QUESTION: Have you got enough points, close enough to the line, to justify the best-fit line you drew? Could you draw a different best-fit line, still pretty close to your points?
Summary: Situation A: You have done a preliminary experiment, and got a few preliminary observations plotted on a graph. You have drawn a best-fit line (a trend line). This is going to be your prediction, which you will test in the main experiment. But are your points sufficient to justify this trend?
The answer is: Almost certainly “No”. If they were, you wouldn’t need to do the main experiment - you’d already have a firm conclusion.
Situation B: You have done the main experiment, and got lots of observations plotted on a graph. You have drawn a best fit line. This is your final conclusion -the actual trend you think is true. But are your points sufficient to justify this trend?
The answer now is: Possibly “Yes”.
What is the difference between these two situations? In Situation A, your observations were not yet accurate enough, and you didn’t have enough of them, to decide what the trend really is. In Situation B, you have much more accurate points (repeats etc.), and more of them; now you can justify a best-fit line.
The more accurate your points are, and the more points, the more they fix what the best fit line really is - the less ‘room for manoeuvre’ there is.
With inaccurate points, and just a few, you could draw just about any best fit line you like. With accurate points, and lots of them, you can only draw one best fit line.
Is your best-line justified? Answer by working out how close your points are to the best line. Work out the percentage difference between the point furthest from your best line, and what it would be, if it was on the line.
In this example, the fifth reading was a time of 7.1 s. If the best-fit line is correct, the reading should have been 7.6 s. The percentage difference between these figures is (7.6 - 7.1)/7.6 X 100 % which is a 7 % difference.
You can even work out the average percentage difference. (*By this stage, you are nearly working out the statistical variance - which you can find in a mathematics book)
Each time, your question is the same: How accurate was the data, and how close is it to your best fit line? The more accurate the data is, the closer it needs to be to the best fit line, for this choice of line to be reasonable.
Why aren’t all your points exactly on the best line you drew? This needs answering. The reason is unavoidable experimental inaccuracy.
Exercise 1: A pupil is doing an experiment timing a pendulum of various lengths. Which of the following are unavoidable experimental inaccuracies? Which are not? (Some are just mistakes, and shouldn’t be mentioned):
(a) The stopclock only reads to the nearest 0.1 s
(b) I may have made mistakes plotting points onto the graph
(c) After I had divided by 5, I rounded up to 3 sig. fig.
(d) It was hard to decide exactly when to start and stop the stopclock
(e) I may have done some of the calculations wrong
(f) The measurement of length was supposed to be to the centre of the pendulum, which was hard to judge
(g) The metre rule could only be read to the nearest mm
Exercise 2:
This is an experiment in which a ball is being timed rolling down a plank. The slope of the plank is gradually increased. The pupil did a quick preliminary experiment. The left-hand graph shows her five inaccurate results. She reckoned that the straight line was a reasonable trend-line, so she used this as her prediction.
Now she did a careful, complete, experiment - with more timings, repeating, and averaging. The right-hand graph shows her nine, much more accurate, results. She concluded that the straight line was the true trend.
(i) On the right-hand graph the points certainly aren’t in a straight line, but she was right to draw a straight trend-line. Why is this OK?
(ii) Why are the crosses smaller on the right-hand graph?
(iii) Write an evaluation of the experiment, showing that the pupil’s conclusion was wrong.
Exercise 3 (non-graphical):
A pupil does an experiment which is supposed to show that doubling the volume of water in the test tube makes it cool down at half the rate. He is measuring the volume of water with a measuring cylinder; he is measuring the temperature with a thermometer.
With his original volume of water, its temperature falls in 2 minutes by 30 oC. When he doubles the volume of water in the tube, its temperature falls, in the same time, by 18 oC.
(i) The pupil cannot reasonably conclude, from his results, that his proposed trend is correct. Explain why, referring to the size of the ± inaccuracy in his result.