The Hidden Brutality of Innumeracy

This is a post about Rape.  It is a post about how numbers relate to the incidence of rape in our country and what understanding and misunderstandings that engenders.  It does not talk about the specifics of rape or any particular case, but if rape is a trigger for you, then I wanted to warn you in advance in case you need to stop reading.

All others, however, should read this, because it’s fucking important.

It is long. But so fucking what.

Rape and sexual asault, in our enlightened society, is remarkably common.  Horribly so.  A while ago when I first heard the term “rape culture,” I reacted against it with the sort of privileged, “Oh come on–that’s a bit extreme–it’s not that bad..” type of attitude.

To me–it just seemed like it could not be that bad.

But it is.  And over the past couple of years, I’ve learned more and more about the topic and my attitudes began to change.

What clinched it for me were the numbers, though.  As I’ve noted before, my surname means something like “skill with numbers” or “skill with calculation,” and numbers have always been important for me.

The rest of this is going to be an analysis of rape & sexual assault numbers presented in a Bureau of Justice Report on “Female Victims of Sexual Assault: 1994-2010” . Go to that page and download the PDF to read the report.

The important question–the one that drove me to write all of this–does not deal with the presentation of the numbers in the report, but rather what they actually mean.  I say this because it can be easy to look at the numbers in this report superficially and come away with the conclusion that things aren’t so bad.

That conclusion, however, is predicated on a lack of understanding of the math behind these numbers.  The reality is that far too many women will be raped or sexually assaulted in their lifetime. By the end of this, hopefully you’ll understand why the numbers show this.

This is what I mean by the hidden brutality of innumeracy.  It is the fact that not understanding the numbers can lead one to disregard this problem as overstated or exaggerated.

This is not the case.

So let’s get started on that..

Going to that report, it actually shows a declining rate of rapes & sexual assaults over the period of 1994-2010.  The chart looks like this:

Taken from Bur. of Justice Report linked above.

Taken from Bur. of Justice Report linked above.

This overall chart shows an almost 60% decline in the overall rate of reported rape & sexual assault attempts over the past 15 years.

This decline is certainly a good thing and I don’t mean to diminish it, but this is not the entire story.  Importantly, it might seem like this decline might mean that there really isn’t that much of a problem with rapes.  After all, how bad of a problem is it when it’s only 2.1 reported rapes/sexual assaults (Completed/Attempted/Threatened with Completed being the largest category) per 1000 females aged 12 & over.

That’s less that .21%!  Or rather–it would mean that 99.79% of women over age 12 are not subjected to rape or sexual assault in any one year.

Sounds pretty safe, correct?

There are a number of problems with that assumption, however.  The most prominent of them is that a woman tends to live for a long time after age 12.  Currently, the life expectancy for a woman is 81 years in the United States, meaning that she has 69 years of possible attempts to get through.

This is where misunderstanding can easily begin to creep in.  If you don’t understand how numbers and mathematics (specifically probability & statistics) works, you can come to some very incorrect conclusions about what the overall experience of rape will be for a woman over her life.  One friend of mine stated almost as much, explaining that he figured that to get the lifetime probability of rape, you would take the 2.1/1000 and make that a .0021% chance per year–and then assume 80 years of life–and multiply them together.

This would give us, he assumed a .168% chance of being raped in a woman’s life–which doesn’t seem so much.

There is an obvious problem here and then less obvious ones too.  The first problem is that 2.1 rapes/1000 is not a .0021% chance–but rather a .21% chance.  Multiply .21% x 80 and you would get 16.8% over 80 years–which is a 1 in 6 chance of being raped/sexually assaulted over that lifespan. (One can see this in the fact that 2 out of 100=2%, not .02%, thus 2.1 out of 1000= .21% not .0021%).

The other problems that crop up are that rape doesn’t happen like that and probabilities don’t work that way.  It does not “add” over time–any more than flipping a quarter will guarantee you a heads on the 2nd flip because you have a 50% chance each time of getting it.

Probabilities of such independent events (the chance of being raped in one year is not connected to the chance 5 years from now in any direct way, just as flipping coins is independent of each other) involve calculations where such chances are multiplied together in different combinations.

Let’s use the coin flipping example to make this concrete.  For flipping a coin–the chance of getting a head is 50% for a fair coin.  The chance for getting 1 head within 2 flips is not 100% though.  Instead, you have to write out the possible outcomes and count. For 2 flips, this would be:

Counting these, one sees that the chance for exactly 1 head in 2 flips is 50% (TH & HT).  The chance of getting two heads in two tries is 25%(HH) and the chance of getting no heads is 25% (TT).

If you keep going with this–you can see how complex it will start to get.  At 5 flips, if you wonder what your chance of getting 1 head will be, you have many different possible combinations of flips (32, in total..), but only a certain number with just one H.  These might be TTTHT or TTHTT, etc.  In total, the chance of getting just 1 H in 5 flips is 5 * (1/2)^5= 5/32==15.625%

The chance of getting 2 Heads within that becomes more complex–because the combinations grow.  Perhaps your flips are HTTTH or HHTTT or HTHTTT.  In total there are 10 different ways to do this.(The Math is actually the Combination of 5 things taken 2 at a time–see math explanations… )

Now–imagine doing this for 60 flips–the combinations grow exponentially.

This is probability.

Is this easy? No–I will admit that it’s a bit complex and can seem overwhelming.

In fact, it’s hard.

But it’s not nearly as hard as surviving a sexual assault.

So let’s do the calculations.

