A1.13 Report calculated results with appropriate accuracy and precision

Ontario Grade 10 Science

Scientific Investigation and Careers

How to choose a sensible number of digits and explain what a calculated result means

In Grade 9, you learned to measure quantities and use measurements in calculations. A calculator can display many digits, but that does not mean every digit is supported by the measurements. For example, if you measure a distance to the nearest centimetre, reporting a calculated distance to one ten-thousandth of a centimetre suggests a level of detail you did not measure. In this lesson, you will learn how to report a calculated result so that its digits, units, and wording match the evidence. The examples use safe, hypothetical data.

What you will learn

Explain the difference between accuracy and precision.
Use the precision of measured values to decide how many digits to report in a calculation.
Round a calculated result without changing its meaning.
Report a result with units and enough context for another person to understand it.

1. From measurements to a trustworthy result

A measurement is a value found by comparing something with a unit, such as centimetres or seconds. A calculated result is found by using one or more values in a calculation. Before using a measurement, check its unit and how finely it was recorded. If a ruler has markings every centimetre, a length recorded as

12.4\ \mathrm{cm}

would claim more detail than those markings show.

Accuracy describes how close a result is to the accepted or actual value. Precision describes how finely a value is stated, or how closely repeated results agree with one another. These ideas are related but not the same. A value can be written with many digits and still be inaccurate. Repeated measurements can be close together but all differ from the accepted value.

A reported result should not claim more precision than the measurements support. Extra digits from a calculator are often only digits produced by the arithmetic. They do not make the original measurements more detailed.

Accuracy is closeness to an accepted or actual value.
Precision is the detail or repeatability of a value.
Calculator digits are not automatically meaningful measurement digits.

2. Match the reported digits to the calculation

For addition and subtraction, the result is usually reported to the same decimal place as the least detailed value in the calculation. A decimal place is a position to the right of the decimal point. For example, a value recorded to the nearest tenth has one decimal place.

For multiplication and division, a useful Grade 10 rule is to report about as many significant digits as the measured value with the fewest significant digits. Significant digits are the digits that show the recorded precision of a value. In a value such as

3.2

, both digits are significant. A trailing zero after a decimal point, as in

3.20

, indicates additional recorded detail.

These rules help prevent false precision. They do not replace judgment: the final result also needs a suitable unit and a clear description. Keep extra digits during the calculation when needed, then round the final result. Rounding earlier can make the final answer less reliable.

To round, look at the digit immediately after the last digit you plan to keep. If it is

5

or greater, increase the last kept digit by one. If it is less than

5

, leave that digit unchanged. Then remove the digits beyond the place you are reporting.

Addition and subtraction: match the least precise decimal place.
Multiplication and division: use the fewest significant digits among the measured inputs as a guide.
Round the final result, not every intermediate result.

3. Report the evidence clearly

A complete result usually includes the calculated value, its unit, and enough words to identify what the value describes. State whether a number is a measured value, a calculated value, or a comparison with an accepted value. This makes the evidence easier to check.

Do not add digits just to make an answer look exact. Do not remove so many digits that useful information is lost. If a value is rounded, make sure the wording does not imply that the unrounded calculator display was measured.

In a school investigation, use only data collected safely and with suitable equipment. Follow your teacher's directions for equipment and materials. The examples here are hypothetical; they are not claims about measurements from a real investigation.

Include units and identify what the result describes.
Make clear when a value is calculated or rounded.
A well-reported result reflects the evidence, not the number of digits on a screen.

Worked example

Adding measured lengths

Hypothetical data: A student records two connected lengths as

12.4\ \mathrm{cm}

and

3.27\ \mathrm{cm}

. What total length should be reported?

Add the values
Add the lengths because the total is made from both connected parts. Keep the units the same.
$12.4\ \mathrm{cm}+3.27\ \mathrm{cm}=15.67\ \mathrm{cm}$
Choose the decimal place
The first measurement is recorded to the nearest tenth of a centimetre. The second is recorded to the nearest hundredth. For addition, the less detailed decimal place controls the reported result, so round to tenths.
$15.67\ \mathrm{cm}\approx15.7\ \mathrm{cm}$
Report the result
State what the number represents and include its unit. The final digit reflects the precision of the less detailed measurement.

Answer: The calculated total length is

15.7\ \mathrm{cm}

.

Check: The result is reported to one decimal place, matching the least detailed measurement.

Worked example

Calculating speed from distance and time

Hypothetical data: A toy cart travels

2.4\ \mathrm{m}

in

3.0\ \mathrm{s}

. Calculate and report its average speed. Average speed is distance divided by elapsed time.

