Lesson 2 Graphs

Refresh What is the answer to 3.00 × 50.0 150 150.00 150. 150.0

Refresh What is the answer to 3.00 × 50.0 150 150.00 150. 150.0

Lesson 15: Graphs Understandings: Applications and skills: Graphical techniques are an effective means of communicating the effect of an independent variable on a dependent variable, and can lead to determination of physical quantities. Sketched graphs have labelled but unscaled axes, and are used to show qualitative trends, such as variables that are proportional or inversely proportional. Drawn graphs have labelled and scaled axes, and are used in quantitative measurements. Applications and skills: Drawing graphs of experimental results, including the correct choice of axes and scale. Interpretation of graphs in terms of the relationships of dependent and independent variables. Production and interpretation of best-fi t lines or curves through data points, including an assessment of when it can and cannot be considered as a linear function. Calculation of quantities from graphs by measuring slope (gradient) and intercept, including appropriate units.

Plotting graphs Name 6-7 things you should you include when you plot a graph.

Plotting graphs When you draw a graph you should: Hopefully these are obvious… Give the graph a title. Label the axes with both quantities and units. Use the available space as effectively as possible. Use sensible linear scales – there should be no uneven jumps. Plot all the points correctly. These might be less obvious A line of best fit should be drawn smoothly and clearly. It does not have to (and normally doesn’t) go through all the points but should show the overall trend. Identify any points which do not agree with the general trend. Think carefully about the inclusion of the origin. The point (0, 0) can be the most accurate data point or it can be irrelevant.

Just to emphasise… (Almost) NEVER join up the dots with a line The ‘best-fit’ line passes as near to as many of the points as possible. For example, a straight line through the origin is the most appropriate way to join the set of points Sometimes a curve will be appropriate

Extrapolation Sometimes a line has to be extended beyond the range of measurements of the graph. This is called extrapolation. Absolute zero, for example, can be found by extrapolating the volume/temperature graph for an ideal gas

Anomalous data Clearly indicate any data points that ‘don’t seem right’. Such points need not be considered when drawing the best-fit line.

Finding the gradient of a straight line The equation for a straight line is y = mx + c. x is the independent variable y is the dependent variable m is the gradient c is the intercept on the vertical axis. The gradient of a straight line (m) is the increase in the dependent variable divided by the increase in the independent variable. The triangle used to calculate the gradient should be as large as possible

Finding the gradient of a curve The gradient of a curve at any point is the gradient of the tangent to the curve at that point The gradient of any line has units; the units of the vertical axis divided by the units of the horizontal axis.

Errors and graphs Systematic errors and random uncertainties can often be recognized from a graph You should always plot average results, this way your graph combines the results of many measurements and so minimizes the effects of random uncertainties in the measurements.

Choosing what to plot Sometimes it is useful to change what we plot to yield a straight line e.g.

Choosing what to plot Similarly ln/log can be employed where necessary e.g. Arrhenius plots – which you will meet in topic 6:

Using spreadsheets to plot graphs There are many software packages which allow graphs to be plotted and analyzed, I strongly suggest you use Excel. The equation of the best -fit line can easily be found, and other properties easily calculated. The closeness of the generated line to the data points is indicated by the R2 value. An R2 of 1 represents a perfect fit between the data and the line drawn through them, and a value of 0 shows there is no correlation.

Using Excel to draw a best-fit line For example, here the ‘polynomial’ line gives an R2 of 1, but it is extraordinarily unlikely to be a true representation of the relationship you are studying.

Your turn Which piece of apparatus is most effective at heating water (line graph closest to R2 = 1)

Key points Always use a best-fit line or curve, rather than joining the dots. Always clearly note any anomalous data Aim to plot data that gives you straight line if possible Avoid excessive use of the polynomial line function on Excel