(Ex-Army Captain) Kyle Smith gets his statistics right

Mull this one carefully. Say a mammogram is 90 percent accurate at spotting patients with breast cancer, and 93 percent accurate at spotting those without breast cancer. Breast cancer afflicts 0.8 percent of women tested. Your mammogram comes back positive. How likely is it that you have breast cancer?

Less than 10 percent, says “The Numbers Game” (Gotham), a chatty, brief, brightly informative and quite possibly essential book by a pair of blokish Brits (not statisticians or mathletes).

This number won’t be surprising to those who read that wonderful text Breaking the Law of Averages.

Read the whole book review here.

Pay especial attention to the examples at the end.

21%:An extra ounce of bacon a day increased the chance of colon cancer by this much in men, according to the American Institute of Cancer Research. While a huge percentage, it neglected to mention what that increase meant in raw numbers. Turns out, five men out of 100 get colorectal cancer in their lifetime. Bacon-eating increased it to six.


  1. Ok, but seriously, who can eat just one ounce of bacon? And is that cooked or raw?

  2. Joy

    At our GP practice there was an alarming number of breast cancer diagnoses. Seven in total out of a practice where on a full day there were only six in total in the satellite clinic. Prior to my starting, there were three that died. A further four were diagnosed within two years. Two of the staff were receptionists and sat at the same desk as they job shared. Within six months of each other they were diagnosed and treated with surgery and chemotherapy. Nothing unusual was suspected when I questioned the lead partner. Everybody had been talking about it, the villagers had also noticed an increased number of cases, but none of the senior staff were prepared to talk about it.
    My impression was initially that there was/is an environmental factor as never in my career in a female dominated profession had I seen so many cases, or even known anyone personally who had the disease amongst the staff population. Suddenly, both of my receptionists, close friends, were diagnosed within six months.
    Being a local centre of excellence for the condition, more testing = more cases diagnosed. The Macmillan nurse told me that they had been warned to expect a higher number of cases, this was due simply to the issuing of more tests. I found this suspicious to say the least.
    It seems that affluent societies have a higher incidence, Epping is an affluent area.
    Smoking and Alcohol consumption seems to have some bearing on incidence, along with hormone therapy such as is administered for fertility treatment, but by far the most important precipitating factor seems to be genetic. Interestingly, none of the cases at the surgery had any of these ‘usual’ causative factors.
    The surgery nestles between the tube line and a road bridge. I read that the cases of breast cancer even in men are far higher in railway workers. This will sound silly, but when everyone around you is suddenly knocked back in such a way, only a fool would not ask why.
    Two years on, I suspect HRT as a strong factor although this is denied vehemently. HRT was a popular drug amongst the staff who obviously had easy access to and were educated about the virtues of taking HRT. I also wonder about the dairy industry. It would be interesting to see what studies have been done, where and into what. This ought not to be the mystery that it currently is.

  3. bgc


    Don’t forget that there is a possibility that this cluster is due simply to chance. If you have enough cases of a condition spread randomly throughout the community then you should expect the occasional cluster to occur even in the absence of local causative factors. This doesn’t mean that there may not be such a factor in the instance you mention, but it should be kept in mind. Especially as it’s quite natural for anyone close to the event to look for causative patterns.

    On the topic of the book – it reminds me a bit of Gerd Gigerenzer’s book Reckoning With Risk from a few years back. He pointed out that most people find it much easier to understand the significance of these tests when actual numbers are used (as in quote in the review) rather than ratios or percentages. What seem to be opaque and confusing calculations suddenly become very clear.

  4. Joy

    Thank you for replying, You may well be right, but that’s certainly what you call clustering. In one 30 by 30 square foot area.
    The local pharmacist reported that he administered far more Tamoxofen and arimidex than his friends who worked in the midlands. Is this because Epping is near a breast cancer unit or is there something else?
    It is established, or rather the Macmillan nurse confirms that Epping in general has a high incidence. This was a hot spot in a hot spot perhaps.

    So if it follows that more testing = more treatment, I’m wondering whether in areas that do not have access to testing, there ought to be a higher incidence of deaths due to lack of intervention? Otherwise the exercise looks pointless.

  5. JH

    Note also that the probability of a false negative (cancer is present but it is not shown on the mammogram) is less 0.1%; see bacon.

    Yum, bacon wrapped pineapple… sweet, salty, scrumptious, classic, timeless!

  6. Joy

    There is another dimension to the mammogram dilemma in that women of a younger age are more likely to be given false negatives due to the nature of the consistency of breast tissue. One irony is that prognosis for younger women is far worse than for older women, however older women have more ‘compliant’ breast tissue and the mammogram is more effective.
    The conclusion I came to was that a different screening procedure is SORELY needed!
    (Sorry, I couldn’t help myself, but the day I have one of those things clamp me horizontally vertically and obliquely it will be a cold day in hell!) Imagine if men had to put their bitsand bobs into a clamp like that! Bad enough just getting them to their appointments.

    The other aspect from a clinician’s perspective is that for some patients there is such a thing as a ‘therapeutic test’ (Named by myself). For some individuals, have wound themselves up to such a pitch that no amount of reassurance or explanation is a good enough replacement for a test. This is particularly true when it comes to the ‘magic’ MRI scan. In extended scope practice the same sort of issues and dilemmas arise for diagnostics of all descriptions. Especially because a high percentage of patients have an understandable fear of cancer. Whichever body part, the fear is there. If not then it is whatever they saw on TV or whatever their neighbour said they had! It can be hard to restore perspective without ultimately resorting to a therapeutic MRI. I quickly learnt that the patient is only ‘better’ when the intervention AT LEAST matches expectation. A whole host of guess work, psychology, and sensitivity is needed to wade through the minefield. If it is apparent that the patient is not reassured or convinced then it is in my opinion the duty of the practitioner to refer for diagnostic tests regardless of cost or likely negative test result. Doctors and Practitioners that use a simplistic, rigid or black and white method of decision making are the least popular with patients but always run on time,

    A couple of number typos, not sure how important these really are now for this article but in the ‘bacon’ linked by JH:
    In “Section 6: How Accurate?” at the end of the first paragraph you say “(922+7)/1000 = 92.7%”. Should this be 92.9%?

    Also in Section 6, in the forth paragraph at the bottom of the page, you say “ We know from the Mammogram Performance Table that 70 + 92 = 992 women…”, shouldn’t this be …70 + 922 = 992

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