Sunday, 13 January 2013

Arithmetic vs logarithmic: Difference between charts plotted using these two scales

Narendra Nathan, ET Bureau Aug 27, 2012,

When data is plotted as a chart, it can be done using two types of scales—arithmetic or semi-logarithmic. The difference in scale can completely alter the shape of the chart even though it is plotted using the same set of data. Consider the three charts given below, which use theSensex data since its inception in 1979 till now. But why does one need to use the semi-logarithmic scale when arithmetic scale is commonly used for plotting charts. This is in order to overcome the inherent weakness of the arithmetic charts.

Arithmetic charts

In arithmetic or linear charts, both X and Y axes scales are plotted at an equal distance. For instance, the Sensex movement from 15,000 to 16,000, that is, an increase of 1,000 points, is treated as equal to the Sensex movement from 16,000 to 17,000, which is another 1,000 points.

This works fine when the data range is small, but will distort the picture when the range is big. Consider the Sensex movement from 20,000 to 21,000, which was a mere 5% increase, but the movement of the first 1,000 points in the Sensex, that is, from 100 to 1,100, was a whopping 1,000% increase. By treating them as equal, the arithmetic chart distorts the picture. This explains why it seems as if the Sensex was almost flat for the first 10 years of its existence in the arithmetic chart.
Logarithmic charts
Logarithmic charts are commonly used in science and engineering when you need the data to be displayed accurately. This is also a necessity when the data that needs to be plotted varies widely. In such charts, the logarithm of the data value (Sensex in the given example) is used as a base to fix the gaps between each data points on the Y axis. This process makes sure that the percentage increase between two data values is depicted clearly. To understand this in a better manner, consider the logarithmic charts given below. Note that the gap between 100 and 200 (100% increase) is equal to the gap between 200 and 400 (next 100% increase). The same gap is maintained for the Sensex increase from 1,000 to 2,000 or from 10,000 to 20,000. Semi-logarithmic charts
In logarithmic charts, both the X and Y axes are plotted using the logarithmic scale. Since there is no possibility of distortion in the X axis (where the date range is plotted), we can continue to use the arithmetic scale while plotting the share price data. In other words, the logarithmic scale is used only along one axis, that is, the Y axis, and therefore, these charts are called semi-log charts.
The semi-logarithmic charts can be of immense help while plotting long-term charts, or when the price points show significant volatility even while plotting short-term charts. This is because the chart patterns will appear as more clear in semi-logarithmic scale charts. For example, the very long-term uptrend line in the Sensex is clearly visible in the semi-logarithmic chart (see Semi-logarithmic scale with trendline), not in the arithmetic chart. Similarly, the Sensex was constrained in a slightly upward moving channel for 13 years in the middle, that is, between 1992 and 2005, and this is clearly visible only in the semi-logarithmic chart (see Semi-logarithmic scale with channel). One can plot the charts in the semi-logarithmic scale to easily identify several other chart patterns, some of which we shall explain in the coming weeks.

Semilogarithmic-scale line graphs

If we use a logarithmic scale on the y-axis and if the x-axis remains the same (arithmetic scale), we create a semi-logarithmic scale line graph. With a logarithmic scale on the y-axis we represent the relative change of y over time rather than its absolute change over time. Semi-logarithmic scale line graphs are used to present and interpret rates of change over time rather than magnitude of change. They also allow showing very different magnitudes and ranges of rates between two lines (e.g. high incidence and low mortality rates for the same disease).

Semi-logarithmic scale paper:
  • On the y-axis, intervals are logarithmic and no longer arithmetic.
  • There are several cycles of tick marks on the y-axis. Each corresponds to an equal distance on the y-axis.
  • The values of one cycle are 10 times greater than the values of the previous cycle.
  • Within a cycle the 10 tick marks are not equally distant (distance from 2 to 3 is different than distance from 3 to 4). Their progression is geometric, not arithmetic.
  • The y axis can cover a large range of y values.
The following characteristics are noteworthy:
  • The slope of the line indicates the rate of change (the relative change) of y over time.
  • A horizontal straight line indicates no change.
  • An upward or downward straight line slope indicates a constant rate of increase or decrease in the measured indicator (e.g. rate) over time.
  • Two parallel lines indicate similar rate of change over time.

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