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This is not the most important election, 1860 was

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Every year we hear the same claim: that this the most important election of America’s history. This year is among the more contentious than most, but the issues dividing the candidates are few. Both, for example, claim they will protect the border and spur the economy. In lieu of issues, there’s name calling. Trump claims Harris is as incompetent buffoon and Harris claims Trump is a fascist dictator. The rancor practically guarantees as they’ll be riots whoever wins but, as these things go, the election is less important, and divisive than ’64 and ’68, and in particular, the election of 1860.

Following the 1860 election, election seven states ceded from the union and we had a Civil War. Even the most bleak prediction for 2024-25 is for a more peaceful transfer of power. The election of 1860 had two major issues on the ballot; one was slavery or rather the expansion of slavery to the territories, and the other was implementation of the Morrill tariffs. These import taxes, proposed by Justin Morrill and passed but not yet implemented, would have raised the average agricultural duty from 15% to to 47%. Duties on durable goods wool rise to 65%, with the burden falling disproportionately on the southern states. Duties on durable goods. There was also a price schedule that would have prevented British shippers from minimizing the effect by falsely claiming a price far below market, something China currently does. In September 1860, Republican Leader Thaddeus Stevens told a New York City audience that “the Tariff would impoverish the southern and western states, but that was essential for advancing national greatness and the prosperity of industrial workers.”

Matching the two sides to the two major issues of the day, there were four major candidates for president in 1860. All of them won states. Lincoln carried the greatest number, 18, and won the most electoral votes, 180. He was for high tariffs and against the expansion of slavery. Second was John Breckinridge, the Southern Democrat, who carried 11 states and got 72 electoral votes. He was for the expansion of slavery and against the higher tariffs. Then there was Stephen Douglas, the Northern Democrat, who was for allowing the expansion of slavery, considering it a “states right,” and also for the higher tariffs. Douglas carried only one state, Missouri, with 12 electoral votes. Finally, there was John Bell, the Constitutional Union candidate, who carried three states, Virginia, Tennessee, and Kentucky, representing 39 electoral votes. He opposed the expansion of slavery and also the increased tariff, but he generally believed that compromise was always possible. This was the worst vote split in US history. The worst split I’ve seen was 1968, when three candidates carried states.

Had either Bell or Douglas won, I suspect that the Civil War could have been avoided, at least temporarily. Virginia, the most important of the slave states, had shown it was willing to accept an anti-slavery president so long as he did not impose high tariffs — tariffs that benefitted the northern industrial workers and manufacturers at the expense of southern consumers and agricultural producers. Lincoln’s victory precipitated the immediate secession of 3 states. Another 4 seceded after inauguration.

The south imagined they could walk away because that’s how they read the constitution before the 13th amendment. They imagined they could win a civil war because they imagined they had British military support. “Cotton was king,” they claimed. The UK prime minister, Lord Palmerston, had told secretary Adams, “We do not like slavery, but we want cotton, and we dislike very much your Morrill tariff.” As it was, the British stayed on the sidelines, in part because of diplomacy. Besides, the gunship Monitor showed that the North could sink most any British ship that entered US waters.

As for 2024, I expect there will be riots whoever wins, but nothing more. The parties are realigning significantly, as happened in 1964-68, and neither side much understands the appeal of the other. This seems like less of a wrenching election than in 1964 and 1968, though. In ’64-’68 US boys were dying in Vietnam in numbers, and black folks and their white friends were being lynched in the south. Nothing like that is happening today. Today’s riots have been fueled by nothing more than name-calling, fear, and the occasional assassination attempt. Mild, even compared to 1968.

Robert Buxbaum, November 4, 2024. Justin Morrill is mostly remembered today for the Land-grant college act of 1862. This created an agricultural -technical college in each state. I taught at Michigan State University, Michigan’s land grant university. I’m generally a fan of tariffs, both as an aid to the domestic economy and as a tool of foreign policy. I present these views here. I got these views from Peter Cooper.

