25.7 - Conclusions & the Limits of Experimental Accuracy

What a conclusion question looks like

• Practical conclusion questions usually have two answer lines: a statement and a justification.
• A typical stem: "State whether the results suggest that resistor P and resistor Q have the same value of resistance, within the limits of experimental accuracy. Justify your statement by reference to your results. [2]"

The two marks

1. The statement (1 mark) - A yes or a no. It must match your own results.
2. The justification (1 mark) - Your reasoning, quoting your actual data. A justification with no numbers in it scores zero.

There is no single "correct" answer decided in advance: two students with different readings can write opposite statements and both score full marks, as long as each justification quotes their own figures to support their statement.

A typical question

State whether the results $R_1$ and $R_3$ suggest that resistor P and resistor Q have the same value of resistance, within the limits of experimental accuracy. Justify your statement by reference to your results.

statement ..............................................

justification ............................................ [2]

The 10% test

"Within the limits of experimental accuracy"

• Every measurement carries small errors, so two genuinely equal values rarely come out identical.
• "Within the limits of experimental accuracy" means two values are close enough that any difference can be explained by small measurement errors.
• The threshold is 10%: if the values are within 10% of each other, they count as equal within experimental accuracy.

Formula

$$\text{Percentage difference}=\frac{\text{difference between the two values}}{\text{one of the values}}\times 100$$

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Worked example - Spring constant by two methods

A student finds the spring constant of a spring by two methods. Timing the oscillations gives $k_1=25.0\text{ N/m}$; measuring the extension gives $k_2=23.5\text{ N/m}$. State whether the two values can be considered equal within the limits of experimental accuracy, and justify your answer by reference to the results.

Describing a relationship

A conclusion about a relationship states how one quantity depends on another and uses data to support it.

Types of relationship

"Increases as" only states the direction. Only state "directly proportional to" if the ratio $\frac{y}{x}$ is constant or the line is straight and passes through the origin.

Writing a relationship conclusion

• A full-mark conclusion names both variables and describes how one changes with the other:

  "As the length of the wire increases, the resistance increases. The ratio $\frac{R}{l}$ is 0.033, 0.033, and 0.032 Ω/cm, so resistance is directly proportional to length."

• Simply writing "the resistance went up" scores no marks because it doesn't name the independent variable or quote numbers.

Justifying from a graph

• Straight line through the origin - $y$ is directly proportional to $x$.
• Straight line not through the origin - $y$ increases linearly with $x$, but is not directly proportional.
• Horizontal line - $y$ is constant.
• Curve - The rate of change varies as $x$ increases.

Common exam questions

Question 1

What does "within the limits of experimental accuracy" mean, and what is the threshold?

It means the two values are close enough that the difference could be explained by the small errors in taking the readings rather than by the quantities really being different. The threshold is 10%: values within 10% of each other count as equal.

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Question 2

Two measurements of the same resistance give 4.05 Ω and 3.81 Ω. Are they equal within the limits of experimental accuracy?

The difference is 0.24 Ω, which is 0.24 ÷ 4.05 × 100 = 5.9% of the larger value. That is less than 10%, so yes, they can be considered equal.

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Question 3

Two values of a focal length come out as 16.0 cm and 13.8 cm. Are they equal within the limits of experimental accuracy?

The difference is 2.2 cm, which is 2.2 ÷ 16.0 × 100 = 14%. That is more than 10%, so no.

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Question 4

A student writes "yes, because the values are very close" as the justification. Why does this score nothing?

Because it contains no numbers. A justification has to quote both values and say how far apart they are, ideally as a percentage compared with the 10% threshold.

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Question 5

Two students get different readings and write opposite statements. Can both be right?

Yes. The statement has to match that student's own results, so an examiner accepts either answer as long as the justification quotes that student's numbers and supports the statement they made.

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Question 6

A student concludes "the readings went up". Why is this not a conclusion?

Because it does not say what was changed to make them go up. A conclusion names the independent variable and the dependent variable and links them, as in "the resistance increases as the length of the wire increases".

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Question 7

What is the difference between saying $y$ increases as $x$ increases, and saying $y$ is directly proportional to $x$?

The first only says which direction $y$ moves in. The second says that doubling $x$ doubles $y$, which you can only claim if $\frac{y}{x}$ is the same for every reading, or if the graph is a straight line through the origin.

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Question 8

How can you tell from a graph that two quantities are directly proportional?

The line of best fit is straight and passes through the origin.

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Question 9

A student measures three resistors and gets 8.6 Ω, 8.8 Ω, and 9.1 Ω, and is asked whether they are the same within the limits of experimental accuracy. What should the answer be?

Yes. The largest difference is between 8.6 and 9.1, which is 0.5 Ω, or about 5.8% of 8.6 Ω. That is under 10%, so all three can be considered equal.