Some probability estimates

associated with violent tornadoes



  Chuck Doswell


Disclaimer: This posting represents only my thoughts and musings on this topic. It has not been subjected to scientific peer review and has no official status. It is provided here only to stimulate thought and discussion on this topic. Thanks to Allan Rosenberg and Mark D. Conner, who sent me e-mail comments on an earlier version

Created: 21 April 1998 Last Updated: 12 April 2005: fixed some outdated links and made some minor revisions

Background information

The database of tornado occurrences has a number of problems, some of which have been described in the scientific literature (e.g., Kelly et al. 1978, Doswell 1980, Doswell and Burgess 1988, Grazulis, 1993). In fact, Kelly et al. (1978) have noted:

An indication of the problems inherent in dealing with the climatology of rare events is shown by the geographical distribution of violent tornadoes (Fig. 6C). Over the 27-year span of this study, only 340 violent tornadoes were reported. Four distinct regions of maximum frequency are evident. However, two of the four maxima can be directly attributed to the occurrence of one or two tornado outbreaks (Galway 1977). Of the 14 violent tornadoes which occurred near Rochester, MN, 10 of them occurred in the outbreaks of 6 May 1965 and 30 April 1967. Similarly, the maximum in Indiana and Kentucky reflects the Palm Sunday (11 April 1965) and the 3-4 April 1974 outbreaks. In contrast, tornado outbreaks do not dominate the climatology of violent storms in the Arkansas and Alabama maxima. Over the 27-year period, no one day in these regions had more than two violent tornadoes.

Thus, it's probably safe to say that the true climatological distribution of violent tornadoes is poorly known and the factual information based on tornado reports may contain a diverse set of errors and biases.

However, it is of some interest to do some simple calculations with the simple climatological data at hand. These serve to give some sense of the data problems, as we shall see.


The chances that a particular house will be struck by a violent tornado, per year

Part 1 - basic numbers

The total land area of the United States is 3,536,342 mi2; if we exclude Alaska and Hawaii, this leaves a total for the 48 contiguous states of 2,959,545 mi2 or roughly 3 million mi2. On average, about 1000 tornadoes are reported annually, of which about 15 are violent (1.5 percent). If they were distributed uniformly over the area of the United States (which they most certainly are not), this would yield an average of one violent tornado per year for every 200,000 mi2 ... let's assume that there is a real possibility of a violent tornado over only about half the land area of the 48 contiguous states, so that means an average of one violent tornado per year for every 100,000 mi2 in the "tornado-prone" area. Put another way, then, on average, a particular square mile in the tornado-prone area has a chance of experiencing a violent tornado in a given year of about 1/100,000 = 10-5. [In Fig. 6c presented by Kelly et al. (1978) the peak values of ~ 0.3 events per year per 10,000 mi2 (or 3 events for every 100,000 mi2) for violent tornadoes is about 3 times higher than the average I've calculated.



Part 2 - for a particular house

A square mile has 52802 = 27,878,400 ft2 or roughly 28 million ft2. A relatively large house has a "footprint" of 2800 ft2, which is about 1/10,000 of a square mile. Let's assume that your house is somewhere within one of those unlucky square miles affected by a violent tornado event. The tornado might devastate the whole square mile, or it might just touch one of the corners (which still might contain your house). What are the chances that your house is going to be hit? This is a tough one to estimate, so whatever I use to estimate it is going to be crude.

According to Schaefer et al. (1986) the median width of 330 F4 tornadoes is 272 m, and 38 F5 tornadoes is 454 m. Since 1 m = 3.28 ft and 1 ft = 1/5280 mi, if we form an average of these two widths weighted by the number of each, this gives a typical violent tornado path width of about 0.18 miles (about 950 ft). The largest possible area within that square mile hit by this "typical" violent tornado then would be if it traversed the square mile from one corner to another, or an area of (2)1/2 X 0.18 = 0.25 mi2. Thus, if the tornado path width is "typical" of observed violent tornadoes, it should affect less than a quarter of the area of any square mile it touches. It's tough to know what to do with this number, so I am going to ignore it, and assume that if a square mile is touched, it's all touched. This will be an overestimate, but it makes the argument simple.