The electoral college favors small, big, and swing states, punishes Alabama and Massachusetts.

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As of this month, the District of Columbia has joined 15 states in a pact to would end the electoral college choice of president. These 15 include New York, California, and a growing list of solid-blue (Democratic party) states. They claim the electoral must go as it robed them of the presidency perhaps five times: 2016, 2000, 1888, 1876, and perhaps 1824. They would like to replace the electoral college by plurality of popular vote, as in Mexico and much of South America.

All the big blue states and some small blue states have joined a compact to end the electoral college. As of 2019, they are 70% of the way to achieving this.

As it happens, I had to speak on this topic in High School in New York. I for the merits of the old system beyond the obvious: that it’s historical and works. One merit I found, somewhat historical, is that It was part of a great compromise that allowed the US to form. Smaller states would not have joined the union without it, fearing that the federal government would ignore or plunder them without it. Remove the vote advantage that the electoral college provides them, and the small states might have the right to leave. Federal abuse of the rural provinces is seen, in my opinion in Canada, where the large liberal provinces of Ontario and Quebec plunder and ignore the prairie provinces of oil and mineral wealth.

Several of the founding federalists (Jay, Hamilton, Washington, Madison) noted that this sort of federal republic election might bind “the people” to the president more tightly than a plurality election. The voter, it was noted, might never meet the president nor visit Washington, nor even know all the issues, but he could was represented by an elector who he trusted, he would have more faith in the result. Locals would certainly know who the elector favored, but they would accept a change if he could justify it because of some new information or circumstance, if a candidate died, for example, or if the country was otherwise deadlocked, as in 1800 or 1824.

Historically speaking, most electors vote their states and with their previously stated (or sworn) declaration, but sometimes they switch. In, 2016 ten electors switched from their state’s choice. Sven were Democrats who voted against Hillary Clinton, and three were Republicans. Electors who do this are called either “faithless electors” or “Hamilton electors,” depending on whether they voted for you or against you. Hamilton had argued for electors who would “vote their conscience” in Federalist Paper No. 68.  One might say these electors threw away their shot, as Hamilton did not. Still, they showed that elector voting is not just symbolic.

Federalist theory aside, it seems to me that the current system empowers both large and small states inordinately, and swing states, while disempowering Alabama and Massachussetts. Change the system and might change the outcome in unexpected ways.

That the current system favors Rhode Island is obvious. RI has barely enough population for 1 congressman, and gets three electors. Alabama, with 7 congressmen, gets 9 electors. Rhode Islanders thus get 2.4 times the vote power of Alabamans.

It’s less obvious that Alabama and Massachussetts are disfavored compared to New Yorkers and Californians. But Alabama is solid red, while New York and California are only sort of blue. They are majority Democrat, with enough Republicans to have had Republican governors occasionally in recent history. Because the electoral college awards all of New York’s votes to the winner, a small number Democrat advantage controls many electors.

In 2016, of those who voted for major party candidates in New York, 53% voted for the Democrat, and 47% Republican. This slight difference, 6%, swung all of NY’s 27 electors to Ms Clinton. If a popular vote are to replace the electoral college, New York would only have the net effect of the 6% difference; that’s about 1 million net votes. By contrast, Alabama is about 1/3 the population of New York, but 75% Republican. Currently its impact is only 1/3 of New York’s despite having a net of 2.5 million more R voters. Without the college, Alabama would have 2.5 times the impact of NY. This impact might be balanced by Massachusetts, but at the very least candidates would campaign in these states– states that are currently ignored. Given how red and blue these states are, it is quite possible that the Republican will be more conservative than current, and the Democrat more liberal, and third party candidates would have a field day as is common in Mexico and South America.

Proposed division of California into three states, all Democrat-leaning. Supposedly this will increase the voting power of the state by providing 4 more electors and 4 more senators.

