Environment: understanding atmospheric and ocean flows

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# 5.1.1 Gradient of a straight-line graph

In this study note you will learn how to calculate the gradient of a straight-line graph.

## Study note: Gradient of a straight-line graph

Figure 28 shows a general trend of Arctic sea ice decreasing with time, though the annual fluctuations can be quite large. To measure this trend, a ‘best fit’ line is constructed as shown on the graph. This is drawn so that approximately the same number of data points lie above and below the line, but where there are significant fluctuations (as here) it may not pass through many or indeed any of these original points.

The average rate of change of ice extent can be deduced by measuring the slope or gradient of this straight line on the graph. To do this, take two convenient points on the line and read off the values on each axis. These points should ideally be widely spaced (to improve accuracy) and will not necessarily correspond to original data points.

In this example, the years 1980 and 2009 have been chosen, and the corresponding values on the vertical scale for the ice extent (according to the best fit line) are 7.8 and 5.3 million square kilometres.

So the time interval is (2009 – 1980) years = 29 years.

Change of ice cover is (5.3 – 7.8) million km2 = –2.5 million km2.

(Note: the minus sign denotes a negative change, in other words a decrease.)

This is easier to interpret if you convert millions to thousands (multiply by 1000), giving a mean rate of decrease of 90 000 km2 per year.

This is the standard method for calculating the gradient of any straight-line graph, often summarised by the formula

where the rise and the run are measured respectively from the change in values on the vertical and horizontal axis scales of the graph for the two chosen points.

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