Using the data above Height range: 6 1to 74 inches Weight range: 95 to 205


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1 Plotting When plotting data, ou will normall be using two numbers, one for the coordinate, another for the coordinate. In some cases, like the first assignment, ou ma have onl one value. There, the second coordinate has no meaning. The data below is provided as an eample. Suppose ou were to plot height vs weight. Number Height Weight Age Number Height Weight Age First ou would need to set up a scale on the paper. Each space on the paper must represent the same amount. So the points will not normall be right on the grid lines. The plot must fit on our paper, so first look at the range of values ou will represent. Using the data above Height range: 6 1to 7 inches Weight range: 9 to 0 Your scale then needs to etend AT LEAST from 61 to 7 inches on one ais and AT LEAST from 9 to 0lb on the other. It does not need to go through zero pounds or zero inches. You know there will never be a need to plot data for a zero height person. It is our choice of what number of inches correspond to the major grid lines. You should make our plot take up enough of the paper that it is eas to see what ou are plotting. It is good to make the grid marks correspond to an even number of units, just to save effort in counting and to avoid making errors. There is no reason to use the same number of squares on the grid for the two aes. In the eample, the range in number of pounds is much larger than the range in inches. There is ever reason to use different numbers of pounds for each scale AND to use the larger dimension for the pounds where ou are likel to need more space. Don t label the points. Linear vs Logarithmic Plot In a normal linear plot, each space represents the same amount. In a logarithmic plot, equal spaces represent the same power of 10. So tpical major intervals on a logarithmic scale are as shown
2 below To use a logarithmic plot b hand, put numbers into scientific notation first. Then look for the sane power of 10 as in the scientific notation. The number goes ABOVE the power of 10. The plot below shows where some points would be plotted. You would NOT generall epect to see the power of 10 on the intermediate tic marks. You would not be epected to label the points. This is just so that ou can see how it should be done , , , , Labels and Aes Each ais should have A title indicating what is being plotted and the units used (e.g. Weight in pounds) Numbers at regular intervals (e.g. 9, 100, 110 ). Tic marks, marks at these same intervals. The numbers and tic marks should continue along the entire ais. Making a Curve from Data As scientists find out information, the do not necessaril know what it means beond that there are a bunch of disconnected facts. In order to make sense of the information, the (and we) tr to find patterns. The idea is perhaps the pattern will show the relationship among all the seemingl disconnected information. One wa we seek patterns is to make graphs of the information we have gathered. If there is a pattern (and if we have graphed the right quantities with respect to one another), the points on the graph will not just fall all over the place. The might fall in a glob (i.e. in a small region of the graph paper), the might fall on a straight line, or the might fall on a curved line. In order to determine the relationship between quantities, we need an equation which describes the
3 best fit line or curve. There are rigorous was to prove that the line or curve is the best. For the purposes of this class, we will usuall use our best judgment, rather than a rigorous computation t o find the best line. Suppose that ou have some data and ou plot it. The result looks like the following figure. The dots are the measured points. The bars represent the uncertainties of the points. (Note that the scales need not start at 0.)  What is the best line or curve to fit this set of data? Is it a straight line like the following?
4 Or is it like? Or like
5 Given this data, the last solution, the parabola, is not justified, since there are no points between 0 and approimatel 0.6 on the descending side of the curve. On the other hand, the data does not rule it out. In general, ou do not use a curve like the one above based on the data shown, unless there is some other reason to know the shape of the curve (e.g. the theor sas so, and the data does not rule out other possibilities). Returning to the straight line fits. Both lines have been plotted on the same graph below. Either of these straight lines could be the best estimate of an honest, independent observer eeballing the data. We could use the difference between these lines to estimate the uncertaint in the fits. Line number 1 Line number But how can we talk about the straight line to discuss whether the are the same or different?
6 Each straight line can be written as the equation: = m + b, which is the usual form, or equivalentl: = /m b/m In the first case, = a + b, the coordinate of each point is related to the coordinate of the same point, b the constants m and b. The constant, m, is called the slope of the line. It tells what angle the line makes with the aes. The constant, b, is called the intercept. It tells what value of results when equals zero. So lets tr to find the equations of the lines in the figure above. Lets consider line number 1 first. The value of for equal zero (on the line, not of an of the points) is. So in our potential equation for the line, we have =0, =, the equation should give =m+b. Substituting for and produces, = m*0 +b =b (es this is a trick, but it s ok ) To find m, choose an other, point on the line. For eample, line number 1 goes through the point (1,), that is =1, =. Substitute into the equation to produce: = m *1 + There is onl one unknown thing left, m. So ou surel can solve for it. Rearranging produces:  =m = m Yes! The slope can be negative. Negative slope means that the bigger that gets, the smaller that gets. Is this true of the data shown here? So ou now have both m and b for line 1. The equation for line 1 is =+. Find the equation for line. The differences in the values of m for the two lines and for b for the two lines is a measure of the uncertaint in the equations which describe the relation of the data. for line : m=0.7, b=.
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