California has petitioned for a different change to the electoral system — one that should empower the Democrats and Californians, or so the theory goes. On the ballot in 2016 was bill that would divide California into three sub-states. Between them, California would have six senators and four more electors. The proposer of the bill claims that he engineered the division, shown at right, so skillful that all three parts would stay Democrat controlled. Some people are worried, though. California is not totally blue. Once you split the state, there is more than three times the chance that one sub-state will go red. If so, the state’s effect would be reduced by 2/3 in a close election. At the last moment of 2016 the resolution was removed from the ballot.

Turning now to voter turnout, it seems to me that a change in the electoral college would change this as well. Currently, about half of all voters stay home, perhaps because their state’s effect on the presidential choice is fore-ordained. Also, a lot of fringe candidates don’t try as they don’t see themselves winning 50+% of the electoral college. If you change how we elect the president we are sure find a new assortment of voters and a much wider assortment of candidates at the final gate, as in Mexico. Democrats seem to believe that more Democrats will show up, and that they’ll vote mainstream D, but I suspect otherwise. I can not even claim the alternatives will be more fair.

In terms of fairness, Marie de Condorcet showed that the plurality system will not be fair if there are more than two candidates. It will be more interesting though. If changes to the electoral college system comes up in your state, be sure to tell your congressperson what you think.

Robert Buxbaum, July 22, 2019.

Flat tax countries: Russia, Mongolia, Hungary

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For no obvious reason, many Republicans and some (few) Democrats are fans of the flat tax. That is a fixed percentage tax on every dollar earned with no deductions, or very few. They see the flat tax as better, or more fair, than the progressive, graduated tax found in the US and most industrial countries. While most Republicans don’t like high taxes, as in Sweden, France, or in the UK, the flat-taxers want a single tax rate: a constant percentage for all. A common version is what Ben Carson described earlier this month, “if you earn ten million dollars your tax will be one million; if you earn ten dollars, your tax will be one dollar.” Herman Caine (R) proposed something similar eight years ago, and (surprisingly) so did Jerry Brown (D).

Ben Carson proposes a 10% flat tax. I'm guessing his source is the Bible.

Ben Carson proposes a 10% flat tax.

As it happens, of the 230 nations on the planet, several already have a flat income tax, and none of them are industrial juggernauts. I will list the larger of these countries in order their tax rate: Mongolia and Kazakhstan, 10% flat tax and hardly any services; Russia and Bolivia, 13% flat tax: moribund, raw-material-based, police-states; Romania and Hungary 16%; Lithuania and Georgia 20%; Zambia 22%; Switzerland, 35% when you include the Cantonal and municipal flat rates, and (topping the list) Greenland at 45%. Not one of these is a productive, industrial powerhouse, like the US, and there is no indication that this will change any time soon.

I suspect that the flat tax enters the minds of conservatives from the Bible, from the 10% of grain that was given to the Levites (Numb. 18:24), and the second 10% eaten of pilgrimage festivals or given to the poor (Deut. 14:22-24). If that’s the source, let me suggest a better modern version is to give out cans of food, or to support ones church. But as a model for government finance, I’d suggest it’s best to leave more in the pockets of the poor, and tax more from the rich. Even in Biblical times, the government (king) levied a substantial tax above the 10%s described above.

A measure of tax rate is the percentage of the total GDP that goes to taxes. As things go, our tax rate isn't particularly high.

A measure of tax rate is the percentage of the total GDP that goes to taxes. As things go, our tax rate isn’t particularly high.

A flat tax does not necessarily imply a low tax, either. Greenland’s flat 45% rate is among the highest in the world, and Israel had a 50% flat tax until fairly recently. It’s also worth noting that personal income isn’t the only thing one can tax. Several countries combine moderate personal income rates with high corporate rates (Venezuela, Zambia, Argentina), or add on a high sales tax, or a transaction tax. Herman Caine’s 9-9-9 tax plan included a 9% transaction tax and a 9% federal sales tax that would have gone on top of whatever the state tax would have been. The revenue collected by the 9-9-9 plan would have been no less than we had, but would be, he claimed, simpler. Cain’s flat tax wasn’t even really flat either, as there was an exclusion, an income level below which you were taxed 0%. That is, he was really proposing a two tier system, with a 0% rate at the first tier. Rand Paul seems to favor something similar today.

