Analysing Data and Evaluating Methods

The readings are in. Now what?

Planning an investigation and carrying it out are only half of the work, and they are the half most people find easier. The other half starts when the results are written down, and it is where a large share of the marks in this unit actually sit. Three judgements have to be made, and none of them is about doing more arithmetic. The first is whether anything in the table is not telling the truth, and if so, which kind of untruth it is, because different faults need completely different responses and one of them cannot be fixed by repeating at all. The second is how many figures you are entitled to write down, which is decided by the instrument you used rather than by what your calculator displays. The third is whether the conclusion you want to draw stays inside what you actually measured, or quietly reaches beyond it. This module covers all three. Two related skills sit elsewhere: designing the investigation in the first place, and judging a claim that somebody else has made using numbers you did not collect. Both have modules of their own, and both are different from what happens here.

Five words for reading a table

Two of these are constantly confused with each other, and the difference between them is a favourite exam question.

What does each table tell you?

What do you do with it?

You take six readings of the same thing. Five are close together and one is far higher than the rest. What should you do with the odd one?

• Repeat that reading if you can, and if the odd value stands, report it and explain how you treated it
• Remove it without comment, because it is clearly wrong
• Include it in the mean, because all the data should be used
• Adjust it so that it fits the pattern of the others

Repeatable and reproducible

These two words are constantly swapped for each other. They ask different questions, and only one of them can be answered by the person who did the work.

Mean without the anomaly

Six readings of the same time, in seconds: 4.0, 4.5, 4.1, 7.9, 4.2 and 4.7. One of them is anomalous. Work out the mean of the readings you would keep, in seconds.

Work through the table in this order

Five things to do with a finished results table. Put them into the order that stops you drawing a pattern out of bad numbers.

• Look down each group of repeats and mark anything sitting far from the rest
• Decide what to do about each anomaly, and record the decision
• Work out a mean for each group from the readings you kept
• Write each mean to the number of figures the instrument justifies
• Only now look for the pattern across the groups

Repeating fixes one of them and not the other

This is the single most useful idea in the whole topic, and it decides what your evaluation should say. Random error is the scatter that comes from small unpredictable differences: a reaction time, a slightly different starting point, a reading taken from a fraction of an angle away. It pushes readings above and below the true value roughly evenly, which is exactly why repeating helps. Take more readings, take a mean, and the pushes in opposite directions cancel each other out. A systematic error is different in kind. It shifts every single reading the same way and by a similar amount: a balance that was not zeroed, a ruler measured from the end rather than from the zero mark, a timer started habitually a little late. Repeating that a hundred times gives you a beautifully consistent set of readings that are all wrong together, and a mean that is wrong by the same amount. Nothing about the repeats warns you, because they agree with each other. The only things that reveal it are checking the equipment, comparing against a known value, or somebody else getting a different answer. So when an evaluation says the results were repeatable and therefore accurate, that sentence is doing something it cannot do. Repeatable readings show the random error was small. They say nothing whatever about whether the whole set is shifted.

Complete the analysis words

A reading sitting far away from the others in its group is _____. Results you get again yourself, with the same method and equipment, are _____. Results somebody else gets working separately are _____. An error that shifts every reading the same way, so that repeating never removes it, is _____.

One table, read properly

A learner times how long an object takes to travel a fixed distance, and does it six times. The readings in seconds are 4.0, 4.5, 4.1, 7.9, 4.2 and 4.7. Work through it the way the marks are given. First, the odd one. Five readings sit between 4.0 and 4.7 and one sits at 7.9, which is not a small difference but nearly double. That is an anomaly, and the honest response is to repeat it if the equipment is still set up, and if it cannot be repeated, to exclude it and say plainly in the write-up that it was excluded and why. Second, the mean of what remains. The five kept readings total 21.5 seconds, and dividing by five gives 4.3 seconds. Notice that including the anomaly would have given 4.9, which is higher than every single one of the five readings the learner trusts, and a mean that no reading supports is a sign something has gone wrong. Third, the figures. The timer read to the nearest tenth of a second, so 4.3 seconds is the honest answer and writing 4.30 or 4.300 would be claiming precision that was never measured. Finally, what the scatter shows. The five kept readings differ from each other by up to 0.7 seconds, which is quite a spread for a timing, and that points at random error in starting and stopping. It says nothing at all about whether the timer itself was running fast.

What the resolution decides

A timer reads to the nearest 0.1 of a second. Which THREE of these follow from that?

• A single reading cannot be trusted to more than one decimal place
• Writing a reading as 4.27 seconds claims precision the instrument never gave
• Two events differing by less than 0.1 of a second cannot be told apart by this timer
• Taking more readings increases what the instrument itself is able to show
• Readings should be written to three decimal places to be on the safe side

Find the conclusion that reaches too far

Five sentences from an evaluation. Tap the ONE that claims more than the data can support.

• Across the five lengths tested, the time taken fell as the length increased.
• The repeats at each length agreed to within about half a second, so the random error was small.
• One reading at the shortest length was excluded as an anomaly, and this is recorded in the table.
• Because the repeats were close together, the readings must also have been accurate.
• A wider range of lengths would be needed before saying whether the pattern continues.

Write the honest conclusion

A conclusion earns marks for staying inside the evidence. Choose the option in each gap.

The analysis run

Five quick questions on analysing results. Three lives.

Marking up a finished table

You have a table of results. At each of five settings there are three repeats. At one setting the three repeats are 2.1, 2.2 and 5.8 seconds. At every setting the three repeats agree closely with each other, and the whole set of means is about half a second higher than the value the class teacher expected. Work through three decisions.

• What do you do about the 5.8 reading?
• Every mean is about half a second high, and the repeats all agree closely. What does that suggest?
• How do you word the evaluation?

Evaluate the investigation

A learner reports that their repeats agreed closely at every setting, so their results were accurate. Their timer was started by hand each time, and every mean came out slightly higher than the expected value. Explain what their results do and do not show, and what you would do about it.

• Explain what close agreement between repeats actually establishes
• Explain why it cannot establish accuracy
• Name the kind of error a consistent shift points to, and give a likely cause here
• Say what would reveal that error, since repeating will not
• Finish with how the conclusion should be worded instead