Goodness me!!

We all know someone who seems to be an especially helpful person and someone who seems to be particularly selfish and unhelpful. In other words we recognize that people vary in traits and influences that result in stable differences in how helpful they are. There are likely to be many influences and traits so by the Central Limit Theorem altruism should be normally distributed. Even if it isn’t we can construct a scale that is.

I found 11 altruism questions in the General Social Survey (GSS) that asked how often respondents had done the following in the last year
- let someone ahead of them in a queue
- give someone directions
- give money to charity
- give money to a homeless person
- help a neighbor when they are away e.g. feed pets
- volunteer to spend some time helping a charity
- give up a seat for someone on a bus or train
- helped carry things for someone e.g. groceries
- loaned a personal item to someone
- returned money when given too much change
- donated blood.

I found that if people did any one of these they also tended to do the others, showing that these aren’t simply random acts of kindness but instances of a general factor of altruism. I formed a scale from these items. If the person didn’t do something they were assigned a score of 0 for that item, a score of 1if they did it only once and a score of 2 if they did it at least twice. Then I added up the scores across all 11 items yielding an altruism scale from 0 to 22. The reliability of the scale is >0.67 which means that over 2/3rds of the variation in the sum is due to variation in factors that each item has in common with others. The average score was right in the middle and there was little bunching of scores at the low or high end so very few people lie outside its measuring range. All in all the scale measures what it was meant to measure reasonably accurately, and does so for all but the most extreme people.

It occurred to me to see what kind of people were more or less altruistic. I found some surprises. For a start it turns out that some things we would expect to matter don’t. For example religion doesn’t matter. It doesn’t matter if you believe in God or not, whether you are a Protestant, Catholic or Jew or how often you pray. It doesn’t matter if you are a religious fundamentalist or liberal, either now or at the age of 16. It’s disappointing not to see a connection given that altruism is probably the central moral requirement of Jesus’ teaching and that the early Christians distinguished themselves by their generosity and helpfulness. It does however matter if you are an active participant in a church group or are a minister, priest or nun. So it is specifically religiosity and theology that doesn’t matter.

Although those who take the trouble to vote are more altruistic than those who don’t political orientation doesn’t matter either. There is no difference between liberals and conservatives or Republicans and Democrats. Neither does it matter if the person has ever cheated on their spouse or seriously violated traffic laws. Clearly being good, in the sense of being helpful, is not necessarily related to fidelity or being law abiding. Finally altruism is not a gender thing - there was no difference between men and women.

There are however large differences by education, class, income and age, and none of these influences account for any other. The more educated and the higher the income and social class the more altruistic the behavior. As intelligence is positively related to all of those intelligence probably plays a role in altruism. Indeed William James (in his The Varieties of Religious Experience) makes the point that when the saintly impulse occurs with a feeble intellect the result is either a loss of all practical interests because contemplating God takes all their mental resources, or it results in cruel fanaticism because the limited mind settles on superficial ideas of what it means to honor God. To get beyond this and tap the saintly impulse to produce exceptional altruism requires surplus mental resources and intellectual depth. The youth turn out to be more altruistic than their elders. The most altruistic group is young high income people with graduate degrees within the middle to upper classes who are active in a church group. It isn’t as though the effect is just through being able to afford more charitable giving either - it extends to behaviors that aren’t about money, like giving up a seat or donating blood. The least altruistic group is the opposite.

Then I took an interest in the opinions of the altruistic. I found that they were more likely than the selfish to be in favor of abortion, to be for the legalization of marijuana, to be opposed to capital punishment for murder and to not consider extramarital or homosexual sex to be always wrong. These are socially liberal opinions. On the other hand the altruistic are also more inclined to think that government should be doing less and not more. The political philosophy that encompasses this combination of social liberalism with economic conservatism is libertarianism which funnily enough is popularly regarded as the least altruistic political ideology.

Curiously men who have paid for sex, and women who have had sex for money, are more altruistic than those who haven’t. One would have thought that women who don’t charge for sex would be more altruistic by definition. Once again we see that people can be sexually sinful and yet have saintly tendencies.

