Handling Data (M1)
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Answers to calculations should be given to an appropriate number of significant figures.
For multiplication and division, give the answer to the same number of significant figures as the value with the fewest significant figures.
For example:
5.9 has the fewest significant figures (2), so the final answer should be given to 2 significant figures.
For addition and subtraction, give the answer to the same number of decimal places as the value with the fewest decimal places.
Question walkthrough
Using an appropriate number of significant figures and rounding answers
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In a multi-step calculation, keep intermediate values unrounded. Rounding too early can change the final answer.
For example:
but
Remember, round only the final answer.
At the end of a calculation, you need to round an answer, usually to a number of significant figures.
Zeroes at the end of a value can be significant figures.
When taking measurements repeat readings are used to improve the reliability of the results. A mean () can then be calculated.
Before calculating a mean, identify anomalous results (or outliers). These should only be excluded if there is a valid reason for doing so.
Question walkthrough
Calculating arithmetic means
Determine the arithmetic mean of three measurements recorded by a student.
When you have taken multiple measurements, one or more may be outliers.
There are no absolute rules for dealing with these results, and you need to consider each case.
Ask yourself:
- Was the suspected anomaly recorded in error?
- Was the suspected anomaly recorded in different conditions from the other values?
If you answer yes to either of these questions, then you can consider the value as anomalous and omit it from the dataset. You should calculate the mean using the remaining values.
For example, a student repeatedly drops effervescent tablets into fresh cups of hot water and measures the time it takes for the reaction to complete.
Question walkthrough
Calculating weighted means
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When you take a measurement, there is an uncertainty because no measuring instrument can show unlimited precision.
The precision offered by the instrument is called its resolution. This is usually determined by the scale on the instrument and is half of the smallest division on the scale.
Absolute uncertainty is the amount by which a measurement could differ from the actual value.
The smallest division on the scale above is
The absolute uncertainty is half of the smallest division.
So the uncertainty in the temperature is
You can calculate the relative uncertainty or percentage uncertainty using the equation:
The percentage uncertainty indicates the absolute uncertainty relative to the measured quantity.
The smallest unit on the balance scale is So, the absolute uncertainty in the measurement is
It is useful to know that when measured quantities are smaller, the percentage uncertainty is greater. An example of this is when a balance is used to measure a mass of the percentage uncertainty will be much greater than a mass of
This is because the number you are dividing the absolute uncertainty by is much smaller.
Question walkthrough
Calculate relative uncertainty in a measurement
Doubles the thermometer’s reading uncertainty to account for two readings (initial and final), then finds the percentage uncertainty in a recorded temperature change.
Always add the absolute uncertainties in the measurements when calculating the percentage uncertainty in the difference between two measurements.
For example, a student records two temperatures, and using a thermometer marked in divisions.
They calculate the percentage uncertainty in the temperature difference.
Question walkthrough
Calculate relative uncertainty when two measurements are added or subtracted
Doubles the thermometer’s reading uncertainty to account for two readings, then finds the percentage uncertainty in the temperature difference between two measurements.