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## Math for Life Sciences I

by: John MacGyver

11

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2

# Math for Life Sciences I MATH 151

John MacGyver
UT
GPA 3.52

Staff

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## Popular in Mathematics (M)

This 2 page Class Notes was uploaded by John MacGyver on Monday October 26, 2015. The Class Notes belongs to MATH 151 at University of Tennessee - Knoxville taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/229826/math-151-university-of-tennessee-knoxville in Mathematics (M) at University of Tennessee - Knoxville.

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Date Created: 10/26/15
How to do a linear analysis of data The first step is to see if indeed this process makes any sense at all That is do the data appear to be linearly related This step is usually accomplished by doing a quotscatter plotquot of the data This means plotting them as xy points on a graph The choice of ranges for the axes matters here a great deal A poor choice for the range of values on one of the axes could lead to all the points falling over a very small portion of the graph making it impossible to tell whether there is any potential relationship in the data set Thus choose your axes so that the range of them is approximately the same as that in your data So if you have a set of body weights ranging from 5 grams to 160 grams it would be reasonable to choose the axis for body weight to run from 0 160 grams or so which is the x horizontal axis and which is the y vertical If you have some reason to expect that one of the measurements is the dependent one eg body weight depends on age rather than the reverse then choose the one that is dependent weight as the y axis and the independent one age as the horizontal axis If you do not have any reason to expect that one of the measurements is caused by the other then the choice of which measurement is plotted on which axis doesn39t matter Just choose one and go with it The next step is to eyeball the data and see if there appears to be any relationship If the scatter plot looks like the points might be described at least approximately by a line eg don39t worry if there is a lot of scatter about any line you might draw on the graph then it is reasonable to proceed with fitting a line The next step then is eyeballing the data and making a guess at a line without doing any calculations or using any program This will not give you an exact answer but it will be useful later in checking to see that the line that is obtained from a computer or other method you choose makes sense To eyeball just quickly draw a line through the data eyeball a rough slope and then write down the formula for the line using the pointslope form that is y y1 m x x1 where m is the slope and the point you have chosen is xlyl or use the pointintersept form if it can easily be estimated where the line crosses the yaxis y m x b where b is the yintersept Be sure in doing this that you keep your units straight so that you know what units each measurement and thus the slope is in What if the data do not appear to be linearly related Then don39t try to fit a straight line Later we39ll discuss ways to transform the data to see if a different relationship might be a better choice If it looks like a linear relationship is a reasonable assumption the next step is to find the quotleast squares fitquot This can be done automatically within Matlab and many calculators can also do this The idea is to choose a line that quotbest fitsquot the data in that the data points have the minimum sum of their vertical distances from this particular line One way that you might code a computer to do this is i Guess at the equation for the line y ax b ii Measure the vertical distance from each data point to the line you have chosen iii Sum up the distances chosen in ii v Change he slo e a and he n erce 0 see f you can reduce he sum obtained in iii v Continue this until you get tired or you39ve done the best you can One of the potential problems with the above is what quotdistancequot to use in ii If you allow some distances to be positive and some negative eg for a point above the line and a point below the line then the distances can cancel which is not appropriate So we want all distances to be positive and the standard way to do this is the just square each vertical distance found in ii and produce the sum of the square distances to get iii The line we would get by going through step v would then be called the quotleast squares fitquot since it chooses the line so as to minimize the sum of the square deviations of points from the line It turns out that it is not necessary to go through the steps 1v above at all It can be proven that the quotbestquot values of the slope a and the intercept b can be obtained from a relatively simple formula that just uses the x and y values for all the data points Matlab does this easily for you using the command quotCpolyfitAW1quot which will produce a vector C in which the first value is the best fit for the slope a and the second is the best fit for the intercept b for the least squares fit of the vector of data W on the vertical axis to the vector of data A on the horizontal axis Think of A as giving a vector of ages in days of bats and the vector W giving the weights of these bats in grams Note that the units of the components of C depend upon the units the data are measured in The first component of C is a slope so it has units grams per day for the bat example and the second component of C the y intercept has the same units as the measurement on the yaxis grams in the case of the bats Once you have a linear least squares fitted line you can proceed to use it to interpolate find the yvalue predicted by the linear fit for an xvalue that falls in the range of the xvalues in your data set or to extrapolate find the yvalue predicted by the linear fit for an xvalue that falls outside the range of the xvalues in your data set Thus if you have values for body weights W in grams and ages of bats A in days 9 15 20 22 34 44 and 49 and it appears that a linear fit to the data is reasonable you can interpolate to find the weight of a bat of age 30 days or you can extrapolate to find the weight of a bat of age 60 days All you do to find these weights is to plug the age into the equation of the line and calculate the associated yvalue weight in grams Matlab makes this easy by using the command quotYpolyvalC30quot which will give the best guess according to the linear fit for the weight of a bat of age 30 days

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