The two advantages of a flat tax are simplicity, and that it reins in the tendency to tax the rich too much, a tendency found with many liberal alternatives. The maximum tax rate was 95% in England under Attlee. Their 95% tax-rate appears in the Beetles’s song, Mr Taxman: “…There’s one for you, nineteen for me; ‘Cause I’m the Taxman.” High rates like this caused the destruction of many UK businesses, and caused The Beetles’s to leave and reincorporate in the Cayman Islands. Bernie Sanders recently proposed a top rate that was nearly as high, 90%, and praised Denmark (60% maximum rate) for its high social services. Sorry to say, Denmark seems to have concluded that their 60% maximum was excessive, and earlier this year reduced their maximum to 47.794%. This is below the maximum US rate if you include New York state and city income taxes. History suggests that if you tax the rich at rates like this, they leave or do other socially unacceptable things, like go black-market. On the other hand, if you tax too little, there is no money for education or basic social services, e.g. for the desperately poor. At one point, I proposed the following version of graduated to negative scheme that manages to provide a floor, a non-excessive top rate, and manages to encourage work at every income level (I’m rather proud of it). And there are other key issues necessary for success, like respect for law, and not having excessive minimum wages or other excess regulations.

Bernie Sanders: tax the rich at 90% of income.

Bernie Sanders: tax the rich at 90%; I doubt this is a good idea.

Whatever the tax structure is, there is probably an optimal average rate and an optimal size for the government sector. I suspect ours is near optimal, but have no real reason to think so (probably just nativism). I’ve found that comparing the US tax rates to other countries’ is very difficult, too. Most countries have a substantial Value Added Tax (VAT), that is a tax applied to all purchases including labor, but we do not. Some countries have import taxes (Tariffs, I’m in favor of them), while we have hardly any. And many countries tax corporate profits (and sales) at rates above 60% (France taxes them at 66.6%). To make any sort-of comparison, I’ve divided the total tax income of several countries by the country’s GDP (I got my data here). This percent is shown in the chart above. The US looks pretty average, though a little on the low side for an industrial nation: just where I like to see it.

Robert E. Buxbaum, November 29, 2015. I imagine myself to be a centrist, since all of my opinions make sense to me. When I change my mind on something, I stay at the center, but the center moves. If this subject interests you,  seems to have dedicated his life to following the flat tax.

Marie de Condorcet and the tragedy of the GOP

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This is not Maire de Condorcet, it's his wife Sophie. Marie (less attractive) was executed by Robespierre for being a Republican.

Marie Jean is a man’s name. This is not he, but his wife, Sophie de Condorcet. Marie Jean was executed for being a Republican in Revolutionary France.

During the French Revolution, Marie Jean de Condorcet proposed a paradox with significant consequence for all elective democracies: It was far from clear, de Condorcet noted, that an election would choose the desired individual — the people’s choice — once three or more people could run. I’m sorry to say, this has played out often over the last century, usually to the detriment of the GOP, the US Republican party presidential choices.

The classic example of Condorcet’s paradox occurred in 1914. Two Republican candidates, William H. Taft and Theodore Roosevelt, faced off against a less-popular Democrat, Woodrow Wilson. Despite the electorate preferring either Republican to Wilson, the two Republicans split the GOP vote, and Wilson became president. It’s a tragedy, not because Wilson was a bad president, he wasn’t, but because the result was against the will of the people and entirely predictable given who was running (see my essay on tragedy and comedy).

The paradox appeared next fifty years later, in 1964. President, Democrat Lyndon B. Johnson (LBJ) was highly unpopular. The war in Vietnam was going poorly and our cities were in turmoil. Polls showed that Americans preferred any of several moderate Republicans over LBJ: Henry Cabot Lodge, Jr., George Romney, and Nelson Rockefeller. But no moderate could beat the others, and the GOP nominated its hard liner, Barry Goldwater. Barry was handily defeated by LBJ.