Although this scale covers most people I wanted to look at more extreme examples of altruistic behavior. Firstly I looked at saints and those who had been beatified. The proportion of Catholics worldwide who achieved either of these distinctions is 1 in 7.4 million and 1 in 2.8 million respectively. Then I looked at the probability of winning a Nobel Peace prize (I used the highest probability across countries). There is at most a 1 in 800 thousand chance. Finally I looked at UK Conscientious Objectors during WW I. My reasoning is that this kind of pacifism in the face of a lengthy prison term or even death by firing squad is an especially courageous example of compassion. 1 in 384 men conscientiously objected, and 1 in 1023 men were jailed for it.

In the graph below I have placed scale scores, and particular items in the scale, where they would be on a normal curve. Less than 2 is equivalent to mental retardation on an IQ scale so I suggest that anyone scoring that low would be an altruistic moron. Considering that they would have to be so unhelpful that even giving directions to someone is too much trouble I think the label fits. The bottom 10% score 5 or less. This is less than one of the less demanding helpful behaviors every 2 months. The average person manages just below one of each of the altruistic behaviors per month. The top 10% manages close to 1.5 altruistic acts per month. Engaging in all the altruistic acts more than once in a year i.e. 2 per month, puts a person at the same degree of rarity as a Nobel Laureate in science would be on an IQ scale. In other words it takes only one act of altruistic kindness every two weeks to be an altruistic genius. This includes one more demanding altruistic activity every 2-3 months.



How does that compare to the more saintly types? Well it seems that being saintly is far more demanding than being a genius. I placed the Conscientious Objectors, Nobel Peace Prize winners, the beatified and Saints on the same graph. As you can see an Objector is roughly on par with an altruistic genius, being an Objector in the face of jail is more demanding, and that the rest are virtually off the scale. Let me try to give you some sense of the relative degree of altruism involved. Giving blood is a bigger deal than giving money to charity and being a Conscientious Objector is a bigger deal than donating blood. If each of these is a step up the altruistic mountain then from Conscientious Objection to Nobel Peace Prize would be a step of the same size i.e. there is similar increase in the demands between a Nobel Peace Prize and Conscientious Objection as there is between Conscientious Objection and blood donation. Put in another way – take the increase between donation to charity and blood donation and multiply that by two more. Similarly the Saint is as far above the regular blood donor as the regular blood donor is above the altruistic moron.

Mother Theresa’s formula for goodness was daily meditation and acts of altruism. My formula to become a Saint (whether or not you are in fact ever beatified or canonized) tells you how many altruistic acts you need. You need to maintain an average of at least 2.6 acts of altruism per month (aim for at least one per week to be safe) and many of those should be demanding i.e. involve a fair amount of inconvenience or self sacrifice. It will help a lot if you become an active member of some group (like a church) that busies itself in altruism and charity work. It would also help if you did some loving kindness meditation regularly or contemplated the lives of Jesus, altruistic saints or Peace Prize winners. Oh and it would be best not to try to be a Saint if you are feeble minded as you are likely to do more harm than good.

The Price of Sex

You’ve heard it said that every man has his price, and no doubt you’ve inferred that so does every woman. It’s easy enough to do research on what the going rate on sex for pay is but that only tells you the prices of those who are currently selling or buying sex. It tells you nothing of the price at which it would be worthwhile for someone outside the market to enter it, and that is after all what we want to know when we talk about everyone having their price. What follows is my attempt to spell out the price distributions of all adult men and women respectively – not just professionals.

The propensity to pay for sex, or have sex for money, must be determined by a large number of relatively independent factors e.g. interest in sex, attractiveness, need for money, availability of alternatives, price, moral qualms, fear of disease, opportunity, the illegality, forgone marriage opportunities, low workload, etc. If that is the case then according to the Central Limit Theorem the disposition will be normally distributed i.e. will fit the famous bell curve. So I can express this behavioral disposition as a trait that is normally distributed.

If there are several groups – like men and women in this case – and the distribution of one group is used as a standard, then the others can be specified in the terms of that standard. The beauty of this approach is that it models supply and demand together i.e. it looks at prices in a situation where women require a certain price knowing what men are willing to pay and men are willing to pay knowing what women require. It doesn’t just ask people what they would charge or pay in isolation.

All we need is stats on the proportion of each group exceeding two or more objective thresholds on prostitution or solicitation. In this case I used the General Social Survey variables - Paidsex and Evpaidsx. These ask if a respondent has paid for (or has been paid for) sex within the last year, and ever since the age of 18, respectively. The first is a higher threshold than the second because it is a subset of it. I also used stats in Steven Levitt and Stephen Dubner’s chapter on prostitution in their book SuperFreakanomics and some prices mentioned in the Governor Eliot Spitzer scandal.

