Tuesday, May 15, 2012

Circular Reasoning

Former student Gabe Krupa, remembering a lecture I gave on the topic of misleading graphs and statistics, alerted me to this graphic showing the fall in Yahoo’s enterprise value. (I didn’t know the term enterprise value, but apparently it’s similar to market capitalization but with a few tweaks.)

As you can see, from 2006 to 2012, Yahoo’s enterprise value fell from $54.9 billion to $17.26 billion. The current value is just under a third of its value six years ago. But that big circle looks a lot more than three times larger than the small circle. In fact, it’s about ten times larger.

As Gabe said in his email to me, the creators of this graphic used a 2/3 reduction in the radius of the circle when they should have used a 2/3 reduction in the area. Since the area of a circle increases with the square of the radius, the graphic drastically overstates the difference in value. (To be more specific, the small circle’s radius is about 31.4% of the big circle’s radius. The square of 0.314 is 0.098, meaning the small circle’s area is 9.8% of the big circle’s area.)

This kind of error was highlighted in Darrell Huff’s How to Lie With Statistics, first published in 1954. The bad news is that media sources still make the same error, whether purposely or accidentally, almost 60 years later. The good news is that apparently some students really do remember what they learned in class, even years later. My thanks to Gabe for bringing this example to my attention six years after taking my course.


Philip Whitman said...

Pie/circle graphs are always confusing, even if you do it right. A chimney graph would be much better for showing a drop in value or number or whatever.

Manjit said...

If students are upgraded to find the area of circle sections by dividing angle of 360 degrees according to section, then pie charts are no more harder for them. This makes them very comfortable and proud so that they get inspiration to do difficult math.

ligneus said...

So if they'd reduced the diameter by 2/3 instead of the radius, that would be OK? [Probably not, the old brain doesn't feel like getting into the math right now!]

Glen Whitman said...

No, because the diameter is just twice the radius, so reducing the diameter by 2/3 is the same as reducing the radius by 2/3. What you want is to reduce the area by about 2/3. Since Area = pi*r^2, you need to reduce the radius by just enough that the *squared* radius decreases by about 2/3. It turns out that reducing the radius to about 0.56 of its original value (which squared is 0.31, the actual ratio we need) would do the trick.