Jianing Min's MATH 6820 Blog

Which Log?

Collect some measurement that has changed exponentially over time.  (You should have at least 10 measurements.)

According to this requirement I collect data from world dataBank, World Development Indicators(WDI), the link is http://databank.worldbank.org/ddp/home.do

In this blog, I construct two graph. first one is a display with log base 10 measurement against  time and the second one is a display with log base 2 measurement against time.  In the following two graph you will see both graphes have log measurement on left vertical scale and actual value on the right vertical scale.

The first graph is with measurement in log10, which has a range from 2.75 to 3.5 on the left vertical scale. However, the second graph is with measurement in log2, which has a range from 9.00 to 12.00 on the left vertical scale. By the nature of Logarithms method, different base does not change the pattern of points but changes only the values at the tick marks because the logarithm of one base is just a constant times the logarithm of another base. The overall trend of this graph is that the GDP of China keep growing as time goes by from 1992 to 2011. Also, the line segments connecting successive plotting symbols are banked to 45 degree in both graphs.

The choice of the base depends on the range of the data values that need to be visually compared. Since my original value has measurement unit  in “Billion” , the largest data only has 3 powers of 10 and the smallest data has 2 powers of 10. In this case, it is inevitable that equally spaced tick marks for log base 10 will involve small fractional powers of 10, as my first graph showed. It is difficult to deal with such fractional powers. In this situation, it is reasonable to convert to log base 2 as the second graph showed. Although the second graph has fractions as well, they are comparably small to the fraction in the first graph. It is easier to deal with powers of 2 than small fractional powers of 1o. Moreover, both graphes has been processed by “statistical scientist’s trick”, which take the original units to be in billion. For log base 2, the original data will make the scale range from 19.0 to 21.5. However, now it only range from 9 to 12.5.

In sum, I think  the second graph with log 2 base is the best choice. The principle of choosing log 2 or log 10 is based on the situation and visual comparison. When the data range though two or fewer powers of 10, the log base 10 scale is not as informative since we must deal with fractional powers of 10. When data go through a small number of powers of 10, log base 2 often provides a useful scale, as I showed in the bottom graph.

This entry was posted on Friday, February 8th, 2013 at 11:26 pm and is filed under Uncategorized and You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.