Then, in 1976; as before the incumbent, Gerald Ford, was disliked. Polls showed that Americans preferred Democrat Jimmy Carter over Ford, but preferred Ronald Regan over either. But Ford beat Reagan in the Republican primary, and the November election was as predictable as it was undesirable.

Voters prefer Bush to Clinton, and Clinton to Trump, but Republicans prefer Trump to Bush.

Voters prefer Bush to Clinton, and Clinton to Trump, but Republicans prefer Trump to Bush.

And now, in 2015, the GOP has Donald Trump as its leading candidate. Polls show that Trump would lose to Democrat Hillary Clinton in a 2 person election, but that America would elect any of several Republicans over Trump or Clinton. As before,  unless someone blinks, the GOP will pick Trump as their champion, and Trump will lose to Clinton in November.

At this point you might suppose that Condorcet’s paradox is only a problem when there are primaries. Sorry to say, this is not so. The problem shows up in all versions of elections, and in all versions of decision-making. Kenneth Arrow demonstrated that these unwelcome, undemocratic outcomes are unavoidable as long as there are more than two choices and you can’t pick “all of the above.” It’s one of the first great applications of high-level math to economics, and Arrow got the Nobel prize for it in 1972. A mathematical truth: elective democracy can never be structured to deliver the will of the people.

This problem also shows up in business situations, e.g. when a board of directors must choose a new location and there are 3 or more options, or when a board must choose to fund a few research projects out of many. As with presidential elections, the outcome always depends on the structure of the choice. It seems to me that some voting systems must be better than others — more immune to these problems, but I don’t know which is best, nor which are better than which. A thought I’ve had (that might be wrong) is that reelections and term limits help remove de Condorcet’s paradox by opening up the possibility of choosing “all of the above” over time. As a result, many applications of de Condorcet’s are wrong, I suspect. Terms and term-limits create a sort of rotating presidency, and that, within limits, seems to be a good thing.

Robert Buxbaum, September 20, 2015. I’ve analyzed the Iran deal, marriage vs a PhD, and (most importantly) mustaches in politics; Taft was the last of the mustached presidents. Roosevelt, the second to last.

An approach to teaching statistics to 8th graders

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There are two main obstacles students have to overcome to learn statistics: one mathematical one philosophical. The math is somewhat difficult, and will be new to a high schooler. What’s more, philosophically, it is rarely obvious what it means to discover a true pattern, or underlying cause. Nor is it obvious how to separate the general pattern from the random accident, the pattern from the variation. This philosophical confusion (cause and effect, essence and accident) is exists in the back of even in the greatest minds. Accepting and dealing with it is at the heart of the best research: seeing what is and is not captured in the formulas of the day. But it is a lot to ask of the young (or the old) who are trying to understand the statistical technique while at the same time trying to understand the subject of the statistical analysis, For young students, especially the good ones, the issue of general and specific will compound the difficulty of the experiment and of the math. Thus, I’ll try to teach statistics with a problem or two where the distinction between essential cause and random variation is uncommonly clear.

A good case to get around the philosophical issue is gambling with crooked dice. I show the class a pair of normal-looking dice and a caliper and demonstrate that the dice are not square; virtually every store-bought die is not square, so finding an uneven pair is easy. After checking my caliper, students will readily accept that these dice are crooked, and so someone who knows how it is crooked will have an unfair advantage. After enough throws, someone who knows the degree of crookedness will win more often than those who do not. Students will also accept that there is a degree of randomness in the throw, so that any pair of dice will look pretty fair if you don’t gable with them too long. I can then use statistics to see which faces show up most, and justify the whole study of statistics to deal with a world where the dice are loaded by God, and you don’t have a caliper, or any more-direct way of checking them. The underlying uneven-ness of the dice is the underlying pattern, the random part in this case is in the throw, and you want to use statistics to grasp them both.