It turns out that 3.4% of men and 0.5% of women exchanged money and sex in the last year, and 1.3% of women and 14.4% of men had done so since turning 18. So from this GSS data I calculated the following normal distributions

Female disposition to have sex for money 0±1
Male disposition to pay for sex -1.739±0.4585

where the female distribution is set as the standard and lower numbers mean a higher propensity. [I give the technical details in the appendix below.]

In other words the average man is more likely to pay for sex than the average woman is to have sex for money, and men are more alike than women when it comes to swapping sex and money. So far there are no surprises.

After mapping these distributions onto prices I put it altogether on the graph you see below. The average woman’s price is therefore $3972.62 but the average man is only willing to pay about one tenth of that - $396.96. Only one in 13423 men is willing to pay the average woman’s price. On the other hand as many as one in 24.3 women are willing to have sex at a price the average man is willing to pay.



You can also see that the asking price at the far right of the graph is $0.72 million. One in 30000 women would require that price. One in a million women would require at least $2 million. This is the roughly the estimated value of life in the US, so one could say that there are women out there that would rather die than have sex for money. On the other hand it’s equally fair to say there are women out there who would pay men for sexual favors. Still most women do price themselves out of the market.

Then there are men. Virtually all men would pay for sex if he had no alternative and his price was accepted. In fact most are willing to pay more than the going rate for street walkers and maybe a third would be willing to pay a high class hooker’s rate of $500.

Where do you think you are on the distribution? If you aren’t in the US then remember to correct for purchasing power parity when translating the prices in the graph.


Appendix

In order to use this information to estimate the distribution of propensity pay for sex, or have sex for money, all we need to do is convert these percentages into normal distribution standard scores – the so called z-scores. All these scores are is the number of standard deviations away from the average one has to be to get that percentage. For women those percentages convert to z-scores of 2.5758 and 2.2262 respectively. In other words a woman who has had sex for money within the last year is at a z-score of 2.5758 on the “propensity to have sex for money” distribution. For men the z-scores are 1.825 and 1.0625 respectively. Now the rest is simple arithmetic.

I took the woman’s distribution as the standard one and set the average at 0 and the standard deviation at 1. You will notice that the difference between the two threshold
z-scores for women (number of standard deviations from the average) is
2.5758-2.2262=0.3496. The difference between the z-scores for the men was 0.7625. Since these differences apply to the same thresholds one can see that the male standard deviation must be smaller than that of the women because more standard deviations fit between the thresholds. The male standard deviation must be 0.3496/0.7625 = 0.4585 or 45.85% the size of the female standard deviation. Now if the female standard deviation was set at 1 then the male standard deviation is 0.4585.

Given that “last year” threshold for men was 1.825 standard deviations away from the average it must be 1.825*0.4585=0.8368 away from the average on the female standard and since this must correspond to the 2.5758 z-score for women on the same standard distribution the male average must be 2.5758-0.8368=1.739 standard deviations away from the female average.

So now we have the female distribution set at 0±1 the male distribution on the same scale is 1.739±0.4585. It should really be expressed as the reverse of this because we want price to go from low to high as we go from left to right so I’ll rephrase the above as

Female disposition to have sex for money 0±1
Male disposition to pay for sex -1.739±0.4585

To convert these to money values I needed two more anchors – two prices for sex that I can link to particular z-scores. From SuperFreakanomics I found that 1 in 3300 women in Chicago are streetwalkers charging $42 on average. This proportion of women corresponds to a z-score of -3.4289. That was one anchor.

Another anchor is afforded by the Governor Eliot Spitzer scandal. He paid as much as $5000 a time for a hooker. The same agency charges up to $6000 a time. I’m taking this as the top of the market because it was the top price of the most expensive agency. Assuming the girl in question has a dozen or so clients this implies that one in 255000 men are willing to pay this price. This is a z-score of 4.4706 on the male distribution, or 0.3108 on the female distribution.

Finally it’s quite usual for values to map linearly onto the log of prices rather than prices themselves so I mapped the z-scores onto $ prices via logs.

