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Graphical Analysis of Data Details on website select Res/Sci method on main menu

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Graphical Analysis b Pictorial representation of a mathematical relationship b Two main approaches b Analytical Based on theory Based on theory b Empirical Based on observation and experience Based on observation and experience

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The straight line (1) b Most graphs based on numerical data are curves. b The straight line is a special case b Data is often manipulated to yield straight line graphs as the straight line is relatively easy to analyse

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The Straight line (2) b Straight line equation b y = mx + c b Intercept = c b slope = m

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Linearisation (1) b Many functions can be manipulated mathematically to yield a straight line equation. b Some examples are given in the next few slides

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Linearisation (2)

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Linearisation (3)

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Polynomials b These are functions of general formula y = a + bx + cx 2 + dx 3 + … y = a + bx + cx 2 + dx 3 + … b They cannot be linearised b Techniques for fitting polynomials exist for example using Excel for example using Excel

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Functions involving logs (1) b Some functions can be linearised by taking logs b These are y = A x n and y = A x n and y = A e kx y = A e kx

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Functions involving logs (2) b For y = Ax n, taking logs gives b log y = log a + n log x b A graph of log y vs. log x gives a straight line, slope n and intercept log A. b To find A you must take antilogs (= 10 x )

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Functions involving logs (3) b For y = Ae kx, we must use natural logs b ln y = ln A + kx b This gives a straight line slope k and intercept ln A b To find A we must take antilogs (= e x )

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Special Graph Paper (1) b It is possible to plot graphs of y=Ax n and y=Ae kx without taking logs b This involves the use of graph paper in which the scale has been adjusted to give the intervals for logs. b The next two slides give examples

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b Log - Log graph paper for y = Ax n b Slope = n = Special Graph paper (2) b The intercept is at b log x = 0 i.e. the value of y at x = 1.0

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Special Graph Paper (3) b Log-linear graph paper for y = Ae kx b Take logs to base 10 b log y = log A + kx.log 10 e b Intercept = A b Slope =k log 10 e

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Using a spreadsheet b Excel has tools to fit lines b Lines can be fitted on the graph b Slope and gradient can be found from data in tabular form b For more details see website

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