How To Show The Relative Effects Of A Graph From A Time Series To A Time-Series

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1 2. GRAPHICAL PRESENTATION OF DATA 2.1 Why do we need to present data graphically? If we take as an example a table of data 1 : x y and ask what features the data show, it is quite difficult to see from the table, but much clearer when the data are plotted as y v x: x Figure From "The Visual Display of Quantitative Information" by ER Tufte, Graphics Press CE8: Quantitative Research Methods Ian Rudy 1

2 Another example is that of a time-series such as the unemployment level in Great Britain 1 : Year Unemployed(thousands) Year Figure 2.2: Unemployed in Great Britain (Source: Annual Abstract of Statistics, 1987 edition) From which the table conveys the sense of increase, but does not convey the drop in the late 197's. 1 From "Annual Abstract of Statistics (1987 edition)'', Central Statistical Office, HMSO. Values are for June in the year shown. CE8: Quantitative Research Methods Ian Rudy 2

3 2.2 Different ways of presenting data The technique used to present the data depends on: * the data itself * what you are trying to show There are a range of techniques (graph types) to choose from, but after some practice it is fairly easy to pick the most appropriate type. Also, modern spreadsheet programs allow one to try several different types in rapid succession and pick the best Bar charts An example of a bar chart is shown below. Note how the table fails to convey the relative magnitudes of the data: the bar chart overleaf clearly shows how there are four countries with large populations and the rest have much smaller populations. The information is conveyed in the heights of the bars, which can be easily compared. Country Population at July 1995 (millions) France 58.1 Belgium 1.1 Germany 81.3 Italy 58.3 Denmark 5.2 Britain 58.3 Netherlands 15.5 CE8: Quantitative Research Methods Ian Rudy 3

4 France Belgium Germany Italy Denmark UK Netherlands Country Figure 2.3: Populations in EEC Countries (Source: CIA World FactBook: Note that the axes have been properly labelled and a title has been supplied, so that it is clear what information the graph is presenting. It is also good practice to include a source to indicate that the data is secondary data. A variant on this simple bar chart is the stacked bar chart, where several quantities can be shown at once: Year Nuclear Oil Coal Figure 2.4: Energy Use (Source: Energy Trends, Department of Energy) If one can use colour, then distinguishing the separate quantities (variables) is no problem. But colour printing is still relatively expensive, and colour photocopies are not common, so that even if you produce a chart in colour someone may wreck it by making a monochrome photocopy. Hence CE8: Quantitative Research Methods Ian Rudy 4

5 in order to avoid these problems, you may prefer to use hatching (parallel lines, such as in Figure 2.5). Even this is not without its problems, because of Moiré effects, when the hatched area appears to shimmer and be distorted. The graph below shows an example of this when too many variables are included. This type of graph is usually known in spreadsheets as an "area chart": it is similar to a bar chart but has the bars for separate time periods joined. Figure 2.5: An Area Chart with Many Variables, Showing Moiré Effects Time series plots These are used when we have values of a variable for several years, months, etc. The unemployment graph above is an example. Note that with time series, individuals are often hoping to prove something and can often do so by choosing an appropriate aspect ratio. The following two graphs (using the same data) demonstrate the effect of this. There is no agreed rule about what to do here. Some would argue that the line on the graph ought to be at roughly 45 to the axes, others would argue that it ought to run from one corner to the opposite diagonal corner, with the horizontal axis about times longer than the vertical. (The ratio of lengths for an A4 page is about 1.4; for a US letter page, it is about 1.3.) Two of the arguments for having the graph more wide than it is tall are that the eye is better at picking out vertical than horizontal fluctuations (because humans are used to watching the horizon) and that a wide graph allows text (such as the axes labels) to be displayed horizontally rather than vertically. 1 From "The Visual Display of Quantitative Information" by ER Tufte, Graphics Press CE8: Quantitative Research Methods Ian Rudy 5

6 Year Year Figure 2.6: Effect of differing aspect ratios Another trick you should look out for is the broken y-axis: Look at our Increase in Sales!! Jan Apr Jul Oct Month CE8: Quantitative Research Methods Ian Rudy 6

