3 Graphing

Graphing skills can be assessed both in written papers for the A-level grade and by the teacher during the assessment of the endorsement. Students should recognise that the type of graph that they draw should be based on an understanding of the type of data they are using and the intended analysis of the data. The rules below are guidelines which will vary according to the specific circumstances. Labelling axes Axes should always be labelled with the quantity being measured and the units. These should be separated with a forward slash (solidus): time / seconds length / mm Axes should not be labelled with the units on each scale marking. Data points Data points should be marked with a cross. Both  and  marks are acceptable, but care should be taken that data points can be seen against the grid. Error bars can take the place of data points where appropriate. Scales and origins Students should attempt to spread the data points on a graph as far as possible without resorting to scales that are difficult to deal with. Students should consider:  the maximum and minimum values of each variable  the size of the graph paper  whether 0.0 should be included as a data point  how to draw the axes without using difficult scale markings (eg multiples of 3, 7, 11 etc)  In exams, the plots should cover at least half of the grid supplied for the graph. Please note that in section M, many generic graphs are used to illustrate the points made. For example, the following three graphs are intended to illustrate the information above relating to the spread of data points on a graph. Students producing such graphs on the basis of real practical work or in examination questions would be expected to add in axes labels and units. 0 5 10 15 20 25 30 35 0 20 40 60 80 100 This graph has well-spaced marking points and the data fills the paper. Each point is marked with a cross (so points can be seen even when a line of best fit is drawn) This graph is on the limit of acceptability. The points do not quite fill the page, but to spread them further would result in the use of awkward scales. At first glance, this graph is well drawn and has spread the data out sensibly. However, if the graph were to later be used to calculate the equation of the line, the lack of a y-intercept could cause problems. Increasing the axes to ensure all points are spread out but the y-intercept is also included is a skill that requires practice and may take a couple of attempts. Lines of best fit Lines of best fit should be drawn when appropriate. Students should consider the following when deciding where to draw a line of best fit:  Are the data likely to be following an underlying equation (for example, a relationship governed by a physical law)? This will help decide if the line should be straight or curved.  Are there any anomalous results? There is no definitive way of determining where a line of best fit should be drawn. A good rule of thumb is to make sure that there are as many points on one side of the line as the other. Often the line should pass through, or very close to, the majority of plotted points. Graphing programs can sometimes help, but tend to use algorithms that make assumptions about the data that may not be appropriate. Lines of best fit should be continuous and drawn as a thin pencil line that does not obscure the points below and does not add uncertainty to the measurement of gradient of the line. Not all lines of best fit go through the origin. Students should ask themselves whether a 0 in the independent variable is likely to produce a 0 in the dependent variable. This can provide an extra and more certain point through which a line must pass. A line of best fit that is expected to pass through (0,0), but does not, would imply some systematic error in the experiment. This would be a good source of discussion in an evaluation. Dealing with anomalous results At GCSE, students are often taught to automatically ignore anomalous results. At A-level, students should think carefully about what could have caused the unexpected result and therefore whether it is anomalous. A student might be able to identify a reason for the unexpected result and so validly regard it as an anomaly. For example, an anomalous result might be explained by a different experimenter making the measurement, a different solution or a different measuring device being used. In the case where the reason for an anomalous result occurring can be identified, the result should be recorded and plotted but may then be ignored. Anomalous results should also be ignored where results are expected to be the same (for example, in a titration in chemistry). Where there is no obvious error and no expectation that results should be the same, anomalous results should be included. This will reduce the possibility that a key point is being overlooked. Please note: when recording results it is important that all data are included. Anomalous results should only be ignored at the data analysis stage. It is best practice whenever an anomalous result is identified for the experiment to be repeated. This highlights the need to tabulate and even graph results as an experiment is carried out. Measuring gradients When finding the gradient of a line of best fit, students should show their working by drawing a triangle on the line. The hypotenuse of the triangle should be at least half as big as the line of best fit. = ∆ ∆ When finding the gradient of a curve, eg, the rate of reaction at a time that was not sampled, students should draw a tangent to the curve at the relevant value of the independent variable (x-axis). Use of a set square to draw a triangle over this point on the curve can be helpful in drawing an appropriate tangent. 25 26 27 28 29 30 31 32 33 34 35 20 40 60 80 100 Δy The line of best fit here has an equal number of points on both sides. It is not too wide so points can be seen under it. The gradient triangle has been drawn so the hypotenuse includes more than half of the line. In addition, it starts and ends on points where the line of best fit crosses grid lines so the points can be read easily (this is not always possible).The equation of a straight line Students should understand that the following equation represents a linear relationship. y = mx + c Where y is the dependent variable, m is the gradient, x is the independent variable and c is the yintercept. Δy = 28 – 9 = 19 Δx = 90 – 10 = 80 gradient = 19 / 80 = 0.24 (2 sf) y-intercept = 7.0 equation of line: y = 0.24 x + 7.0. Testing relationships Sometimes it is not clear what the relationship between two variables is y 2 ∝ x. A quick way to find a possible relationship is to manipulate the data to form a straight line graph from the data by changing the variable plotted on each axis. For example:  Raw data and graph This is clearly not a straight line graph. The relationship between x and y is not clear. A series of different graphs can be drawn from these data. The one that is closest to a straight line is a good candidate for the relationship between x and y. This is an idealised set of data to illustrate the point. The straightest graph is y2 against x, suggesting that the relationship between x and y is. More complex relationships Graphs can be used to analyse more complex relationships by rearranging the equation into a form similar to y=mx+c. Example one: testing power laws A relationship is known to be of the form y=Axn , but n is unknown. Measurements of y and x are taken. A graph is plotted with log(y) plotted against log(x). The gradient of this graph will be n, with the y intercept log(A), as log(y) = n(log(x)) + log(A) Example two The equation that relates the rate constant of a reaction to temperature is = − This can be rearranged into ln() = − ( 1 ) + So a graph of ln() against ( 1 )should be a straight line, with a gradient of− and a y-intercept of �