sums will set you free
how to teach your child numbers arithmetic mathematics
understanding graphs and charts
There are all manner of ways in which you can illustrate and present data in order to make it easier for yourself and others to understand.The terms ‘charts’ and ‘graphs’ are often used interchangeably or sloppily, but some people like to reserve the term ‘graphs’ for diagrams with numbered axes and for continuous scales.
The choices of graphs and charts, and the scales used on those illustrations, mean it is very easy to mislead and confuse. So graphs and charts are widely popular with all manner of con artists and swindlers.
You see this number line (from minus and zero, dealing with no-thing and less than nothing!) is counting forward and back from zero.
And from fractions, decimals and percentages 2, you can see numbers counting up from zero in two different directions. As you see in this second diagram, the horizontal line, or axis, is now marked with an X and the vertical axis is now marked with a Y.
We can plot values for different equations and draw the lines, whether straight or curved, involved.
In this last set of graphs, note that each line crosses the y axis at y = 1 when n = 0.
Pie charts are the most basic type of graph, used to illustrate disparate collections of objects, such as how many apples, oranges and children are there. These are collections separated by quality.
The complete pie chart holds 100% of the items.
Thus, if there are 12 children, 5 apples and 7 oranges,
the whole pie is made up of 12 + 5 + 7, or 24, objects.
A bar chart has broad lines, or bars, whose lengths are proportional to the values that they represent. The greater the height or length, the greater the value. The bars can go horizontally or vertically, and can be drawn to appear three-dimensional.
In comparison with the planets, the radius of the Sun is 685,000 km, almost ten times that of Jupiter.
When only a small portion of one or both of the scales are used, often the axis/axes are ‘abbreviated’ by omitting part of the scale, either near the zero mark or further along the axis. This shown by inserting a zigzag,
or a break to mark the portion of the scale that is omitted.
Using different graph paper, it is possible to plot special graphs, such as ones with a logarithmic scale. here is such a graph (click on the image to go to a full-size version at Is Intelligence Distributed Normally? By Cyril Burt, 1963).
To show how uncertain data is graphed, we are using graphs that illustrate global temperatures. Measuring the whole world’s temperature is necessarily problematic, with temperatures being taken by many people over centuries, in many places - sea, land and air - with differing instruments and methods. Standardising such data can only be achieved to a level of confidence, a probability of accuracy, as is illustrated just below.
In the next graph, you will notice that each year’s temperature bar estimate [in red] is overlaid by what looks like a long letter ‘I’ [in black]. That usually indicates that the probability is that we have a 95% ‘confidence’ that the real temperature fell between the two cross bars of the ‘I’. Notice that as you go further back in time, the range of uncertainty (the vertical bar of the ‘I’) increases for obvious reasons.
The blue line is a typical moving average, smoothing out natural variations.
Sometimes, you will see such ranges of uncertainty indicated by ‘fans’, often giving more than one uncertainty level. This is especially common in graphs giving future forecasts in complex domains, like economics and climatology.
Notice carefully how the line narrows to a single point at about the year 2000. This is our present day best estimate. Notice also how the fans spread as they go both forwards and backwards [in grey] in time, with our past estimate and with our future forecasts becoming more and more uncertain the more distant the time.
Here is another type of fan graph. Graphs on economic forecasting are particularly interesting because the numbers are ever changing their real meaning as governments manipulate inflation. The forecasts are commonly heavily politically loaded. For example, governments facing an election, and companies trying to sell shares, regularly project growth far above any realistic level, and other parties make self-interested ‘forecasts’. Thus, it is hardly surprising that a great number of economic forecasts turn out to be false.
Note that many graphs either do not mark that a graph scale does not start from zero, or do not do so obviously. This makes it easy to misinterpret data being provided on such a graph.
Here are two graphs that illustrate how graphs can be used to mislead. The graph on the left is drawn confusingly, even though displayed on a supposedly reputable web-site. The graph on the right shows another way of showing closely related data to the left graph, but with an extra point added before and after, thus presenting a very different picture. An explanation of the changes.
Notice that the graph to the right, the left-hand, y-axis is stretched relative to the equivalent axis on the left-hand graph. This has the visual effect of making the changes look much bigger on the right-hand graph, or much smaller on the left-hand graph.
What matters is whether a graph is clear and does not mislead (of course, it is easy to be misled if you do not have much experience of reading graphs). Temperatures below 0°C do not tend to interest humans most of the time, unless they are Eskimoes/Inuit. It is not misleading to concentrate on temperature variations when considering global warming, though maybe it is misleading when discussing the temperature of the sun, or when making ice cubes - context is always relevant.
Note that the left graph is illustrating global temperatures, a phenomenon which has short-term variations and a long-term trend. By showing data for just eight years, instead of centuries, a very misleading notion is being given of global temperature change. In fact, the long-term trend is that global temperatures are increasing, as can be seen in this graph above.
[Note: GMG = Guardian Media Group]
GMG plc’s revenues as depicted in their annual report:
Observe how GMG have put the latest year at the top of the graph, when it is usual to ‘read’ downwards.
And here is the same data re-arranged by Guido Fawkes:
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