well, the results will not be reliable. But randomness irons all of this out.

First, the polling place must be random. Imagine writing down the locations of all of the polling places in your state on little strips of paper. Then put all of these papers into a bowl, reach in and choose one. That’s the basic process, though this is done with computer programs now.

Then the polling times must be well represented. If a pollster only surveys people who voted in the morning, the results could be skewed to people who vote on their way home from their night-shift or don’t work at all or who are early risers, right? So, care is made to survey people at all times of the day.

And finally, it’s important to randomly select people to interview. Most often, this can be done by simply approaching every third voter who exits the polling place (or every other voter or every fifth voter; you get my drift).

Questions

But the questions being asked — or I should say the ways in which the questions are asked — are at least as important. These should not be “leading questions,” or queries that might prompt a particular response. Here’s an example:

Same-sex marriage is threatening to undermine religious liberty in our country. How do you plan to vote on Question 6, which legalizes same-sex marriage in the state?

(It’s easier to write a leading question asking for intent rather than a leading exit poll.)

Questions must be worded so that they illicit the most reliable responses. When they are confused or leading, the results cannot be trusted. Simplicity is almost always the best policy here.

Interpreting the Data

It’s not enough to just collect information. No survey results are 100 percent reliable 100 percent of the time. In fact, there are “disclaimers” for every single survey result. First of all, there’s a confidence level, which is generally 95%. This means exactly what you might think: Based on the sample size, we can be 95 percent confident that the results are accurate. Specifically, a 95% confidence interval covers 95 percent of the normal (or bell-shaped) curve.

The larger the random sample, the greater the confidence level or interval. The smaller the sample, the smaller the confidence level or interval. And the same is true for the margin of error.

But why 95%? The answer has to do with standard deviation or how much variation (deviation) there is from the mean or average of the data. When the data is normalized (or follows the normal or bell curve), 95% is plus or minus two standard deviations from the mean.

This isn’t the same thing as the margin of error, which represents the range of possibly incorrect results.

Let’s say exit polls show that Governor Romney is leading President Obama in Ohio by 2.5 percentage points. If the margin of error is 3%, Romney’s lead is within the margin of error. And therefore, the results are really a statistical tie. However, if he’s leading by 8 percentage points, it’s more likely the results are showing a true majority.

Of course, all of that depends — heavily — on the sampling and questions. If either or both of those are suspect, it doesn’t matter what the polling shows. We cannot trust the numbers. Unfortunately, we often don’t know how the samples were created or the questions were asked. Reliable statistics will include that information somewhere. And of course, you should only trust stats from sources that you can trust.

Summary

In short, there are three critical numbers in the most reliable survey results:

• 1,000 (sample size)
• 95% (confidence interval or level)
• ±3% (margin of error)

Look for these in the exit polling you hear about tomorrow. Compare the exit polls with the actual election results. Which polls turned out to be most reliable?

I’m not a statistician, but in my math books, you’ll learn math that you can apply to your everyday lives and help you understand polls and other such things.

P.S. I hope every single one of my U.S. readers (who are registered voters) will participate in our democratic process. Please don’t throw away your right to elect the people who make decisions on your behalf. VOTE!

The post Exit Polling: A statistics refresher appeared first on Math for Grownups.

There’s no denying the math that goes on in elections. There are polls, ad buys, the number of minutes each candidate has spoken during debates — and yes, the electoral college. Whatever you may think of our dear map, it is how elections are decided in this country — for the most part.

There’s no reason to expect a repeat of Election Day (and the weeks following) 2000 this year. So I thought it would be a good idea to review the electoral map — from a mathematical perspective — so that we can better understand its power. First some history.

During the Constitutional Convention in 1787, the founding fathers quickly rejected a number of ways to select the country’s president: having Congress choose the president, having state legislatures choose and direct popular vote. The first two ideas were tossed based on fears of an imbalance of power — giving Congress or the states too much control. They also worried that a direct popular vote would be negatively influenced by the lack of consistent communication. In other words, without information about out-of-state candidates, voters would simply choose the candidate from their own states. And then there was the very real fear that a candidate without a sufficient majority would not be able to govern the entire nation.