Furthermore, the chance of actually experiencing the violent winds is probably on the order of 1 in 100; that is, only about 1 percent of the total damage area experiences the strongest winds. Just for simplicity, let's assume that if your particular square mile gets hit, the chance that your house is affected by the violent winds within that event is roughly 1 in 100 = 0.01 ... it might be lower than that (see above). That means that for any given year, the chances of your house being hit are about 10-5 X 0.01, or about one in 10 million (10-7).

Consider looking at this a slightly different way. Schaefer et al. (1986) used the path length and width estimates to arrive at a figure for the affected area, by intensity category. The median is probably a better estimate of the typical event than the average when the distributions are skewed by having long "tails," which these distributions tend to have. Schaefer et al. (1986) give a median area of 6.50 km2 for 288 F4 events, and a median area of 24.15 km2 for 34 F5 events. If we make use of the fact that 1 km2 = 0.386 mi2, and combine the areas for both F4 and F5 events by a weighted average of the two, this gives a median area of 3.2 mi2 for violent tornadoes. Thus, the average of 15 violent tornadoes every year strike about 45 mi2 annually. Assuming that these are all within the tornado-prone area of 1,500,000 mi2, this gives an annual frequency of about 3 X 10-5 [in agreement with the peak figures given by Kelly et al. (1978), as described above]. There is a discrepancy of a factor of 3 between this method and the first ... a more or less negligible discrepancy in these hand-waving arguments ... at least they are within the same order of magnitude.

From there on, the analysis is about the same as the first, still giving pretty low annual probabilities (order of 10-7) for a particular house in the tornado-prone area experiencing violent windspeeds in any given year.

Part 3 - A different path

Now, let's turn this around. Let's assume every reported violent event means that at least one house experienced violent windspeeds during that tornado. After all, for a tornado to be rated violent, this means that someone observed damage that they have associated with violent windspeeds (see Doswell and Burgess 1988, and my Pet Peeves discussion of the Fujita scale .. Item B.11). Let's assume further that there are 45 million homes in the tornado-prone area, so the chances of any particular home being struck by violent windspeeds, according to this analysis, is about 15/45,000,000 = 3.33... X 10-7, or about 3 chances in 10 million annually. As above, given the roughness of these arguments, it should be noted that 3 chances in 10 million are within a factor of 3 of the figure derived in the previous section: 1 in 10 million. Thus, it can be argued that these are roughly consistent calculations. Note: the peak annual probabilities given for violent tornadoes in Schaefer et al. (1986 - their Fig. 13) are comparable to this figure.


The results in Schaefer et al. (1986) don't look exactly like those in Kelly et al. (1978), although there certainly are similarities. This may be due, in part, to additional data. The data in the tornado database are not yet very stable (see Schaefer et al. 1993), especially in the violent tornado category. Note the maxima in Arkansas, Alabama-Mississippi.


To buy or not to buy?

If I assume that the figure of 1 chance in 10 million annually is crudely representative of the odds of experiencing F4-F5 winds, then what about over the lifetime of a family's residency in the home? I'm going to assume that lifetime is about 30 years. Allan Rosenberg reminds me that if the annual frequency is Fv, then the probability over N years is not exactly N X Fv; when each year is statistically independent of any other year and the probability is constant from one year to the next, the proper thing to use is the Binomial distribution.

A statistical diversion: Suppose that the probability of having a tornado in a given year is q, and that each year is statistically independent from any other year. What is the probability of having x tornadoes in n years? Under these conditions, it can be shown that this can be found from the Binomial distribution:

pn{x } = {n ! / [x ! (n -x )]!} q x (1-q )n-x,

where the "!" stands for factorial. For the odds of 1 tornado in 30 years, x = 1, n = 30, so that n ! / [x ! (n -x )]! = 30, which means that p30{1} = 30 q (1-q )29. For q = 10-7, this becomes 30 X 10-7 X (0.9999999)29, which turns out to be quite close to 30 X 10-7, a number that we could have found the easy way.