Responsible Democracy

South Africa has just had a local election and the US is already starting to gear up for the 2012 Presidential election. Almost everyone who has something to say about it will tell it’s very important to vote. About 80% of the US population considers it a very important obligation for a citizen to vote but here’s the thing – of those who consider it so import only 57% think it’s also very important to be informed about matters relevant to your vote. It’s no surprise then that voters are quite astonishingly ignorant.

Some examples of the degree of ignorance – in 2000 26% thought Al Gore was a Conservative and 23% thought George Bush was a Liberal; 40% and 60% respectively don’t know that defense spending and social security are the top two budget items; only 50% knew which Party controlled the Senate before the election; barely 35% could identify British Prime Minister as the post held by Tony Blair, and 58% knew little or nothing of the Patriot Act. Some 25-29% of US voters are literally ‘know nothings’ in that they did no better than pure guessing on a 31 item political questionnaire featuring items like those above. If some easy questions, mainly personal information on candidates, are dropped the proportion of ‘know nothings climbs to 35%. Even this overestimates their knowledge because the remaining questions were still fairly basic. For example social security is virtually never mentioned as a racial issue. Yet because blacks have a lower life expectancy, and they pay the same social security rates, there is a substantial regressive redistribution of income from black workers to white retirees. If this sort of knowledge and insight were included in political knowledge assessments the level of ignorance would be shown to be far worse.

How can one judge whether the candidate you vote for is good for you, or any one else, if you don’t even know who the conservative and liberals are in the election, or anything about their programs? For all you know the other guy’s program might be much better for what you value. How would you know which candidates, if any, aim to deal with anything really important if you don’t actually know enough to say what is or isn’t import? Mostly people use shortcuts, like relying on Party affiliation or opinion leaders and activists. Unfortunately it’s difficult to gain much information about Party policy effectiveness from experience with a few governments, and opinion leaders tend to have more extreme opinions than the general voter and have an incentive to exaggerate the importance of their pet issues. Sometimes people vote on the basis of the state of the economy but many are even ignorant about that. For example about 60% couldn’t tell you whether unemployment is improving or worsening. Even if they knew the state of the economy they wouldn’t be able to say whether the state had anything to do with the previous government’s policies, and even if they knew that they would still need to know if the other Party wouldn’t do better.

Some commentators have claimed this widespread ignorance isn’t a problem because the ignorant voters constitute random noise and the votes of the informed few therefore make the overall vote correct. This assumes that the votes of the ignorant are truly random i.e. make errors equally in all directions and that the informed voters represent the interests of the general electorate. The evidence is strongly against either assumption being true. The very shortcuts voters use creates systematic biases e.g. favoring known politicians over new ones or loud special interest activists over good sense. Informed voters are decidedly not representative - they are very slanted in terms of gender, race, income, education, ideology and age, and of course lobbyists with special interests tend to be very knowledgeable politically.

There are also many issues about the rationality of voter ignorance or of voting itself and various paradoxes produced by voting systems, but I won’t discuss these now. Suffice it to say that a lot of ignorant voters aren’t just random noise which ballot distills the good sense of informed voters. Ignorant voters tend to be systematically biased away from their own optimum positions i.e. support bad policies and programs. To the extent that politicians give them what they want they therefore cause real harm. Voting is far from being an important obligation if you are ignorant. If you don’t know the facts or understand the issues and how government works, voting is downright irresponsible.

OK so how much irresponsibility is there going around? Look at the graph below.



The pink line shows how well informed voters think they are by IQ (the self judgments are weighted by some knowledge testing). Like knowledge and understanding in every field political knowledge increases with IQ so we should be grateful that the probability of voting (the dark blue line) also increases with IQ. Indeed the yellow line shows that the probability of being politically savvy if you vote rises with IQ. The purple line curving downwards shows the probability of thoroughly ignorant people taking the trouble to vote. The peak probability of this curve is in the same IQ range where crime is most probable - apt for choosing to be irresponsible. Finally the light blue line is the probability of voting if you are well informed plus of not voting if you are ignorant – hence the label, ‘responsible’. Note that as many as 1 in 5 of even the very brightest among us is politically ignorant, and is prepared to vote irresponsibly. Happily responsible voting never drops below 50% at any IQ. The overall figure for responsible voting behavior is about 57%. So just more than 2 in 5 people know a lot and fail to make that knowledge count, or they know little to nothing and inflict their ignorance on us politically.

