While averages are a great way of summarizing data, in practice the key concern for statisticians is how many averages? Is one average too simplistic? Would one average for each subgroup be more informative? This is the principal concept discussed in Chapter 3 of Numbers Rule Your World.
Consider this paragraph that summarizes the finding of a recent British study linking the consumption of red meat to the risk of bowel cancer (my italics):
People who consume more than about 90 grams (3.2 ounces) a day are at greater risk of getting bowel cancer, which kills 16,500 people in Britain every year, the Department of Health said, urging them to reduce their intake to 70 grams. Forty-two percent of men eat more than 90 grams every day on average, compared with 12 percent of women.
Notice that men as a group differs significantly from women in terms of consuming a lot more red meat. And also note that the guideline prescribed one level of consumption (70 grams per day) for both genders.
It is also the case that men are much more likely to get bowel (colon or rectal) cancer than women in the UK, as seen here:
So, you wonder if in fact there should be two guidelines, a stricter rule for men than for women.
By issuing one optimal level of consumption for both genders, that is, one average instead of two, the health authority in the UK has made a statistical assumption of "additivity". They are stating that the effect of red-meat consumption is independent of gender: given the base rate of cancer for each gender (which is far from equal as shown above), an equal increase in consumption of red meat causes an equal increase in risk of bowel cancer. This condition is also known as "no interaction effect".
Is this plausible? Medical experts can tell us. The point for consumers is to recognize that an underlying assumption has been used when issuing this guideline.
It is very instructive to examine the possible "impact" of this change in guidelines.
From the Cancer Research UK website linked above, we learn that there are about 30 million men in the UK, so about 12-13 million consume more than 90 grams of red meat a day, the level of consumption which the Bloomberg article identified as posing significantly higher bowel-cancer risk.
However, only 20,000 cases of such cancer is reported per year among men (with about 16,500 deaths across both genders). So, it is a certainty that almost all of the 12 million men who consume a risky amount of red meat will not develop bowel cancer even if they don't change their eating habit. (It's a kind of gamble who among these millions will be the unlucky ones.)
So, how many people might benefit from this new guideline?
Among men, we know the upper limit is 20,000 per year. The reality is much smaller than that. Firstly, red-meat consumption is only one of many risk factors for bowel cancer and so a portion of the 20,000 are surely men who do not currently consume more than 90 grams per day; for these men, the new guideline is irrelevant. Secondly, some of those who reduce their consumption now will still get bowel cancer because the statistics is only promising reduced odds, not certainty.
Thirdly, the maximum benefit assumes perfect compliance. In practice, not all men will choose to follow the guideline. In fact, the most at-risk people are the most likely to be eating more than the optimal amount of red meat, and they are precisely the ones who are the least likely to comply with the guideline. The overall risk of bowel cancer is very low which also deters compliance. So, a big haircut on the potential effect for non-compliance.
That may not leave us with many saved lives. And the possible effect on women is still smaller because the number of cases for women is smaller.
Nobody says curing people is easy!