However, at these low probabilities, the issue is apparently not worth consideration, so let's use the simple method. In 30 years of living in that house, there are roughly 3 chances in 1,000,000 of having that home flattened by the F4-F5 winds in a violent tornado. This is important, because for frame homes that are secured to their foundations, the chances of riding out (i.e. without serious injury or death) a tornado up to F3 intensity in an interior room of the home are pretty good ... interior walls should still be standing. It is only in F4 and F5 tornadoes that avoiding becoming a casualty during a direct hit by a violent tornado when aboveground in an interior room (provided the home is reasonably well-constructed and secured to the foundation) becomes doubtful. Having a special shelter built into a new home to withstand violent tornado hits aboveground costs about $1000-$3000. Retrofitting such a shelter into an existing home would be more expensive. Assuming it's possible to build a below-ground tornado shelter near the home, it probably would be cost roughly $1000-$3000, as well. Given the low odds of experiencing a violent tornado, it is not obvious how to make the decision to have or not to have a tornado shelter built. The decision has to be a personal one. Peace of mind might be worth something to you, even though the odds of actually experiencing the violent winds in a violent tornado are pretty tiny.

There might be local "tornado alleys" or other factors where the chances might increase by as much as another 2-3 orders of magnitude, although there is no objective evidence for this at the moment. If over the 30-year lifetime of your house, you had about 3 chances in 10,000 of having your home wiped off the foundation by a violent tornado, would you buy a shelter then? What about with 3 chances in 1000?

A comparative diversion:

For the sake of comparison, there are roughly 40,000 motor vehicle fatalities per year. In 30 years, that amounts to 1,200,000 deaths from being on the roads. If the population of the U.S. is assumed, for simplicity's sake, to be 240,000,000, your odds of dying in a traffic accident during a given year are about 40,000 / 240,000,000 = 1.667 X 10-4 (about 2 chances in 10,000). If I make the same simplifying assumption that I can simply multiply this annual figure by the number of years to get the probability over a 30-year period, this becomes about 5 chances in 1000. What do you do at present to reduce the odds of your dying in a motor vehicle? Do you use your seat belts? Would you spend $1000-$3000 more per vehicle to increase your safety? Do you know what fraction of the cost of a motor vehicle presently is tied up in government-mandated safety features (in spite of which we see the current fatality figures)? What might the fatality figures be without those mandated safety features? ...

Some additional musings

When warnings are issued

If the probability of experiencing a tornado (especially a violent tornado) is so low, then it conceivably could be argued that the most cost-effective strategy is simply to ignore the threat. This situation can be radically altered, however, if you consider the odds of experiencing a tornado when a tornado warning has been issued. Sure there are false alarms ... false alarms are a natural consequence of trying to have warnings out whenever the situation is serious, given the realities associated with our ability to predict tornadoes. However, the probability of experiencing a tornado goes up several orders of magnitude when warnings are issued. Even if communities only experience tornadoes 1 percent of the time (1 time out of 100 warnings) when warnings are issued, this represents an increase over the annual frequency by something like 4 orders of magnitude ... that's a factor of 10,000! If on any given situation where a warning has been issued for your community, you have 1 chance in 100 of experiencing a tornado in a tornado warning, what should you do? I've already discussed this at some length in my Tornado Realities essay - I'm content to leave the decision up to anyone who reads this. Does it make sense to be prepared for an event which is unlikely, but if it occurs can be catastrophic beyond most people's imaginations?


Suppose we consider the impact of all tornadoes on the United States in our recorded history. The total number of tornado fatalities since the Europeans first arrived in this country is probably less than the death toll from motor vehicle accidents in one year (on the order of 40,000). The Viet Nam war was responsible for about 58,000 American fatalities. In comparison with traffic fatalities or the Viet Nam war, isn't this concern about tornadoes a lot of hoopla over something that really isn't very important? How much should we be spending on tornado forecasting and research, anyway?