Two important numbers to understand when trying to use statistics are the average and the standard deviation. For an honest pair of dice, you’d expect an average of 1/6 = 0.1667 for every number on the face. But throw a die a thousand times and you’ll find that hardly any of the faces show up at the average rate of 1/6. The average of all the averages will still be 1/6. We will call that grand average, 1/6 = x°-bar, and we will call the specific face average of the face Xi-bar. where i is one, two three, four, five, or six.

There is also a standard deviation — SD. This relates to how often do you expect one fact to turn up more than the next. SD = √SD2, and SD2 is defined by the following formula

SD2 = 1/n ∑(xi – x°-bar)2

Let’s pick some face of the dice, 3 say. I’ll give a value of 1 if we throw that number and 0 if we do not. For an honest pair of dice, x°-bar = 1/6, that is to say, 1 out of 6 throws will be land on the number 3, going us a value of 1, and the others won’t. In this situation, SD2 = 1/n ∑(xi – x°-bar)2 will equal 1/6 ( (1/6)2 + 5 (5/6)2 )= 1/6 (126/36) = 3.5/6 = .58333. Taking the square root, SD = 0.734. We now calculate the standard error. For honest dice, you expect that for every face, on average

SE = Xi-bar minus x°-bar = ± SD √(1/n).

By the time you’ve thrown 10,000 throws, √(1/n) = 1/100 and you expect an error on the order of 0.0073. This is to say that you expect to see each face show up between about 0.1740 and 0.1594. In point of fact, you will likely find that at least one face of your dice shows up a lot more often than this, or a lot less often. To the extent you see that, this is the extent that your dice is crooked. If you throw someone’s dice enough, you can find out how crooked they are, and you can then use this information to beat the house. That, more or less is the purpose of science, by the way: you want to beat the house — you want to live a life where you do better than you would by random chance.

As a less-mathematical way to look at the same thing — understanding statistics — I suggest we consider a crooked coin throw with only two outcomes, heads and tails. Not that I have a crooked coin, but your job as before is to figure out if the coin is crooked, and if so how crooked. This problem also appears in political polling before a major election: how do you figure out who will win between Mr Head and Ms Tail from a sampling of only a few voters. For an honest coin or an even election, on each throw, there is a 50-50 chance of head, or of Mr Head. If you do it twice, there is a 25% chance of two heads, a 25% chance of throwing two tails and a 50% chance of one of each. That’s because there are four possibilities and two ways of getting a Head and a Tail.

pascal's triangle

Pascal’s triangle

You can systematize this with a Pascal’s triangle, shown at left. Pascal’s triangle shows the various outcomes for a coin toss, and shows the ways they can be arrived at. Thus, for example, we see that, by the time you’ve thrown the coin 6 times, or polled 6 people, you’ve introduced 26 = 64 distinct outcomes, of which 20 (about 1/3) are the expected, even result: 3 heads and 3 tails. There is only 1 way to get all heads and one way to get all tails. While an honest coin is unlikely to come up all heads or tails after six throws, more often than not an honest coin will not come up with half heads. In the case above, 44 out of 64 possible outcomes describe situations with more heads than tales, or more tales than heads — with an honest coin.

Similarly, in a poll of an even election, the result will not likely come up even. This is something that confuses many political savants. The lack of an even result after relatively few throws (or phone calls) should not be used to convince us that the die is crooked, or the election has a clear winner. On the other hand there is only a 1/32 chance of getting all heads or all tails (2/64). If you call 6 people, and all claim to be for Mr Head, it is likely that Mr Head is the true favorite to a confidence of 3% = 1/32. In sports, it’s not uncommon for one side to win 6 out of 6 times. If that happens, it is a good possibility that there is a real underlying cause, e.g. that one team is really better than the other.