So, these fine men drew up a fourth option: a College of Electors. The first design, which is outlined in Article II of the U.S. Constitution, was pitched after four Presidential elections, after political parties emerged. Much of the original system remained, but the 12th Amendment to the Constitution instituted a few changes to reflect the country’s new party system. Here what the electoral college looks like today:

• The Electoral College consists of 538 electors.
• Each state is allotted the same number of electors as it has Congress members (Senators and Representatives)
• Therefore, representation in the Electoral College is dependent on each state’s population. More populous states have more electoral votes; less populous states have fewer electoral votes.
• The 23rd Amendment to the Constitution gives the District of Columbia 3 electoral votes, event though it is not a state.
• Each state has its own laws governing how electors are selected. Generally, electors are selected by the political parties themselves.
• Most states have a “winner takes all” system, which means that the candidate with the majority of the direct popular votes in the state gets all of the electoral votes.
• However, Maine and Nebraska have a proportional system, which means the electoral votes can be divided between candidates.

Whew!

Some basic calculations allow the media and election officials and the candidates themselves to make really good predictions on election night in most situations. But the electors don’t officially cast their votes until the first Monday after the second Wednesday in December. Then, on January 6 of the following calendar year in a joint session of Congress, the electoral votes are counted, and the President and Vice-President are declared. (Got all that?) Almost always, though, the losing candidate concedes the election on election night or the next day, making the electoral vote and counting a mere formality.

The thing that makes this complex is that each state has a different number of electoral votes. In order to win the presidential election, a candidate must secure at least 270 electoral votes. And that’s why you’re probably seeing a red and blue (and purple?) map in your newspaper, on television and online.

In my state, there is no question which candidate will take all of the electoral votes. Maryland has been staunchly Democratic for several decades. And there’s no mystery about Texas, which is about as red as a state can get. But if it were a contest between Maryland’s and Texas’ electoral votes, Governor Romney would win. That’s because Texas has 38 electoral votes, while Maryland has 10.

Right now, there are lots and lots of predictions out there concerning how the electoral college will vote. (Personally, I think Nate Silver0 of the New York Times is the most reliable source. Dude has a killer math brain, correctly predicting the electoral college outcomes in 49 of the 50 states in the 2008 election. In that same election, he correctly predicted all of the 35 Senate races.) But there’s little doubt about many of the states. A few swing states will certainly claim this election: Colorado, Florida, Iowa, New Hampshire, Ohio, Virginia and Wisconsin. Mathematically speaking, we’re talking about 89 votes:

• Colorado: 9
• Florida: 29
• Iowa: 6
• New Hampshire: 4
• Ohio: 18
• Virginia: 13
• Wisconsin: 10

Now out of those, which states would you guess the candidates really want to win? Yep, the ones with the highest number of electoral votes. So to them, the most important states in these last days of the campaign are Florida, Ohio and Virginia. (Where do you draw the line? I chose more than 10 electoral votes.)

If you live in one of these three states, you are acutely aware of this fact. Unless you don’t have a television set or listen to the radio or have a (really) unlisted phone number.

So what does this mean? Right now, it means that President Obama is likely to win the election. There are scenarios that show the opposite outcome — and there are even a few that produce a tie. However, most political analysis says that it’s Obama’s to lose at this point. This is despite the fact that most polls show the popular vote at a statistical dead heat (in other words, any lead by either candidate is well within the margins of error).

Because our founding fathers made a decision that we wouldn’t elect our presidents with a direct popular vote. What matters in these last days are the popular votes in the swing states — most importantly Florida, Ohio and Virginia — though there are scenarios that give Mitt Romney the edge without winning all of the swing states.

If you are a complete geek about election numbers, do visit Silver’s FiveThirtyEight blog at the New York Times. His math is good, regardless of what some conservative pundits have claimed in recent weeks.

