### When A Picture is Worth √1000 Words

This morning @WSJ posted a link to the story about Microsoft’s announcement of its plans to lay off 18,000 employees. This picture (as captured on my iPhone)...

...accompanied the tweet, which is presumably available through their paywall link.

While I’m really sorry to hear about the Microsoft employees who will be losing their jobs, I am simply outraged at the miscommunication in the pictured graph. (This news appeared to me first on Twitter, and the seemingly typical response on Twitter is hyperbolic outrage.)

Here’s the problem as I see it: the graph communicates one-dimensional information with two-dimensional images. By doing so, it distorts the actual intensity of the information the reporters are supposed to be conveying in an unbiased manner. In fact, it makes the relationships discussed appear much less dramatic than it actually is.

For example, look at Microsoft’s (MSFT) revenue per employee compared to Apple’s (AAPL). WSJ reports MSFT is $786,400/person; APPL, $2,128,400. The former is 37% of the latter. But for some reason, WSJ communicates the intensity with an area, a two-dimensional measure, whereas intensity is one-dimensional. Our eyes are pulled to view the length of the side of the square as a proxy for the measurement being communicated. The sides of the squares are proportionally equal to √(786,400) and √(2,128,400); therefore, the sides of the squares visually communicate the ratio of the productivity of MSFT:AAPL as 61%. In other words, the chart visually overstates the relative productivity of MSFT's employees compared to that of AAPL's by a factor of 1.62.

If the numbers are confusing there, consider this simpler example. The speed of your car as measured by your speedometer is an intensity. It’s one dimensional. It tells you how many miles (or kilometers, if you’re from most anywhere else outside the US) you can cover in one hour if your car maintains a constant speed. Your speedometer aptly uses a needle to point to the current intensity as a single number. It does not use a square area to communicate your speed. If it did, 60 miles per hour would look 1.41 times faster than 30 miles per hour instead of the actual 2 times faster that it really is. The reason for this is that the the sides of the squares used to display speed would have to be proportional to the square roots of the speed. The square roots of 60 and 30 are 7.75 and 5.48, respectively.

For your own personal edification, I have corrected the WSJ graph here:

Do you see, now, how much more dramatic the AAPL employees' productivity is over that of MSFT's?

This may not seem like a big deal to you at the moment, but consider how much quantitative information we communicate graphically. The reason is that, as the cliché goes, a picture is figuratively worth a thousand words. I firmly believe graphical displays of information are powerful methods of communication, and a large part of my professional practice revolves around accurately and succinctly communicating complex analysis in a manner that decision makers can easily consume and digest. But I’m also keenly aware of how analyst and reporters often miscommunicate important information via visual displays, either by design, inexperience, or by trying to be too clever. I see these transgressions all the time in the analyses I’m asked to audit.

The way we communicate information is not just a matter of style for business reporters. We often make prodigious decisions based on information. If information is communicated in a way that distorts the underlying relationships involved, we risk making serious misallocations of scarce resources. This affects every aspect of the nature of our wealth - money, time, and quality of life. The way we communicate information bears fiduciary responsibilities.

For discussion sake I ask,

- How often have you seen, and maybe even been victimized by, graphical information that miscommunicates important underlying relationships and patterns?
- How often have you possibly incorporated ineffective means of graphically communicating important information? (Pie charts, anyone?)

If you want to learn more about the best ways to communicate through the graphical display of quantitative information, I highly recommend these online resources as a starting point:

Labels: bias, data analytics, visual display of quantitative information

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