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A1.8 · Analyse evidence, solve quantitative problems, and evaluate error

Learn to analyse evidence, solve quantitative problems, and evaluate error through clear examples and targeted practice.

Ontario Grade 11 Physics

Scientific Investigation Skills and Career Exploration

A practical approach to A1.8: analyse evidence, solve quantitative problems, and evaluate error

Physics conclusions should be supported by evidence. Evidence can include measurements and observations, or results from a clearly identified model or simulation. A calculation connects quantities; it does not make the evidence exact. In this lesson, you will practise checking data, choosing a relationship, showing calculations, and judging what the results can support. All numerical examples are hypothetical practice values, not reported laboratory measurements.

What you will learn

1. Establish what the evidence says

A measurement is a value obtained by measuring a physical quantity. Examples include a time in seconds or a distance in metres. An observation is a recorded feature of what happened, such as a cart moving past a marked line. A prediction is a value expected from a model. A proposed procedure describes what someone could do. A proposal is not evidence that the procedure was carried out.
Before calculating, identify the physical system: the object or group of objects being studied. List the known values and the unknown quantity. If direction matters, choose a positive direction and use it consistently. A scalar has magnitude only. Time and distance are scalars. A vector has magnitude and direction. Displacement and force are vectors.
Check whether the evidence answers the question. Confirm that values have units and that important information is not missing. If one value looks unusual, keep it visible and investigate it. Do not remove it without a reason. It could be a recording mistake, or it could reflect a real feature of the situation.

2. Choose a relationship and solve

A physics relationship connects quantities in a model. Average speed is total distance divided by elapsed time. Speed is a scalar. Average velocity uses displacement, which includes direction, divided by elapsed time. Read the question carefully so you choose the quantity it asks for.
Use SI units where possible. For distance and time, these are metres and seconds. Substitute the values with their units. Algebra can be used to rearrange a relationship, but the final unit should still match the quantity being found. For example, dividing metres by seconds gives metres per second.
Significant figures are the digits that communicate the precision of a measured value. Do not report far more digits than the supplied values support. Keep extra digits during intermediate calculations if useful, then round the final answer sensibly. For a vector answer, state the direction or give a sign tied to the chosen positive direction.
vavg=ΔdΔtv_{\mathrm{avg}}=\frac{\Delta d}{\Delta t}

3. Analyse patterns and distinguish error from uncertainty

When several values are available, look for a pattern rather than relying on one value. A table organizes values and units. A graph can show a pattern clearly. Label each axis with the quantity and unit. A trend is the general pattern shown by the data. A point far from the trend should be checked, but it is not automatically wrong.
Error is the difference between a measured or calculated result and a value chosen for comparison. The comparison value is called a reference value. Error can be expressed as a signed difference or as the size of the difference. Error does not automatically mean carelessness. It describes a difference; further evidence may be needed to explain why it occurred.
Uncertainty is a limit on how precisely a value is known. For example, a tool’s markings may limit how finely a reading can be made. A person’s reaction time may limit how precisely a time can be recorded. These are possible sources of uncertainty, not themselves the definition of error. A result can have uncertainty even when no reference value is available to calculate error.
Repeated measurements show whether values vary. The mean is their sum divided by their number. The range is the largest value minus the smallest. A wider range indicates more variation in that set. It can help describe consistency, but it does not prove that the mean is correct. Repeated values may all be affected in a similar way, such as by a tool that reads consistently high.
If a suitable reference value is supplied, percentage error compares the size of the difference with the reference. State which value is the reference. Do not report percentage error when no suitable reference is given. To improve a method, suggest a change linked to a likely limitation, such as using a timer with finer readings or repeating a timing measurement.
% error=∣xresult−xreference∣∣xreference∣×100%\%\text{ error}=\frac{|x_{\mathrm{result}}-x_{\mathrm{reference}}|}{|x_{\mathrm{reference}}|}\times100\%

4. Make a conclusion that fits the evidence

A conclusion answers the question using the evidence and calculation. Include the value, unit, and direction if relevant. State whether the evidence is strong or limited and give a reason, such as variation among repeated values or a timing limitation.
Keep observations separate from explanations. “The recorded times differed by 0.05 s” reports a difference in the data. “The difference may be due to reaction time” proposes a possible cause. Do not present that cause as a measured fact unless the evidence supports it.
Finish by checking the units, sign or direction, significant figures, and physical reasonableness. If an answer fails one of these checks, review the data, relationship, or arithmetic. A defensible conclusion is limited to what the available evidence can support.

Worked example

1. Calculate average speed from a stated distance and time

A practice record states that a cart travelled 12.4 m in 3.10 s. Treat these as supplied values, not measurements made in this lesson. Find its average speed.
  1. Define the system and quantities
    The system is the cart during the stated trip. Distance and time are scalars, so direction is not part of the requested average speed. The known values are distance and elapsed time; the unknown is average speed.
  2. Choose the relationship
    Average speed is total distance divided by elapsed time. This relationship matches the scalar distance given in the question.
    vavg=ΔdΔtv_{\mathrm{avg}}=\frac{\Delta d}{\Delta t}
  3. Substitute and check
    Divide the distance in metres by the time in seconds. The unit is metres per second. The supplied values have three significant figures, so report three significant figures.
    vavg=12.4 m3.10 s=4.00 m/sv_{\mathrm{avg}}=\frac{12.4\ \mathrm{m}}{3.10\ \mathrm{s}}=4.00\ \mathrm{m/s}
Answer: The cart’s average speed is 4.00 m/s.
Check: The unit is correct for speed. Three significant figures match the supplied values. A few metres per second is reasonable for the stated trip. Speed does not require a direction.