If you ask me, I think we should spend a large fraction of the gross domestic product on tornado and flash flood research, so I am probably not the right person to ask! But who is the right person to set national priorities about such things? Most people, and especially including politicians have axes to grind and probably can't take an objective look at priority-setting. When disasters happen, we as a nation tend to go through a lot of finger-pointing and angst for a while and then the concern dies down and the issue falls into the cracks until the next disaster.

In a democracy, does anyone care about what the national priorities (as set by, say, the politicians) really are? $70+ million was spent producing "Twister" and it grossed some ridiculous amount of money for the producers. For that $70+ million (to say nothing of the gross receipts from the movie), you could fund VORTEX for 70+ years, or put on a two-year program with some really high-resolution observations. People spend billions on entertainment every year ... sports, music, movies; more billions are spent on cosmetics, gambling, cosmetic surgery, drugs, astrology and psychics, etc. As a nation, we spend our money mostly on non-essentials that we choose to support with our hard-earned dollars, even as we struggle to find national, state, and local resources and the will to solve collectively some really serious problems: the coming fossil fuel shortages, killing 40,000+ in motor vehicle accidents, destruction of the environment, our abysmal public education system, decaying national infrastructure, etc.

Who decides where our money (i.e., that available to governments) gets spent and, more importantly, on what basis ? It's just as much a mystery to me as the atmosphere itself. I didn't feel very guilty about having the federal government pay my salary and support my work when I was still working for the government, if I consider that a mediocre second baseman in the major leagues gets millions per year! At least what I do has the possibility of returning some tangible value for that expense - saving lives and property (if anyone listens!). If I find it to be entertaining and fun to be in tornado and flash flood research, it also turns out to provide something useful to forecasters ... and forecasts have real value, even if we have been incompetent at figuring out what that value actually is (see the 1998 article on the value of forecasting by Harold Brooks and me).

Harold and I have estimated that since the inception of public tornado forecasting, something on the order of 13,000 lives have been saved. This number is probably not strictly justifiable, but we have some basis for it - the basis was introduced by Al Moller, Harold Brooks, and me in a 1999 paper. If you can accept a figure of $8 million per life [not my figure so don't ask me to justify it!], this means on the order of $104 billion [in inflation-adjusted 1997 dollars] has been saved by public tornado forecasting and warnings since 1953. Compared to the Defense Budget, this is small potatoes - how much have they spent since 1953? - but it certainly more than pays for all of the NWS and NSSL since 1953. This argument is expanded upon here.


NOTE: Articles I've authored or co-authored (denoted by an asterisk, below) are available here.

*Doswell, C.A. III (1980): Synoptic scale environments associated with High Plains severe thunderstorms. Bull. Amer. Meteor. Soc., 60, 1388-1400.

*______, and D.W. Burgess (1988): On some issues of United States tornado climatology. Mon. Wea. Rev., 116, 495-501.

*______, and H.E. Brooks, 1998: Budget cutting and value of weather services. Wea. Forecasting. 13, 206-212.

*______, A.R. Moller, and H.E. Brooks, 1999: Storm spotting and public awareness since the first tornado forecasts of 1948. Wea. Forecasting, 14, 544-557.

Galway, J.G (1977): Some climatological aspects of tornado outbreaks. Mon. Wea. Rev., 105, 477-484.

Grazulis, T. (1993): Significant Tornadoes. 1680-1991. Environmental Films, St. Johnsbury, VT, 1326 pp.

*Kelly, D.R., J.T. Schaefer, R.P. McNulty, C.A. Doswell III and R.F. Abbey, Jr. (1978): An augmented tornado climatology. Mon. Wea. Rev., 106, 1172-1183.

Schaefer, J.T., D.L. Kelly, and R. F. Abbey (1986): A minimum assumption tornado-hazard probability model. J. Clim. Appl. Meteor., 25, 1934-1945.

______, R.L. Livingston, F.P. Ostby, and P.W. Leftwich (1993): The stability of climatological tornado data. The Tornado: Its Structure, Dynamics, Hazards, and Prediction (Geophys. Monogr. 79), Amer. Geophys. Union, 459-466.