Statistics don't lie,
but their interpretation and presentation can.
You've probably heard the cliché about the three kinds of lies: Lies, damned lies, and statistics. Statistics don't really lie, they have no ability to do so, they're just data. However, depending on how statistics are interpreted and presented they can mislead. If done deliberately, it would be a sort of lie, but I think a lot of people don't see the flaws of interpretation they use themselves.
Part 1. Something from Nothing
Our first examples concern two populations of green beings and purple beings on an imaginary planet we'll call Envia. The purples are on average richer than the greens. In both groups, most tend to fall in the average wealth ranges with fewer very poor and very rich beings at the extremes. The distributions of each expressed in chart form:
On planet Envia any being with 9k or more in the bank is considered rich. (9k might not seem like wealth to you or me, but on Envia the cost of living is low.) As a percentage of their population there are more rich purples (13%) than rich greens (4%). Some will say there is a 9 point gap in the wealth status between the two. Some would say as a percentage the purple rich outnumber the greens by 13 to 4, or a little over 3 to 1.
Now let's say there's a period of improved productivity and the price of everything goes down. It takes less money now to be rich. So we change the threshold of rich status from 9k to 7k. Here are the results:
Now 50% of the purples are rich and 30% of the greens. As a percentage the purple rich now outnumber the greens by 5 to 3, less than 2 to 1, contrasted to the previous 13 to 4 which is more than 3 to 1. Congratulations, you've narrowed the gap. The greens are happy and proud.
On the other hand, comparing percentages you could say there is now a 20 point gap (50% to 30%) contrasted to the previous 9 points (13% to 4%), an 11 point increase. Sorry, you've actually widened the gap. The greens are unhappy and upset.
In actuality nothing significant has changed. By changing the threshold of rich you've altered the relationships of the mathematical artifacts of the two charts without changing the relationship of the two populations at all.
In both cases the comparisons between percentages above or below a threshold is a mathematical artifact of the chart, the real relationship is in the unchanged totality of the data. In this case 18% of greens are below the wealth overlap of the populations and 18% of purples are above. The relative percentages in each collumn are the same, the relative differences are the same. These don't change no matter where you place the threshold of rich.
Changing a threshold may well have a greater impact on one group rather than another, but it doesn't improve the overall performance of the groups in comparison to each other. Keep this in mind when someone claims a narrowing or widening of some achievement gap in wealth, education or the like. It might all be nothing more than mathematical artifacts or misinterpretation of the data.
Part 2. Point of View
Our next examples concerns two populations of green beings and purple beings on an imaginary planet we'll call Cannibalia. The Greens outnumber the Purples by 9 to one so in a sample population of 1,000 beings there are 900 Greens and 100 Purples. One percent of each population are cannibals so in a sample population of 1,000 beings there are 9 Green cannibals and one Purple cannibal.
Let's say in a year a cannibal of either color eats 10 beings a year selected purely at random, which means a cannibal of either color will eat 9 Greens and one Purple in a year. Out of a sample of 1,000 beings there will be 90 Greens eaten and 10 Purples.
At this point there are true things you can say about this situation. Depending on how you want to slant it you can make it sound good or bad for a particular color group:
1. A Purple cannibal eats 9 times as many Greens as a Green cannibal eats Purples.
2. Though outnumbered 9 to one, Purple cannibals eat as many Greens (9) as Green cannibals eats Purples (9).
3. There are 9 times as many Green cannibals as Purple cannibals.
4. A Purple is nine times more likely to be eaten by a Green (9 in 100 or 9%) as a Green is to be eaten by a Purple (9 in 900 or 1%).
Points one and two make it seem Purple cannibals are more anti-Green than Greens are anti-Purple. Statement three suggests Greens are more cannibalistic. Scenario four implies Greens are more anti-Purple than Purples are anti-Green. But both have the same propensity for cannibalism, one percent, and both select victims purely at random. The seeming disparities are not a reflection of differing cannibalistic tendencies but are due to different population sizes.
What's particularly interesting is while the Greens outnumber the Purples 9 to 1, the inter-color cannibalism rate is the same. Greens eat nine Purples and Purples eat nine Greens. When selecting victims at random this will always be the case no matter how different the populations sizes, from 50/50 to a thousand to one or more.
Instead of being equal opportunity cannibals, say cannibals only eat beings of the other color. One percent of each color are cannibals eating 10 beings a year. In a sample population of 1,000 beings the cannibalization looks like this:
At this point there are true things you can say about this situation. Depending on how you want to slant it you can make it sound fair, bad or worse for a particular color group:
1. Greens and Purples are equally likely to be cannibals (1%) and eat an equal number of beings each per year (10).
2. Green cannibals eats 9 times as many Purples as Purple cannibals eat Greens (90 to 10).
3. A Purple is almost 90 times more likely to be eaten by a Green (90 out of 100 or 90%) than a Green is to be eaten by a Purple (10 out of 900 or 1.1%).
Point one suggests cannibalism is equal both ways. Point two makes Greens seem worse than Purples. Point three makes Greens out to be very much worse. The seeming disparities are not a reflection of differing cannibalistic tendencies but are due to different population sizes.
As you can see, even accurate statistics can mislead. If you interpret and present the data in a particular way you can make it scary or not. You can claim success or failure from the same set of numbers. You can claim progress or regression when there's been no change at all. You can make things out to be fair, bad or horrible with the same data. For politicians and rabble-rousers it's a dream come true, statistical proof of whatever they want.
The numbers don't always mean what they are perported to show. The proof may be in the pudding, but be on your toes, the books might be cooked.