A Logarithmic Safety Index

How safe is an activity? We desire not a vague answer such as “very safe” (what does this mean?), but rather, a concrete, numerical answer such as 1 death in 100,000 cases. But 100,000 is a large number. Imagine some safety report throwing out numbers in this range—how are we to see from a glance the difference between 100000 and 1000000? Mathematician John Allen Paulos (1988, 127-132) suggested we follow geologists by creating our own Richter scale: a logarithmic safety index.

It would work as follows. Consider an activity that results in a certain number of deaths per year, for example, riding in or driving a car. Paulos in 1988 quoted that one American in 5,300 died each year due to car crash. The safety index is thus 3.7, the logarithm of 5,300. By the way, we shall use not the natural logarithm, whose base is e, but the decimal logarithm, whose base is 10. That is, 103.724… = 5300, so the logarithm of 5,300 is 3.7. But what about more recent data? According to the United States Census Bureau’s 2010 Statistical Abstract, the number of persons killed in fatal crashes in 2007, the most recent year with complete data, was 41,059. The US population at the time was 301,621,000. So in 2007, one American in 7,346 died from some car crash, and the safety index is thus the logarithm of 7,346, or 3.9. From a quick glance, we can tell that the 2007 safety index for car crashes, 3.9, is higher than the 1988 safety index, 3.7.

What does safety index mean? Because we are defining the index as the ratio of amount dead per population, that is, 41,059 in 301,621,000 is the same ratio as one to 7,346, a higher safety index means a safer activity. (This is the opposite result of the Richter scale, on which a higher number is more dangerous.) For instance, let us find the safety index of walking outside and hoping to avoid being struck by lightning. According to the National Oceanic and Atmospheric Administration (NOAA)’s website, an average of 58 Americans each year are killed by lightning. We again use the 2007 population figure. 302 million divided by 58 is still a large number: 5.2 million. This means roughly one American in 5.2 million died from being struck by lightning in 2007. The logarithm of 5.2 million, and thus the safety index, is 6.7. This is far greater than the corresponding safety index for car crashes in 2007, which is 3.9, meaning it is much less likely for one to be killed by lightning than by car accident. The difference between the indices is 2.8, or roughly three. A difference of three in logarithm is a difference of three orders of magnitude in actual value, or 1000. Roughly speaking, a person is about 1000 times more likely to be killed in a car crash than by lightning.

Of course, we could have just divided the actual figures for each: 41,059 deaths from car accidents versus 58 deaths by lightning; the difference is three orders of magnitude. Such numbers, however, are so apart that it is difficult to list and compare them. Plus, they are not normalized for comparisons of different sample sizes. In the previous examples, we have been looking at data for the United States only—what about for the world? According to the websites of the World Health Organization and the Population Reference Bureau respectively, in 2004 there were 1.183 million deaths from road accidents worldwide and the world population was 6.396 billion. Thus one person in 5,407 died from road accidents worldwide in 2004, and the safety index is 3.7. Of course, we could not simply compare the 1.183 million deaths in the world with 41,059 deaths in the United States, but when we scale down both figures appropriately, the results are comparable. Driving in the United States in 2007 had a safety index of 3.9, so it was safer on average to drive in the United States in 2007 than to drive in the world in 2004. And driving in the world in 2004 was on average equally as safe as driving in the United States in 1988; both had a safety index of 3.7.

What if we categorize by age group? Does driving at 21 have a lower safety index than driving at 30? In 2007 there were 52 fatal accidents by a 21-year old driver per 100,000 licensed drivers, for a safety index of 3.3, whereas in the age 25-34 bracket, the number of fatal accidents per 100,000 drivers was 30, for a safety index of 3.5. The safety index increases further in the 35-44 bracket, to 3.6. Note that these figures (3.3, 3.5, 3.6) are lower than 3.9, the safety index of a fatal car accident in 2007. This is explainable by the fact that the lower figures are all within the group of licensed drivers: the chances of being killed in a car accident decrease for you if you are not a licensed driver. (Huh? That doesn’t mean you’re a bad driver; it most likely points to the fact that those who are not licensed drivers are less likely to be in a car, especially with a young driver, and thus less likely be involved in an accident.)

Now let us take a look at disease. In 2004 the world suffered 1.46 million deaths from tuberculosis; in 2006 the United States suffered 644 (according to the Center for Disease Control). The safety index for tuberculosis in 2004 across the world was 3.6, while in 2006 across the United States, it was 5.7. You are 100 times less likely to die from tuberculosis if you are in the United States than in the world. But, this is not the same as the statement, “Given that you have tuberculosis, you are 100 times less likely to die from it if you are in the United States than in the world.” In 2006 the United States had 13,727 cases of tuberculosis, out of which 644 died. Thus in the United States, given that the person has tuberculosis, the safety index was 1.3. In the world, in 2005 there were 8.81 million cases of tuberculosis, and 1.58 million deaths. This gives a safety index of 0.7. So, you have tuberculosis, you are still less likely to die from it in the United States than in the world, but not 100 times less likely—rather, 3.8 times less likely. By the way, the safety indices for other transmittable diseases, e.g. malaria and influenza, are also higher in the United States than for world averages.