Set up the calculation
Use the given relationship for average speed. Dividing metres by seconds gives metres per second.
$v=\frac{d}{t}$
Calculate before rounding
Substitute the hypothetical distance and time. The calculator value is kept for now so that rounding does not affect a later step.
$v=\frac{2.4\ \mathrm{m}}{3.0\ \mathrm{s}}=0.8\ \mathrm{m/s}$
Check the precision
Both measured inputs have two significant digits. The calculated value, $0.8$ , has only one significant digit. It is better to report the same level of detail as the inputs by writing the trailing zero after the decimal point.
$0.8\ \mathrm{m/s}=0.80\ \mathrm{m/s}$

Answer: The cart's calculated average speed is

0.80\ \mathrm{m/s}

.

Check: The trailing zero shows that the reported speed has two significant digits, consistent with the two measured inputs.

Worked example

Reporting an average

Hypothetical data: Three classroom trials give calculated values of

4.6\ \mathrm{s}

,

4.8\ \mathrm{s}

, and

4.7\ \mathrm{s}

. Find the average and report it appropriately.

Find the mean
The mean, often called the average, is found by adding the values and dividing by how many values there are. Here, there are three trials.
$\frac{4.6\ \mathrm{s}+4.8\ \mathrm{s}+4.7\ \mathrm{s}}{3}=4.7\ \mathrm{s}$
Choose a sensible precision
Each trial is recorded to the nearest tenth of a second, and the average is exactly $4.7$ seconds. Reporting it to the nearest tenth keeps the same level of detail as the trial values.
$4.7\ \mathrm{s}$
Describe the average
Name the quantity and state that it is an average of three trials. This tells the reader how the reported value was obtained.

Answer: The average time for the three hypothetical trials is

4.7\ \mathrm{s}

.

Check: The sum is

14.1\ \mathrm{s}

, and dividing by three gives

4.7\ \mathrm{s}

.

Common mistakes and how to avoid them

Copying every digit shown by a calculator into the final answer.

Correction: Use the measured inputs to decide how much precision is supported, then round the final result.

Thinking that a very precise-looking value must be accurate.

Correction: Precision describes detail or repeatability. Accuracy describes closeness to an accepted or actual value.

Leaving off the unit or failing to say what was calculated.

Correction: Include a suitable unit and identify the quantity, such as average speed or total length.

Rounding each value before finishing a multi-step calculation.

Correction: Keep useful digits during the calculation and round the final reported result.

Lesson summary

A reported result should match the precision of the measurements used to calculate it.
For addition and subtraction, report to the least detailed decimal place.
For multiplication and division, use the input with the fewest significant digits as a guide.
Round at the end and include units and a clear description.
Accuracy and precision are different: one concerns closeness to an accepted value, and the other concerns detail or repeatability.

Check your understanding

Question 1

Hypothetical measurements are

8.36\ \mathrm{cm}

and

1.2\ \mathrm{cm}

. What total should be reported?

$9.56\ \mathrm{cm}$
$9.6\ \mathrm{cm}$
$9.560\ \mathrm{cm}$
$10\ \mathrm{cm}$

Show answer and explanation

9.6\ \mathrm{cm}

The values are added, so report to the least detailed decimal place. The value

1.2

is recorded to tenths, giving a total of

9.6\ \mathrm{cm}

.

Question 2

A result is repeated closely, but it is not close to an accepted value. Which description fits best?

It is precise but not accurate.
It is accurate but not precise.
It is both accurate and precise.
It cannot be described using accuracy or precision.

Show answer and explanation

It is precise but not accurate.

Close agreement among repeats indicates precision. Being far from the accepted value means the result is not accurate.

Question 3

A calculator displays many digits after dividing two measured values. What should you do?

Report every displayed digit.
Round the final result to a precision supported by the measured inputs.
Always round to a whole number.
Remove the unit to make the result more precise.

Show answer and explanation

Round the final result to a precision supported by the measured inputs.

The display may contain digits that the measurements do not support. Use the input precision as a guide, round the final result, and keep the appropriate unit.

Key terms

Measurement: A value found by comparing a quantity with a unit.
Calculated result: A value found by using one or more values in a calculation.
Accuracy: How close a result is to an accepted or actual value.
Precision: The detail shown by a value, or how closely repeated results agree.
Decimal place: A position to the right of the decimal point.
Significant digits: Digits that show the recorded precision of a value.
Mean: An average found by adding values and dividing by the number of values.

Continue through SNC2D

View the complete SNC2D Ontario Grade 10 Science curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Science (SNC2D), expectation A1.13. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

A1.13 · Report calculated results with appropriate accuracy and precision

Ontario Grade 10 Science

How to choose a sensible number of digits and explain what a calculated result means

What you will learn

1. From measurements to a trustworthy result

2. Match the reported digits to the calculation

3. Report the evidence clearly

Adding measured lengths

Calculating speed from distance and time

Reporting an average

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SNC2D

About this lesson and its review