Graphical Analysis of Data

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Presentation transcript:

Graphical Analysis of Data Details on website http://sst.tees.ac.uk/external/U0000504 select “Res/Sci” method on main menu

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

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

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

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

Linearisation (2)

Linearisation (3)

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

Functions involving logs (1) Some functions can be linearised by taking logs These are y = A xn and y = A ekx

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)

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)

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

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

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

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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