Scientists construct models of the systems or phenomena they are studying in order to make predictions about future events or to reconstruct events which may have occurred long ago. One of the simplest modeling activities is to curve-fit a graph. In curve-fitting, an equation is developed which will duplicate the data curve. Once this equation is found, predicting a future point on the curve is as easy as plugging in the new x-value and calculating the new y-value.
Activity: We would like to know what the atmospheric concentration of CO2 will be in the years 2025, 2050, and 2100. We are assuming that the current conditions (emission levels of CO2, etc.) will not change. To do this, you will curve-fit the Mauna Loa CO2 plot shown on the first page of this section. The data for this graph can be downloaded in Excel 5.0 spreadsheet format or as a text file.
When a problem is difficult, scientists often make simplifications or approximations in order to make their work easier; divide and conquer. For the CO2 curve, let's make a simplification. The wiggle in the curve seems to repeat itself every year, so let's assume it will continue to do this and ignore it. To find out more about the wiggle, see Seasonal Vegetation Changes activity. Plot the yearly average CO2 value only.
There are many ways to fit data. Generally, the x-values are manipulated in some way, and the data is replotted to see if it makes a straight line. For instance, all the x-values might be squared. If this manipulation gives a straight line, then the equation for this data would be y = mx2 + b. The value m would be the slope of the straight line and b would be the y-intercept.
There are also formulas which can be used to determine which mathematical function most closely matches your data. This is what spreadsheets do when you let them automatically fit a curve for you. Ask your teacher which method to use.
The data will be easier to work with if you renumber the years. Let 1958 be year zero, 1959 year 1, 1960 year 2, etc. The year 2000 would then be 41. Why is this easier? Because 22 = 4 and 19602 = 3841600.
Because there are thousands of mathematical equations to choose from, there is no one right solution to this problem. Some equations will fit the curve better than others, but there are several equations which work pretty much equally well. How do you know which one to use? In science, we always pick the easiest solution or explanation. Never make your work harder than you have too!
When you think you have found an equation that fits the data, be sure to test it. Select several years for which we know the CO2 value. Plug the values into your equation and see how close they are to the actual values.
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