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**Graphical Analysis of Data**

Details on website select “Res/Sci” method on main menu

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

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**The straight line (1) Most graphs based on numerical data are curves.**

The straight line is a special case 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) Straight line equation y = mx + c Intercept = c**

slope = m

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Linearisation (1) Many functions can be manipulated mathematically to yield a straight line equation. 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 These are functions of general formula**

y = a + bx + cx2 + dx3 + … They cannot be linearised Techniques for fitting polynomials exist for example using Excel

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**Functions involving logs (1)**

Some functions can be linearised by taking logs These are y = A xn and y = A ekx

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**Functions involving logs (2)**

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

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**Functions involving logs (3)**

For y = Aekx, we must use natural logs ln y = ln A + kx This gives a straight line slope k and intercept ln A To find A we must take antilogs (= ex)

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

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**Special Graph paper (2) Log - Log graph paper for y = Axn Slope = n =**

The intercept is at log x = 0 i.e. the value of y at x = 1.0

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**Special Graph Paper (3) Log-linear graph paper for y = Aekx**

Take logs to base 10 log y = log A + kx.log10e Intercept = A Slope =k log10e

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**Using a spreadsheet Excel has tools to fit lines**

Lines can be fitted on the graph Slope and gradient can be found from data in tabular form For more details see website

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