Responsible voting increases with intelligence so it would be informative to take the policy preferences and votes of the most intelligent within every demographic group of interest and then weight those votes to reflect the relative size of each group. That way we could estimate the intelligent, informed and least biased vote in a way that represents everyone’s interests.

Let me show you another exercise where the intelligent vote tells us something interesting. Look at the next graph. Each dot is the ratio of the proportion of voters with IQs above 120 voting for a Democrat or Republican presidential candidate, over the proportion of the total vote that went to the same candidate. This is for whites only because blacks tend to vote exclusively Democrat.



Don’t make too much of the fact that the Democrat line is mostly above the Republican line. The truth is that if we looked at a stupidity ratio instead we would see the same thing because Republican voters tend to cluster around moderate IQs and Democrat voters are disproportionally either very bright or very dull.

What I want to show is that the ratio at any point predicts the Party of the next president quite accurately. The rule “if the Republican line gets very close to or rises above the Democrat line for any election it will win the next presidential election – other wise the Democrats will win” gets 5 out of 7 correct. If I add a rule saying that “if the previous election had the Republican line at least close to the Democrat line then the Republicans would win the next presidential election regardless of how much the Democrat line exceeds the Republican line in this election” then the hit rate is 7/7. Victor Serebiakov (former Mensa International Chairman) said something similar in a report on the voting of Mensa members in both the UK and the US (going back further than I did). He showed that the Mensa vote tended to anticipate changes in the general population vote very well. Based on my two rules it looks like the Democrats will win the next US presidential election. Currently election futures markets are saying the same thing.

Is the Liberal Lifestyle for Everyone?

In traditional Western culture if you weren’t an aristocrat you couldn’t afford to be sexually (or even romantically) impulsive. The cost of out of wedlock births was so harsh that the illegitimacy rate (proportion of unpartnered women of childbearing age giving birth out of wedlock) stayed close to 4% from 1200-1800, even though the average marriage age was close to 25 and contraception was essentially non existent. Today we have a radically different situation. In the UK today the rate is 30% and among white women in the US it is 25%.

The effect of removing social indoctrination and sanction from the masses is to leave those who are more impulsive, or have less ability to foresee the consequences of their actions, with much less guidance. So staying out of trouble depends far more on intelligence, and perhaps coolness, than it used to.

Have a look at the graph. The ability of a woman to navigate 2 or more sexual partners since turning 18 and not falling pregnant depends very strongly on IQ. The relationship is very similar to that of passing high school, only very fractionally tougher. Note that even when a woman has already made the mistake of an out of wedlock birth there is still some relationship between IQ and making the same mistake again.



Murray and Herrnstein show that the chances of a single lower IQ mother raising a child out of poverty are slim, and that her chances of neglecting or abusing the child in some way are appreciable. From The Bell Curve chapter on Poverty I inferred that as her IQ increases the difference a husband makes to her chances of avoiding poverty for her child drops rapidly. Notice from the graph that women with IQs over 120 manage to maintain the old 4% illegitimacy rate in spite of the more liberal milieu. Not only do intelligent women not need the old indoctrination and sanctions to avoid illegitimacy but if they do happen to have an illegitimate child they seem to be more able to manage without a husband too.

Individualistic self determination seems to work fine for them so the modern liberal elite continue to argue for the breaking down of old restrictions. They point out that they can get by without the old rules. Yesterday it was premarital sex and illegitimacy and today gay marriage, swinging and polyamory. They are probably tired of being expected to be role models for the stuffy restricted behavior of the rest.

All this is fair enough but when combined with the egalitarian tenor of the times whatever rights you argue for yourself needs to apply to everyone. This has disastrous consequences for those without the self control and intelligence to handle it. Though it is un-PC it’s hard to avoid the conclusion that the liberal lifestyle is not for everyone. It is also clear that some people seem to be able to thrive without some of the core social institutions we have taken for granted. Perhaps anarchism would only work in a country with a mean IQ some 15 points higher than the highest observed to date.

IQ and Getting Away With Murder

Let’s face it - criminals are usually stupid. The average IQ of white men doing jail time is well below the population average, and for those doing time for violent crime it’s even lower – 92 and 84 respectively. Why is that? Is it because the stupid are simply so much more likely to commit serious crime? Yes that is part of the explanation but it isn’t all of it.

