Presentation of Data

The writing up of practical reports is an essential skill in physics.  You might have got brilliant results in a required practical, but that's no good if you don't tell anyone about them.  I have written at length about practical reports in the induction notes in Tutorial 7 There is more stuff about graphs in the induction notes Tutorial 5 and Tutorial 6.

A sure-fire way of getting data that are unreliable is to jot them down on a scrap piece of paper with no semblance of order.  While you may know what the data items refer to when you are taking them, it is very easy to forget.  Or the bit of paper gets lost.  It is depressing to hear the excuse for an experiment not being handed in, "Sean's got my results".

Data are recorded on a table of results, and each student must have its own set of data.   This should be drawn up before you start your experimental work.  The picture below shows a neat table of results.   While it may not be of an experiment at AS level, it still shows the principle:

Your data must be recorded to the precision of the instruments.  The averages from repeat readings must be recorded to the same number of significant figures as the data.

If you process data, the number of significant figures must be no more than the minimum number of significant figures.  Let us look at the resistance of a length of resistance wire:

 Length / m Current / A Potential difference (V) Average / V Resistance  / W Reading 1 Reading 2 Reading 3 0.100 0.25 2.30 2.25 2.32 2.29 9.0 0.200 0.25 0.300 0.25 0.400 0.25 0.500 0.25 0.600 0.25 0.700 0.25 Etc…

• The p.d. readings are consistent with the readout of a digital voltmeter.

• The current was kept constant at 0.25 A.  The number of significant figures is consistent with the precision of an analogue ammeter.

• The average is consistent with the number of significant figures of the p.d. readings.

• The resistance is calculated to two significant figures.  It calculated to 9.04 W, but is written as 9.0 W to make it consistent with the current which is recorded to two significant figures.

When you draw the graph, remember the rules.  Graphs should always:

• Be large – they should occupy the whole of a side of A4 paper.

• Have a title to tell the reader what the graph is about;

• Have sensible calibration of axis, with simple scales;

• Have both axes labelled with the quantity and units.

And do them in pencil, please.  Also be a good chap and don't produce a graph like this:

Graphical Skills

On their own numbers do not mean a lot.  A table of numbers can be confusing.  A graph allows us to see a picture of how the numbers relate to each other.

1. Always use a sharp pencil and a ruler.

2. Draw the axes

3. Label the axes with the quantity and the units

4. When you plot Quantity 1 against Quantity 2, you put Quantity 2 on the horizontal axis.

5. Look for the highest value in each range.  You calibrate (put numbers on) your axes to the nearest convenient step above your highest value.

6. Use a sensible scale.

7. Plot your points with crosses (+ or ×).  Points get lost.

8. Join your points with a line, but not dot-to-dot!

It can be difficult to decide whether a set of results is a straight line or a curve.  If it’s clearly a straight line, draw your line of best fit with a ruler.  If the graph is a curve, then try to make a smooth curve.  A flexi-curve can help you with this.

If a point is way out from the rest, then it’s probably an anomalous result.  If you can, recheck the data or do that part of the experiment again.  If not, ignore it.

The table below shows some data to plot:

 Voltage / V Current /mA 0 0 1 20 2 30 3 65 4 98 5 174 6 280

This graph is nonsense.  Can you see why?  Although graphs drawn like this are quite useless, they are depressingly common.

The correct graph is shown below:

Notice:

• Axes labelled with quantities and units;

• Scales are sensible;

• Line of best fit drawn through the points.  No dot-to-dot.

When we read a point off the graph within a range, we are interpolating.  When we extrapolate a graph, we are extending it beyond the plotted range, making a reasoned guess as to where the line is going to go.

All graphs in physics are line graphs.  A set of results is not very easy to use to judge the way quantities are related to each other.  The graph shows it instantly.

Graphs should always:

• Be large – they should occupy the whole of a side of A4 paper.

• Have a title to tell the reader what the graph is about;

• Have sensible calibration of axis, with simple scales;

• Have both axes labelled with the quantity and units.

You can draw a graph in portrait (long side vertical), or landscape (long side horizontal).   Sometimes it’s quite clear which way it should go.  Other times it’s a matter of what seems best.

In general, the independent variable goes on the horizontal axis (the x-axis, or ordinate), and the dependent variable on the vertical axis (y-axis or abscissa).  When you are told to plot quantity 1 against quantity 2, quantity 2 goes on the horizontal axis.

Sometimes the scales go the other way round, for example voltage (independent variable) against current (dependent variable).  This enables us to work out the resistance from the gradient.

A line of best fit needs to be drawn, because the data points will always be slightly out.

Consider whether the origin is a valid data point.  For example, at zero volts, we get zero amps.  So in a voltage-current graph, the origin is a data point in its own right.

A straight-line graph that goes through the origin shows that the two quantities are directly proportional.

In this case:

• If the force is doubled, the extension is doubled.

• If there is zero force, there is zero extension.

Descriptions like “…if weight increases, the extension increases…” are too vague and will not gain any credit.

This graph does NOT go through the origin:

# This is a linear progression (i.e. it’s a straight line), but is NOT proportional.  If the x-value is doubled, the y-value is not doubled.

 Question 1 A student draws a graph as below and joins the points with a “line of best fit” as shown.     (a) The title of his graph is _________________ against _____________________. (b) What is wrong with the line of best fit? (c) Draw the correct line of best fit. (d) In the exam, he writes that Quantity A is directly proportional to Quantity B.  Discuss how you would mark his answer.

Sometimes the data form a curve.  Draw a smooth curve, not doing dot-to-dot.

Do not force a straight line through the curved progression.

Decide whether the origin is a valid data point.  If it is, include it.

Sometimes it is not at all easy to decide whether the graph is a curve or a straight line.  In this case, you should take more data points.

You may well get a data point that does not fit in with the rest of the data.  This is called an anomalous result.

If possible, check out anomalous results by doing a repeat.  Your new data will most likely fit in with the other points, and the anomalous data can be discarded.

# Gradient = rise ÷ run = (change in y-value) ÷ (change in x-value)

The gradient will give a reading that could be used directly, for example resistance is the gradient of the graph of voltage against current.

For other relationships, you may need to do some further processing to get the value that you want.

Consider this graph:

We know the formula for resistivity as R = rl/ATo get the area, we would need to measure the wire with a micrometer.

 How would you work out the resistivity from the graph of resistance against length?

# With a curved graph, measuring the gradient requires the use of a tangent:

First thing to do is to draw a line at 90 degrees to the curve, a normal line.  Then draw the tangent at 90 degrees to the normal.  Then work out the rise and run of the tangent, making sure that your triangle is at least 8 cm in its shortest dimension (of course).

In the Exam:

It is important that you understand that the exam not only tests knowledge with understanding, but also tests a range of other skills.  One of these is to be able to interpret and transfer data from a variety of sources such as tables and graphs.  Therefore graphical skills are essential for success and are widely used to decide whether a candidate gets a grade A or B, or at the other end, an E or U.

Therefore you will need to get a lot of practice with graphs!