What about the safety index of homicide? The United States suffered 18,573 homicides in 2006. (If this is alarming, remember 41,059 in the US died in car accidents in 2007.) This amounts to one American in 16,152 being murdered, for a safety index of 4.2. In the world in 2005-6, approximately 198,000 homicides occur each year, or one person in 32,323, for a safety index of 4.5. Thus, with respect to homicide, the United States is not the most safe. That said, in the United States you are still over twice as likely to die from a car accident as from a homicide.

So far we have been talking about general cases, in which almost anyone could have been the victim. On the other hand, babies can’t drive, but that hardly ruins the result for driving—passengers are included in deaths from car crashes. (For the age group paragraph, I used special statistics.) What would ruin the result is a dangerous but rare activity. Suppose that in one year 18 people in the United States play Russian roulette (chance of death is one in six). On average, three of them would die. So, because one American in 100 million dies of Russian roulette per year, does that mean the safety index is 8? In a way, yes, but a more relevant number is the safety index of Russian roulette among Russian roulette players. This gives one in six, for a safety index of 0.8, an extremely low index, and thus extremely dangerous activity.

For historical contemplation, let us find the safety index of World War Two. For this war, we shall throw out the one-year clause of the safety index, as we are applying this over a greater amount of time. Not surprisingly, the Soviet Union, which lost 27 million dead, has the lowest safety index, 0.8. China lost in absolute numbers the next greatest, but its overall population was so large that the safety index is at 1.4. Two states, Poland and Germany, which stand at 0.8 and 1.0 respectively, are between the Soviet Union and China in terms of safety index. Note that France, Italy, Great Britain, and the United States each suffered less than one million killed; the United States, with a high population and low death toll, had the highest safety index of the listed countries.

Table 1. Death Toll of World War Two*

Country 1939 Population Death Toll Safety Index
TOTAL 2.3 billion 60 million 1.6
Soviet Union 169 million 27 million 0.8
Poland 35 million 5.8 million 0.8
Germany 70 million 6.6 million 1.0
China 518 million 20 million 1.4
Japan 71 million 2.7 million 1.4
France 42 million 580,000 1.9
Italy 44 million 450,000 2.0
Great Britain 48 million 450,000 2.0
United States 131 million 420,000 2.5

*1939 population in millions; death toll in millions. The total category is for the entire world, not only for nations involved. Death toll includes military and civilian, including Holocaust. For estimate ranges we use upper bound. Most figures are only approximate; sources vary greatly from one another.

It might be interesting to compare the safety index of World War Two to a modern war. One that comes to mind is the Persian Gulf War, for which results are easy to tabulate.

Table 2. Death Toll of the Persian Gulf War

Country 1991 Population Death Toll Safety Index
Iraq 19 million 158,000 2.1
Kuwait 2.2 million 1,000 4.3
United Arab Emirates 2.4 million 6 5.6
United States 253 million 294 5.9
Saudi Arabia 18 million 18 6.0
United Kingdom 58 million 47 6.1
Egypt 54 million 11 6.7
Syria 13 million 2 6.8
France 57 million 2 7.5

If we simply look at the safety indices of various countries in the Gulf War, we see that they are much higher than in World War II, and there are of course obvious historical explanations—WWII was a total war fought over six years, while the Gulf War was in essence a lightning war (not to be confused, however, with Blitzkrieg). Recall that the safety index for being killed by lightning was 6.7. From the table, the safety index for France in the Persian Gulf War was 7.5, meaning it was safer, for the French at least, to be in the war than at home, as lightning doesn’t happen much in Iraq or Kuwait.

Overall the safety index allows us a quick and sensible way of looking at large numbers. If the safety index of some activity is over six, then it’s pretty safe. It means the chance to die is one in a million or less. Being killed by lightning is one such example, with a safety index of 6.7. If it’s less than four, it’s fairly dangerous: a car crash, at 3.9, or tuberculosis (worldwide), at 3.6. Below two, it’s pretty grim. Japan during World War Two had a safety index of 1.4. Below one, something is seriously wrong: the Holocaust on Poland, the Soviet Union invaded by Nazi Germany, and a game of Russian roulette—they all have a safety index of 0.8.

This logarithmic safety index will probably never permeate the public domain. If it could, it could have done so 22 years ago, when Paulos proposed it. Nonetheless, it is interesting to view data in a logarithmic manner. In chemistry, the pH scale is logarithmic. In geology, the Richter scale is logarithmic. In physics, the decibel system for sound intensity is logarithmic. And these systems would make little sense without it, if they were only in plain numbers. Similarly, part of the public confusion these days in mathematics, and particularly probabilities involving risks, may be resolved through the use of a more natural, more intuitive system—the logarithmic system.


Paulos, John Allen. 1988. Innumeracy. New York: Vintage Books (1990).

United States Census Bureau. The 2010 Statistical Abstract: The National Data Book. http://www.census.gov/compendia/statab/ (accessed Mar. 17, 2010).