For a start a fairly large fraction of crime goes unpunished. The national clearance rate for murder in the US is only about 62% right now. For lesser crimes it’s much lower. Furthermore the clearance rate has been dropping for some time in spite of better law enforcement technology. Part of the reason is rapid urbanization. In small rural areas where everyone knows everyone it’s much easier to narrow down the suspect list but in large urban areas it’s easy to stay anonymous. So since so many people are getting away with serious crime are the bright more likely to get away with it?

Murray and Herrnstein’s The Bell Curve shows a graph of the probability of being in the top 10% of self reported crime, by IQ (with the effect of social class controlled). It slopes slightly upward. Another line sloping downwards shows the probability of having been jailed by IQ. So, brighter men are both more likely to be in the top 10% of self reported crime, and less likely to have been jailed. Clearly brighter men do less jail time per unit of crime i.e. they are more likely to get away with it.

That shouldn’t be a surprise. Validity studies show that IQ has a very general validity. In brief, on any task requiring thought, almost all the variance in performance explainable by cognitive ability factors is explained by whatever it is that IQ tests measure. Usually this is called the g factor. Overcoming barriers when committing crime and making sure not to leave evidence that would arouse the suspicion of the police, requires definite problem solving. Higher IQs are better for successful problem solving.

Wouldn’t it be interesting to know your chances of being a successful Evil Genius? I mean just how hard would it be to be Professor Moriarty? To get an idea I had a go at putting numbers to this speculation. I produced a curve showing the probability of not doing jail time given that a man was in the top 10% of self reported crime. While this is interesting it’s more interesting to show the probability of getting away with a serious crime like murder because the police will be sure to at least try to solve it.

Unfortunately not many people admit to murder in self reported crime surveys, so I had to think of another way to estimate the odds of getting away with murder for various IQs. My approach involved serial killers and some creative statistics. It was lots of fun. (See the Appendix below for technical details.) You see the results in the graph below. To give some perspective I also included curves showing the probability of finishing high school, junior college, college and graduate school by IQ.

As you can see as a problem getting away with a single murder in the US is much like school or college in its intellectual demands. It’s just shy of junior college in difficulty. Being in the top 10% of self reported minor crime and avoiding jail time is shows a similar pattern but is closer to high school in difficulty. That suggests serious crime isn’t particularly demanding as intellectual tasks go. Being a successful serial killer i.e. getting away with 3 separate murders, is slightly tougher than a basic college degree but still a lot easier than obtaining a post graduate or professional degree.



Let’s try to define the minimum requirements for an Evil Genius IQ. I suggest two.

a) The same IQ as other geniuses, Nobel laureates say, which is an IQ of 146 on average.
b) Getting away with a murder at the standard 5% level of significance we apply to research results when we want to say it wasn’t due to chance. This implies a 95% chance of getting away with a murder and would require an IQ of 152.

A convenient compromise would be an IQ of 148 which falls neatly at +3 standard deviations above average and is neatly at the 1 in 1000 level. Combining this with the likelihood of someone with that IQ murdering someone in their lifetime, 1 in 8217 men, I estimate that there are only around 12 male and 1 female Evil Geniuses in the US today.


Appendix.

I assume that each killer has a unique and unchanging probability of avoiding being convicted for any murder. Call this E - for escape. I assume each murder is an independent event i.e. I don’t count mass murders. So the chance of getting away with n murders is Eⁿ, which is (1 – prob(getting caught)).

I reasoned that serial killers who managed to kill more victims before being caught are more talented at avoiding identification and capture, and that this talent is strongly related to their IQs. I also assumed a constant probability of being eventually caught applied to all serial killers. Initially I guessed that to be 0.62, the national murder case clearing rate.

So Eⁿ = 0.38 or
ln(E) = ln(0.38)/n and
E = exp(ln(0.38)/n)

Radford University’s Serial Killer Information Centre data base provided the IQ scores and number of victims for 48 white male serial killers in the US. Using these numbers and regressing IQ linearly on

E/(1-E) = m*IQ + c

I obtained an equation relating IQ to probability of getting away with a single murder. IQ did prove to be strongly correlated to victim number (r=0.44). I adjusted the constant c so that the equation would give me a probability of being caught of 0.62 for a single murder when IQ was the typical convicted murderer’s 84.