7 Jan Apr Jul Oct Month Figure 2.7: Broken y-axis to exaggerate effect of change When drawing this kind of graph using a computer, one is nowadays faced with a huge number of different styles for the lines: single lines, broken lines, just symbols (no lines), symbols and lines: don't over-complicate the graph so that its main message is obscured. Also, be careful not to use too many variables on one graph xy plots xy-plots are similar to time series plots, but now the x-axis is used to display a variable other than time. These are used to show relationships between two variables. Figure 2.1 is an example. xy-plots are more common in science/technology areas, less common in management. Symbols tend to be used more than lines, as often there is no particular connection between successive results in a table. We will see some xy plots when looking at correlation analysis, later on in the year. CE8: Quantitative Research Methods Ian Rudy 7

8 2.2.4 pie charts Nearly everyone learns about pie charts at school. They can be useful, but beware of them: they can convey an idea of only the relative importance of several factors, and the slices are more difficult to compare than rectangles: Engine/Cylinder Cooling 63% Tyre Drag 6% Air Drag 6% Brakes/Idling 7.5% Accessories/ Transmission/Axles 5.5% Air Pumping/Engine Friction 14% Figure 2.8: Areas of Energy Loss in a Typical Car (Source: Bending and Eden "UK Energy" (1984)) CE8: Quantitative Research Methods Ian Rudy 8

9 An alternative to the standard pie chart, which I think is better, is a component bar chart: Industry Domestic Other final uses Transport Figure 2.9: Energy Consumption by Final User in the UK Pie charts (and other types of chart as well) are vulnerable to the current fashion for 3D effects. For example, Figure 2.8 would be shown as: Engine/Cylinder Cooling 63% Tyre Drag 6% Air Drag 6% Brakes/Idling 7.5% Accessories/ Transmission/Axles 5.5% Air Pumping/Engine Friction 14% Figure 2.1: 3D Version of Figure 2.8 CE8: Quantitative Research Methods Ian Rudy 9

10 In most cases the 3D effect is unnecessary and may actually mislead the viewer because the area of the resulting block is not proportional to the number being represented. 2.3 Good and bad charts There is considerable scope for deceiving, either accidentally or deliberately, with graphs. The next few sheets show some examples of this. Tufte's book contains many examples of good and bad graphs, and he introduces the idea of a lie factor which measures just how misleading a graph is: lie factor = size of effect conveyed by a graph. true size of the effect Tufte also considers the concept of the data-ink ratio. This is: data-ink ratio = data - ink total ink used to print the graphic. Sometimes, the data-ink ratio is very low because the graph has been decorated with unnecessary ink. At the other end of the honesty range, it is possible to use graphs (or graphics in general) to convey information in an extraordinarily effective way. The last sheet of the set gives another example from Tufte's book, which he says "may well be the best statistical graphic ever drawn". It depicts the fate of Napoleon's army in Russia: in one sense the graphic is a map of Russia, showing the path of the army. Paths of retreat are indicated by solid black bands. The width of the band is proportional to the number of troops left in the army. The graph manages simultaneously to tell us about the paths the army took, and the loss of life as time went on. Notice the extraordinary large losses of troops at some of the river crossings. CE8: Quantitative Research Methods Ian Rudy 1

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15 2.4 The use of histograms to convey continuous data Histograms are similar to bar charts: the difference is that in a histogram, the x-axis variable is a continuous numerical variable, such as income, price, or population. That is, it is allowed (in principle) to take any numerical value, rather than just one of a small number of numerical or text values. When producing a graph for this kind of data (for example earnings in the UK), one would want to group people into one of a number of earnings classes: Earnings ( ') Number of workers (millions) If we divide each of the 5-6, 6-7 and 7-8 boxes in two, so that the boxes are the same width, then there is no problem in producing a histogram: Earnings ( ') Figure 2.11: Histogram (equal box widths) But when the width of the boxes does vary, a problem arises. Consider what happens if we group together the workers in the 4, to 8, categories and plot a new graph as follows: CE8: Quantitative Research Methods Ian Rudy 15

16 Earnings ( ') Figure 2.12: Incorrectly Drawn Histogram (unequal box widths) The graph is clearly misleading: it is useless at conveying the relative fractions of workers in each earnings range. The moral of this is that you should not use absolute numbers for the y-axis unless the boxes are all of the same width. If the boxes are not all of the same width, then we should adjust their height so that the number of people corresponding to each box is conveyed by its area and not its height: Earnings ( ') Figure 2.13: Correctly Drawn Histogram (unequal box widths) This idea will be very important when we consider distributions, later in the course. CE8: Quantitative Research Methods Ian Rudy 16