EDITOR’S UPDATE: Sam Wang of the Princeton Election Consortium also has great analysis. Hurricane Sandy has messed with his servers, so the site looks pretty rudimentary, but he is updating his site regularly. It’s pretty cool to compare Silver’s and Wang’s conclusions — especially on a day-by-day basis.

I also highly recommend a really slick interactive tool put out by the New York Times. It graphically illustrates ways in which the electoral votes could swing the election in either way, based solely on the math. Unlike Silver’s blog, this section does not offer a prediction of who will will win, but describes the various scenarios for each candidate.

Whatever you think of the candidates and the issues, vote. No matter what, vote. Our votes — even outside swing states — matter. It’s our responsibility as U.S. citizens to declare our preferences. And in my mind, if you don’t vote, you can’t complain.

Coming on Monday… a look at the polls themselves. What makes a good poll? How should we average folks interpret polls? Can they really tell us what’s going on?

What are your thoughts on the math of the electoral college? (I get it. These discussions can get heated. Please be respectful in your comments. I will not approve or will delete any comments that I deem outside the bounds of civility. Thank you for playing nice.)

The post Why National Polls Don’t Matter: Electoral college math appeared first on Math for Grownups.

So don’t even tell me what the heat index is. I really don’t want to know. But I have always been fascinated with how it is calculated. What are the variables that affect the heat index? Let’s take a look.

The heat index is how it really feels when the humidity is figured in. (Those of you who live in a climate with dry heat have no clue about this. Count yourselves lucky.) When the humidity is high, the heat index goes up, producing a hot, sticky mess that makes my hair frizzy and sours my otherwise lovely temperament.

The thermometer may say 95 degrees Fahrenheit, but if there’s significant humidity, it might feel like it’s 105. But of course meteorologists don’t guess at this number. There’s an actual formula that’s used to find the heat index.

Before we get to that, let’s consider the variables involved. According to the National Oceanic and Atmospheric Administration (NOAA), there are 20 (yes, twenty) variables that are used to calculate the heat index. These range from vapor pressure to the dimensions of a human to ventilation rate to sweating rate (ew). Because most of these are very specific to each person, a mathematical model was used to determine an appropriate range for each. This allows meteorologists to use a (relatively) simple formula for finding the heat index:

HI = -42.379 + 2.04901523T + 10.14333127R – 0.22475541TR – 6.83783(10^-3T²) – 5.481717(10^-2R²) + 1.22874(10^-3T²R) + 8.5282(10^-2TR²) – 1.99(10^-6T²R²)

Pretty, right? It’s actually not that hard to understand, if you break down the pieces. First, let’s define the variables.

HI = heat index

T = ambient dry bulb temperature (in Fahrenheit)

R = relative humidity (integer percentage)

So there are basically three variables, one being what we are looking for — the heat index. If you were to use this formula, you would need to know two things: the ambient dry bulb temperature (which is merely the ambient temperature as measured by a thermometer) and the relative humidity.

If you put to work the logical part of your brain that notices connections and patterns (yes, you do have one), the math becomes clear. When the temperature and relative humidity go up, so does the heat index. How do you know that? Look at the equation. It’s full of addition and multiplication. In fact, aside from the negative exponents (which actually yield smaller numbers), the equation is based solely on increasing values.

(That is, unless you have negative values for T and R. But in that case, you wouldn’t be figuring the heat index, right? A negative T means a negative air temperature, which is really cold in Fahrenheit. And I’m not sure that relative humidity can be negative at all.)

Now, almost nothing is absolute in weather prediction and measurement, right? And this equation is no exception. As NOAA points out, this equation is created by multiple regression analysis, which means it is not exact. (Basically, in this process, the mathematicians are fitting points to the closest line. Think of a bunch of points on a graph and how you can draw a predictable line or curve that is closest to all of those points.) There is in fact an error of ±1.3 degrees Fahrenheit. But what’s 1.3 degrees when you’re looking at a heat index of 102? Either way, it’s still darned hot.

How do you manage the heat? Do you head inside or hide in a cool, dark place? Share your ideas in the comments section.

The post How Hot Is It? Calculating the heat index appeared first on Math for Grownups.