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

Ontario Grade 11 Chemistry

Scientific Investigation Skills and Career Exploration

A Grade 11 guide to interpreting results and judging their reliability

A balance may show slightly different masses when the same sample is weighed more than once. Each reading is evidence: an observation recorded so it can be examined. Evidence can support a conclusion, but it does not automatically prove that the conclusion is correct. Check what the measurements show, calculate carefully, and consider how measurement limits may have affected the results. This lesson uses a practice dataset. Its values are supplied for learning and are not results from a completed investigation.

What you will learn

Describe how observations and measurements can support a chemistry conclusion.
Use averages, differences, and percent error to analyse numerical evidence.
Distinguish accuracy from precision and identify possible sources of error.
Report calculated results with units and appropriate significant digits.

1. From observations to evidence

An observation is information gathered with the senses or an instrument. A description such as “a white solid formed” is qualitative because it describes a quality rather than giving a number. A balance reading such as

4.92\ \mathrm{g}

is quantitative because it includes a number and a unit.

A measurement is a quantitative observation made with a tool. A balance measures mass, while a graduated cylinder measures liquid volume. A recorded number without its unit is incomplete. The unit tells you what kind of quantity was measured.

Visible changes can provide evidence about what may be happening among particles. For example, a new solid appearing in a mixture is an observation. It may be evidence that a substance with different properties has formed. A particle-level explanation is a model used to explain the observation; it is not the observation itself.

Before using results, check that they address the question and that their units match. Look for patterns, unusual values, and differences among repeated readings. A conclusion should fit the evidence and should not claim more than the evidence supports.

Record the measured value and its unit.
Separate observations from explanations of what caused them.
Use patterns and differences in results to judge what the evidence supports.

2. Summarize and compare measurements

An average, also called a mean, summarizes repeated measurements with one central value. Add the measurements and divide by their number. All values must use the same unit. The mean does not show how close the individual readings are to one another, so inspect the readings too.

A reference value is a value supplied for comparison, such as an accepted value stated in a problem. The absolute difference is the positive size of the gap between a measured result and the reference. Percent error expresses that gap as a percentage of the reference value.

In these calculations, error means the difference between a measured result and a reference value. It does not necessarily mean that someone made a mistake. The calculation gives the size of the difference, but it does not identify its cause.

Use matching units before comparing values. In percent error, the units cancel because both the difference and the reference value use the same unit. The percentage has no unit. Keep extra digits during intermediate calculations, then round the reported result appropriately.

\bar{x}=\frac{\sum x_i}{n},\quad \Delta x=|x_{\mathrm{measured}}-x_{\mathrm{reference}}|,\quad \%\mathrm{error}=\frac{\Delta x}{x_{\mathrm{reference}}}\times100\%

Find the mean by adding the values and dividing by their count.
Find an absolute difference by subtracting the smaller value from the larger one.
Percent error compares the difference with the reference value.
Keep units consistent when comparing measurements.

3. Evaluate accuracy, precision, and possible error

Accuracy describes how close a result is to a reference value. Precision describes how close repeated measurements are to one another. These are different ideas. Readings can be close together but still far from the reference. That pattern shows precision but not accuracy.

Random error is unpredictable variation that can make repeated readings differ in either direction. For example, a person might read a scale slightly differently each time. Repeating measurements and finding a mean can help summarize variation, but repetition does not remove every source of error.

Systematic error shifts measurements in a similar direction each time. A balance that does not read zero when empty could make every mass too high or too low. Repeating measurements with the same uncorrected balance may give closely grouped results, even though they are not accurate.

To evaluate error, describe what the results show, name a possible source only when reasonable, and explain how it could affect the measurement. Do not say a possible cause definitely occurred unless the evidence supports that claim. If the cause is unknown, say so.

Significant digits communicate the precision of a measured or calculated value. For addition and subtraction, round the reported result to the least precise decimal place in the values used. For multiplication and division, round to the fewest significant figures among the measured values. Do not round intermediate values too early. Keep extra digits for a later calculation, then clearly report the final value.

Accuracy compares a result with a reference; precision compares repeated readings with one another.
Random error can vary from one reading to the next.
Systematic error can shift readings in a similar direction.
Separate what the evidence shows from what you think may explain it.
Use the appropriate rounding rule and do not report more precision than the measurements support.

Worked example

Find a mean and compare it with a reference

Practice dataset: three mass readings are

4.92\ \mathrm{g}

,

4.96\ \mathrm{g}

, and

4.94\ \mathrm{g}

. A problem supplies

4.90\ \mathrm{g}

as the reference value. Find the mean and percent error, then describe what the results suggest about accuracy and precision. These values are for practice, not reported measurements from a completed investigation.

