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The last couple of weeks I have been reading Charles Murray‘s new book Coming Apart in my spare time. It is a worthwhile book to read, written by a careful researcher. One thing that struck me as strange for such a well researched and informative book was the use of graphs starting at some number such as 0% and rising to say 5% and comparing it to graphs starting at say 0% and running to 100%. A serious problem arises because a change from 1% to 5% is a 400 percent increase but a change from 95% to 99% is only a 4% increase.

Murray does discuss graphic representation and how the way the numbers on the side and bottom scales affect how the results appear and are therefore interpreted. He is careful to label his scales to be comprehensible but in the old fashioned linear way. Perhaps he isn’t at fault for this because I spent a couple of hours looking for visual scales which would present percentage data in a way that made better visual sense. Perhaps they don’t exist, so I was compelled to create one and perhaps a whole new way for the visualization of data.In the old style percentage chart seen above from page 160, Murray presents a standard and well done linear percentage-style graphic view of the information but it misrepresents the data. We have all seen thousands of charts like this, but they are all flawed. The reason is that in some sense the data is logarithmic and when going from low percentages like 1% to mid-range percentages like 50%, the data goes through a fundamental change of character. I made the chart below, which would correct that distortion. It consists of two logarithmic charts running in opposite directions and meeting at 50% midpoint with their end points set at 1% and 99% respectively. The extremes are not at 0% and 100% because logarithmic graphs run to infinity. The chart seen above could be extended indefinitely by adding more logarithmic cycles at each end. When a single cycle is  added to each end it would make the chart run from 0.1% to 99.9%. [Click here 0.1% to 99.9%. for a more recent and better drawn chart.]

A log scale running from 1% to 99%

A hand compiled log scale running from 1% to 99% with 50% at the midpoint.

On Murray’s scale, at the top, the value of about 2% in 1955 shows a barely perceptible change to 4% in 1965 and yet that represents a 100% increase in the value being graphed. If the linear chart were scaled from 0% to 3% it would show a spectacular rise. It is shown from 0% to 30% and barely shows the data, and if it were scaled from 0% to 100% the data would be near invisible. On page 161 Murray writes, …

“The shaded area contains the decades we are studying 1960-2010, when the percentage of nonmarital births rose steeply throughout. But before that, hardly anything had changed since the first numbers were collected in 1917. Studies of the white family in earlier eras indicate that the line hugging the bottom of the graph 1917 to 1960 would have been flat all the way back to the Revolution. White children were conceived outside marriage at varying rates in different social classes, but hardly ever born outside marriage in any class.”

A view with lines for showing relationships more clearly

A view with extra value lines drawn in for showing the relationships more clearly.

It was hard to see the difference between 1955 and 1965 so I drew the thin lines on the chart above to make the substantial differences more visible, but they still aren’t very clear.

Charles Murray - Coming Apart page 160 redrawn

Charles Murray - Coming Apart - page 160 chart redrawn using a log-percentage scale

[Update: A newer and better drawn chart is found at - 0.1% to 99.9%.]
This is the same data as presented by Murray above but redrawn using this new Log-Percentage scale and what it shows is that the processes he was writing about were in progress long before 1962 where he begins his book. The doubling of effect between 1920 and 1935 is almost invisible on his chart but clearly shown to be just as steep then as between 1960 and 2010, which is what his book is about. Therefore,   Murry’s own data from his chart above directly contradicts the fundamental premise of the book – that the problems he writes of began in the early 1960s. Perhaps they began much earlier.

What this demonstrates is not that his data or methods were wrong in any way, at least by the old standards of display of data, but that this new method of log-percentage graphic display shows the information more clearly and therefore it is better. Projections into the future on the old style linear chart don’t make sense, but on this new-style log-percentage chart a straight line projection does make sense and we could predict the data line will hit 50% in 2020 and 60% in 2030 unless some substantial new force comes into play.

The log-percentage scale adds clarity to the presentation of data.

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