L2 - Graphical Techniques

objectives

Understandings

• 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-fit 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.

Dependency

• Dependency in graphing means the relationship that the two variables have with one another.
• There are only a few possible shapes of graph lines with which students should be familiar.
• It should be possible for a student to state the type of relationship between independent variable and the dependent variable simply by looking at the graph.
• This does pre-suppose that the graph has sufficient points plotted. Clearly two points can always be joined by a straight line!
• The variable that is chosen and controlled is called the independent, or input variable.
• This should always be plotted along the x-axis.
• The dependent, or output variable is plotted along the y-axis.

No Relationship

Proportionality

• Proportionality between variables may be direct or inverse.

Complex Relationship

Periodic Relationship

Random

drawing graphs

Choice of Axes

• In principle the choice of axes is usually the easiest part of graphical data analysis.
• The independent (input) variable is chosen as the x-axis and the dependent (output) variable as the y-axis.
• However, it is quite possible that a student performs an experiment and, understanding the theory behind the process, decides to plot either variable as an inverse, or logarithmic value.
• In this case the concept derived from the independent variable still goes on the x-axis.

Choice of Scale

• The scale is chosen to maximise the data extraction potential of the graph.
• A decision must be taken as to whether or not to start the axis scale at zero.
• There are many cases where it is simply not relevant to do so.
• However, there are also cases where it is imperative. The decision depends entirely on the graph plotted.

Example: If you are measuring pressure in Nm-2and have values ranging from 1.01 x 105 to 10 x 105, it would be reasonable to add the factor 105 to the units of the scale.

• If the units being plotted are inconveniently small or large, the axis can have a factor built in to make the numbers of the axis scale more comfortable.
• In this case the scale measures from 1 - 10 and the units are Nm-2 (x 105)
• On a purely common sense note, the scale should be chosen with plotting in mind.
• Graph paper is divided into squares usually subdivided into 10 smaller units.
• If your scale sets the value of each 'big' square at 3 units, it will be difficult to plot points.

Plotting the Points

• Once your scale has been chosen it should be easy to plot the points carefully for every x and y value.
• At this stage you should consider whether to include error bars to cover uncertainties in the event that you wish to draw a line of best fit.

Straight Line Graphs

• Plotted points will generally give a good indication of whether or not they can be joined by a straight line.
• It is quite probable that any line will not pass through all of the points.
• In this case a line of best fit must be drawn
• This means a line that can be drawn smoothly, going through as many of the data points as possible.

Curved Graph Lines

• If the plotted points appear to form a curve then they can be joined by a smoothly sketched curve.
• Once again it is quite likely that not all of the points lie on the curve.
• This is normal experimental deviation and can often be resolved by using error bars

Error Bars

Gradient - Straight Line

Units of Gradient

• The units of a gradient depend on the units of the x-axis and y-axis. As the gradient is calculated by dividing the y-difference by the x-difference then the units of gradient are the units of the y axis divided by the units of the x-axis.
• It is important to remember that the axis may be multiplied (or divided) by a predetermined factor, such as 1 x 105, for example.

Gradient of a Curve

• Although the gradient of a curve is constantly changing, the value of the gradient at any specific point on the curve can be found by taking a tangent to the curve and measuring its gradient in the same way as before.
• A tangents is a line drawn just touching a curve and running perpendicular to an imaginary line bisecting the curve which gives two identical angles.

The Intercept