Find the mean
Add the three readings and divide by their count, which is three. The mean is a mass, so keep grams in the calculation.
$\bar{x}=\frac{4.92\ \mathrm{g}+4.96\ \mathrm{g}+4.94\ \mathrm{g}}{3}=4.94\ \mathrm{g}$
Find the absolute difference
Subtract the smaller mass from the larger mass. Both values are recorded to the hundredths place, so the difference is reported to that place as well. The result, $0.04\ \mathrm{g}$ , has one significant figure.
$|4.94\ \mathrm{g}-4.90\ \mathrm{g}|=0.04\ \mathrm{g}$
Calculate percent error
Divide the difference by the reference value and multiply by one hundred percent. Grams cancel. Since the difference has one significant figure, report the percentage to one significant figure.
$\frac{0.04\ \mathrm{g}}{4.90\ \mathrm{g}}\times100\%=0.8\%$
Interpret the evidence
The readings are close to one another, so this small set shows good precision. The mean is slightly above and close to the reference. The calculation does not tell you why the mean differs. Do not claim a specific cause without further evidence.

Answer: The mean mass is

4.94\ \mathrm{g}

. The absolute difference is

0.04\ \mathrm{g}

, and the percent error is 0.8%. The readings show good precision, and the mean is close to but above the reference.

Check: The mean lies between the smallest and largest readings. The absolute difference is positive. The percentage is small compared with the reference, consistent with the mean being close to it.

Common mistakes and how to avoid them

Reporting a number without a unit.

Correction: Include units for measured quantities and calculated quantities that have units, such as

4.94\ \mathrm{g}

.

Calling similar repeated readings accurate without comparing them with a reference.

Correction: Similar readings show precision. Accuracy requires comparison with a reference value.

Rounding an intermediate value and then using it in a later calculation without considering the effect.

Correction: Keep extra digits during intermediate calculations. Round the final reported value to a precision supported by the measurements.

Giving percent error the original measurement unit.

Correction: The units cancel when the difference is divided by the reference, so percent error has no unit.

Stating a possible cause of error as though it were confirmed.

Correction: Describe it as a possibility unless the evidence establishes that it happened.

Lesson summary

Evidence includes observations and measurements; interpretation explains what they may mean.
Use means and differences to summarize and compare numerical results.
Percent error expresses a difference from a reference as a percentage of that reference.
Accuracy compares with a reference, while precision describes agreement among repeated readings.
Evaluate error by describing the results, considering plausible causes, and separating evidence from speculation.
Round reported values according to the precision of the measurements, while retaining extra digits during calculations.

Check your understanding

Question 1

Which statement best describes precision?

Repeated readings are close to one another.
A result is close to a supplied reference value.
A measurement has a unit.
A result has no possible error.

Show answer and explanation

Repeated readings are close to one another.

Precision describes how closely repeated readings agree with one another. Closeness to a reference describes accuracy.

Question 2

A mean is

8.4\ \mathrm{g}

and the reference value is

8.0\ \mathrm{g}

. What is the absolute difference?

$0.4\ \mathrm{g}$
$1.05\ \mathrm{g}$
$4.0\ \mathrm{g}$
0.4%

Show answer and explanation

0.4\ \mathrm{g}

Subtract the smaller mass from the larger mass:

8.4\ \mathrm{g}-8.0\ \mathrm{g}=0.4\ \mathrm{g}

. The difference retains the unit grams.

Question 3

A balance may have been set incorrectly, shifting all readings in the same direction. Which type of error is this?

Systematic error
Random error
Percent error
A unit conversion

Show answer and explanation

Systematic error

A systematic error shifts results in a similar direction. Random error varies unpredictably among readings.

Key terms

Evidence: Observations or measurements used to support a conclusion.
Quantitative observation: An observation expressed with a number and, when appropriate, a unit.
Mean: The sum of a set of values divided by the number of values.
Reference value: A comparison value supplied or established for a problem.
Absolute difference: The positive size of the gap between two values.
Accuracy: How close a result is to a reference value.
Precision: How close repeated measurements are to one another.
Random error: Unpredictable variation that can make repeated measurements differ.

Continue through SCH3U

View the complete SCH3U Ontario Grade 11 Chemistry curriculum and lessons

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Chemistry (SCH3U), 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.

Official curriculum reference

Report a correction or ask a question

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

Ontario Grade 11 Chemistry

A Grade 11 guide to interpreting results and judging their reliability

What you will learn

1. From observations to evidence

2. Summarize and compare measurements

3. Evaluate accuracy, precision, and possible error

Find a mean and compare it with a reference

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SCH3U

About this lesson and its review