Worked example

2. Describe repeated times and their spread

A practice data set lists three times for the same event: 2.41 s, 2.46 s, and 2.43 s. Find the mean and range. Also find half the range as a simple description of the spread in this set.
  1. Identify the values
    The system is the event being timed. Time is a scalar. These are supplied practice values, not measurements from a completed experiment.
  2. Calculate the mean
    Add the three times and divide by the number of values. Keep seconds as the unit.
    tˉ=2.41 s+2.46 s+2.43 s3=2.433… s≈2.43 s\bar{t}=\frac{2.41\ \mathrm{s}+2.46\ \mathrm{s}+2.43\ \mathrm{s}}{3}=2.433\ldots\ \mathrm{s}\approx2.43\ \mathrm{s}
  3. Calculate the range and half-range
    Subtract the smallest time from the largest. Half the range is a simple description of the spread of these values. It is not a reference-based error calculation.
    range=2.46 s−2.41 s=0.05 s;range2=0.025 s≈0.03 s\text{range}=2.46\ \mathrm{s}-2.41\ \mathrm{s}=0.05\ \mathrm{s};\quad\frac{\text{range}}{2}=0.025\ \mathrm{s}\approx0.03\ \mathrm{s}
Answer: The mean time is 2.43 s, and half the range is 0.03 s. The values can be summarized as a mean of 2.43 s with a half-range spread estimate of 0.03 s.
Check: Both quantities have units of seconds. The rounded mean is consistent with the hundredth-second entries. The small range shows that these three values are close together; it does not show whether they differ from a reference value.

Worked example

3. Compare a result with a reference

A calculation gives a force magnitude of 9.4 N. A reference value for this practice question is 10.0 N. Find the percentage error relative to the reference.
  1. Define the comparison
    The system is the object whose force is reported. Force is a vector, but this question supplies magnitudes and asks for a comparison of magnitudes. The directions cannot be determined from these values.
  2. Use the percentage relationship
    Percentage error is the absolute difference between the result and reference, divided by the reference magnitude, then multiplied by 100%.
    % error=∣Fresult−Freference∣∣Freference∣×100%\%\text{ error}=\frac{|F_{\mathrm{result}}-F_{\mathrm{reference}}|}{|F_{\mathrm{reference}}|}\times100\%
  3. Substitute and check
    The absolute difference is 0.6 N. Dividing by the 10.0 N reference and multiplying by 100 gives the percentage error.
    % error=∣9.4 N−10.0 N∣10.0 N×100%=6.0%\%\text{ error}=\frac{|9.4\ \mathrm{N}-10.0\ \mathrm{N}|}{10.0\ \mathrm{N}}\times100\%=6.0\%
Answer: The result has a percentage error of 6.0% relative to the supplied 10.0 N reference.
Check: The newton units cancel, leaving a percentage. The result is 0.6 N below the reference, while percentage error reports the size of the difference. The comparison does not identify the cause of the error.

Common mistakes and how to avoid them

Treating a prediction or suggested procedure as measured evidence.
Correction: Label each item as a measurement, observation, prediction, or proposal. A proposal is not evidence that a procedure was carried out.
Using error and uncertainty as if they mean the same thing.
Correction: Error is a difference from a reference value. Uncertainty describes a limit on how precisely a value is known.
Dropping units during substitution or reporting an answer without a unit.
Correction: Write units beside the values and check that the final unit fits the requested quantity.
Giving a scalar answer when the question asks for a vector, or adding a direction to a scalar.
Correction: Check whether direction is part of the quantity. State a positive direction before using signed vector values.
Calling every difference between repeated values a careless mistake.
Correction: Describe the observed variation first. Identify a possible cause only when it fits the stated method.

Lesson summary

Check your understanding

Question 1

A student proposes timing a cart but has not done it yet. Is the proposed time a measured result?
  1. Yes; a proposed procedure is already evidence.
  2. No; it is a plan, not a measurement.
  3. Yes, if the student includes a unit.
  4. No; physics cannot use time measurements.
Show answer and explanation
No; it is a plan, not a measurement.
A proposed procedure describes what could be done. It does not provide measured evidence until measurements are actually collected.

Question 2

A distance of 18 m is travelled in 6.0 s. What is the average speed?
  1. 3.0 m/s
  2. 108 m/s
  3. 0.33 m/s
  4. 3.0 m
Show answer and explanation
3.0 m/s
Average speed is distance divided by time: 18 m divided by 6.0 s gives 3.0 m/s. The unit and two significant figures are appropriate.

Question 3

Three times are 1.8 s, 2.0 s, and 2.2 s. What is their range?
  1. 0.2 s
  2. 0.4 s
  3. 2.0 s
  4. 6.0 s
Show answer and explanation
0.4 s
The range is the largest value minus the smallest: 2.2 s minus 1.8 s equals 0.4 s.

Key terms

Evidence
Measurements or observations used to support a conclusion.
Scalar
A quantity with magnitude but no direction, such as time or distance.
Vector
A quantity with both magnitude and direction, such as displacement or force.
SI units
Standard science units, including metres, seconds, kilograms, and newtons.
Significant figures
Digits that communicate the precision of a measured or calculated value.
Error
The difference between a result and a value chosen for comparison.
Uncertainty
A limit on how precisely a value is known.
Range
The largest value in a data set minus the smallest value.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation A1.8. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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