First things first–let’s use the correct numbers:
a) Instead of 80 years, the actual number is 69 years from age 12 to age 81.  (Doing the incorrect calculation above, this would yield a 14.49%  (or even more inaccurately, a .1449%) chance for reference.

b) The 2.1 /1000 is an overall average for all women over aged 12, but it does not actually equal the chance per year for all ages. This should make some sense in that it is more likely that younger women are attacked than elderly women (although that does happen).  Because of this, we will use the following table which shows how the rate changes over the lifetime of a woman.

Taken from the Bur. of Justice Link above--pg 3.

Taken from the Bur. of Justice Link above–pg 3.

Interestingly, the easiest way to calculate the probability of a woman being raped/sexually assaulted over her life (using these numbers) is to calculate the chance that she IS NOT EVER raped or sexually assaulted.

This is because you don’t have to deal with any of those combinations of possible multiple rapes over the years (think of the Heads/Tails example above–it is like making the calculation of there being only all tails every single time TTTTT<–which there is only one way of achieving).

Instead, you are just calculating the probability for the one outcome where there is one long string of “not-raped” outcomes.

For this “simple” calculation we need only to pay attention to the complications produced by the changing rape/assault rates for the different periods of a woman’s life.

The actual calculation itself would then look like this:

Chance of NOT EVER BEING RAPED= (Chance of not being raped in a year)^# of years, where “^” equates to “being raised to this power”  Thus, the chance for a woman not being raped from age 12-17 would be (using the table above):

(.9959)^6  ==.975651

(.9959 is  1 – .0041) and 6 comes from the 6 years (12, 13, 14, 15, 16, 17) that this rate is experienced.

To find the rate for a woman’s entire life, we’d need to do this for all periods of her life and multiply them together.  This calculation would then be:

Chance of never being raped/sexually assaulted=(.9959)^6* (.9963)^17*(.9985)^30*(.9998)^16

Chance No Rape=.975651*.938928*.955966*.996805

Chance of No Rape= .872930 == 87.293 %

Thus, the chance for a woman not suffering from an actual, attempted, or threatened rape or sexual assault in her lifetime is 87.293%, which is about 7 in 8.  

The conclusion you draw from this is that there is a 1 in 8 chance that a woman will suffer those things at least once in her lifetime.

How acceptable is that to you, my readers?  Is that low or high?

Is it acceptable to think that you–and male readers I want you to join in here too–have a 1 in 8 chance of being sexually assaulted by a male over your lifetime.

Is that acceptable?

Now let me complicate this even further.  Importantly, these numbers are all based on statistics for reported numbers of rapes/sexual assaults.  The Buruea of Justice Report, however, notes that rape is notoriously underreported due to numerous social factors–including fear of reprisal, the belief that the police could not or would not do anything to help, the belief that it was just a personal matter and/or not important enough to report.

In 2010, the Bureau of Justice report estimated that only 36%  of rapes were reported, by 64% were not.

That changes everything with our numbers.

It means that the actual overall incidence of rape was not 2.1 /1000, but rather 5.83 /1000 . ( You get this by taking 2.1 * (100/36).  This scaling factor is 2.778.)  Applying that scaling factor to the table above, we would get the following rates:
Age 12-17 is 11.39 /1000
Age 18-34 is 10.28/1000
Age 35-65 is 4.17/1000
Age 65+ would be .556 /1000, but I’m gonna keep it at .2 /1000 due to the data being so sparse.

Taking these rates and doing the same calculation above for the entire life of a woman, you’d get:

Chance of never being raped: (.98861)^6*(.98972)^17*(.99583)^30*(.9998)^16

Chance Not Raped=.933577*.83890*.882178*.996805 == .68869

Chance Not Raped= 68.869%

So, using these rates of rape/sexual assault–which correspond to the actual rates of rape according to the Bureau of Justice, the chance, for a woman, of NEVER BEING RAPED over her lifespan is 68.87%–or a bit more than 2 in 3.

The flipside is that women have almost a 1 in 3 chance of being raped/sexually assaulted by a man AT LEAST ONCE in their lives.

Does this seem like a problem to anyone.

It seems like a problem to me.

To conclude, I might note that only 15 years ago, the reported rates of rape/sexual assault were nearly 250% higher than they are now, while the reporting rates were even lower.  You could do these calculations if you wanted, and then think about what women have suffered throughout much of history.  So things have likely improved from utterly horrible to merely pretty awful.
This is what the numbers say.  1 in 3 chance over their lifespan.  To make this concrete, take 10 of your female friends.  The chance that none of them has been raped or sexually assaulted is 2.4% on average.


Is this acceptable?

Take 20 of your female friends–this chance drops to .06%–6 in 10,000 that none of them have been raped or sexually assaulted.

Is this acceptable?

Not to me.

What about you?


About Prof. Woland

I contain multitudes. Come meet us.
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3 Responses to The Hidden Brutality of Innumeracy

  1. WOW that’s some amazing mathmatics! I can say personally that rape has affected my immediate family multiple times among at least three of us and only one of our rapes was ever reported. So far my daughters have been saved from these statistics, but they are young and have many years to go, leaving my heart heavy with concern for them and their future. I definitely agree that this is utterly unacceptable, but the real question is, how do we stop the inevitable from happening? I have a son and have hopefully trained him up to understand that rape is absolutely wrong but not all Mothers do this. It would be useful for high schools and colleges to teach the truth about what rape does to women as individuals and to society as a whole! Thanks for sharing this information. Mz Jai

  2. Reblogged this on Sensuous Enemy and commented:
    I write lyrics form personal experience and once wrote a song about being raped at age 12 titled “Das Grauen” located here on our bandcamp site:
    I happen to be of the statistics Dr. Kundert talks about in his recent blog “The Hidden Brutality of Innumeracy”. If you have the chance to read this you will be blown away, by what he reveals, using math, to show the hidden truths about rape in our American society!

  3. Pingback: Random convergence on privilege | The Philosophy of NOM

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