And now we get to how significant is significant. If you threw 4 heads and 2 tails out of 6 throws we can accept that this is not significant because there are 15 ways to get this outcome (or 30 if you also include 2 heads and 4 tail) and only 20 to get the even outcome of 3-3. But what about if you threw 5 heads and one tail? In that case the ratio is 6/20 and the odds of this being significant is better, similarly, if you called potential voters and found 5 Head supporters and 1 for Tail. What do you do? I would like to suggest you take the ratio as 12/20 — the ratio of both ways to get to this outcome to that of the greatest probability. Since 12/20 = 60%, you could say there is a 60% chance that this result is random, and a 40% chance of significance. What statisticians call this is “suggestive” at slightly over 1 standard deviation. A standard deviation, also known as σ (sigma) is a minimal standard of significance, it’s if the one tailed value is 1/2 of the most likely value. In this case, where 6 tosses come in as 5 and 1, we find the ratio to be 6/20. Since 6/20 is less than 1/2, we meet this, very minimal standard for “suggestive.” A more normative standard is when the value is 5%. Clearly 6/20 does not meet that standard, but 1/20 does; for you to conclude that the dice is likely fixed after only 6 throws, all 6 have to come up heads or tails.

From skdz. It's typical in science to say that <5% chances, p <.050 are significant. If things don't quite come out that way, you redo.

From xkcd. It’s typical in science to say that <5% chances, p< .05. If things don’t quite come out that way, you redo.

If you graph the possibilities from a large Poisson Triangle they will resemble a bell curve; in many real cases (not all) your experiential data variation will also resemble this bell curve. From a larger Poisson’s triange, or a large bell curve, you  will find that the 5% value occurs at about σ =2, that is at about twice the distance from the average as to where σ  = 1. Generally speaking, the number of observations you need is proportional to the square of the difference you are looking for. Thus, if you think there is a one-headed coin in use, it will only take 6 or seven observations; if you think the die is loaded by 10% it will take some 600 throws of that side to show it.

In many (most) experiments, you can not easily use the poisson triangle to get sigma, σ. Thus, for example, if you want to see if 8th graders are taller than 7th graders, you might measure the height of people in both classes and take an average of all the heights  but you might wonder what sigma is so you can tell if the difference is significant, or just random variation. The classic mathematical approach is to calculate sigma as the square root of the average of the square of the difference of the data from the average. Thus if the average is <h> = ∑h/N where h is the height of a student and N is the number of students, we can say that σ = √ (∑ (<h> – h)2/N). This formula is found in most books. Significance is either specified as 2 sigma, or some close variation. As convenient as this is, my preference is for this graphical version. It also show if the data is normal — an important consideration.

If you find the data is not normal, you may decide to break the data into sub-groups. E.g. if you look at heights of 7th and 8th graders and you find a lack of normal distribution, you may find you’re better off looking at the heights of the girls and boys separately. You can then compare those two subgroups to see if, perhaps, only the boys are still growing, or only the girls. One should not pick a hypothesis and then test it but collect the data first and let the data determine the analysis. This was the method of Sherlock Homes — a very worthwhile read.

Another good trick for statistics is to use a linear regression, If you are trying to show that music helps to improve concentration, try to see if more music improves it more, You want to find a linear relationship, or at lest a plausible curve relationship. Generally there is a relationship if (y – <y>)/(x-<x>) is 0.9 or so. A discredited study where the author did not use regressions, but should have, and did not report sub-groups, but should have, involved cancer and genetically modified foods. The author found cancer increased with one sub-group, and publicized that finding, but didn’t mention that cancer didn’t increase in nearby sub-groups of different doses, and decreased in a nearby sub-group. By not including the subgroups, and not doing a regression, the author mislead people for 2 years– perhaps out of a misguided attempt to help. Don’t do that.

Dr. Robert E. Buxbaum, June 5-7, 2015. Lack of trust in statistics, or of understanding of statistical formulas should not be taken as a sign of stupidity, or a symptom of ADHD. A fine book on the misuse of statistics and its pitfalls is called “How to Lie with Statistics.” Most of the examples come from advertising.