Handling Data (M1)
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You need to be able to give answers to calculations using a number of significant figures that is appropriate for the number of significant figures in the values used.
You also need to be able to round the answer correctly. For example, look at this calculation:
Each value has a different number of significant figures. You can only give an answer to the same number of significant figures as the lowest number of significant figures in any of the values.
In this case, 5.9 has only two significant figures, so you can only give the answer to two significant figures. This calculation does not give an exact value, so you need to be able to round the answer correctly. When calculating results from measurements that you have made, you can only give the results to the limits of the measurement with the lowest resolution (or precision).
If the value were 5.90, then it would have three significant figures. If a measurement is accurate to three significant figures and the final digit is zero, the zero must be included in the value.
Question walkthrough
Using an appropriate number of significant figures and rounding answers
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In a multi-step calculation, you must use the unrounded answer from one calculation in another.
If you use a rounded answer in the next calculation, the final answer may be different.
For example, if you rounded the first part of the calculation:
to two significant figures (9.212 rounds to 9.2), the final answer would be 8.3 instead of 8.4.
When using a scientific calculator:
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 in science, you usually take at least three readings. You then need to calculate the average value of the readings, or the arithmetic mean of the readings. You then use this value in subsequent calculations.
You can use this formula to calculate the arithmetic mean
Sometimes you obtain readings that are anomalous (or outliers), and you need to consider what to do with them. There are no absolute rules for dealing with them, and you need to consider each case.
Question walkthrough
Calculating arithmetic means
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When you have taken at least three readings of a measurement, one reading may be anomalous or an outlier.
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 other 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.
If something happens at random, it means that we do not know what will happen. However, we may know the probability (or the chance) of certain outcomes.
For example: If you roll a fair six-sided die, there are six outcomes (1, 2, 3, 4, 5 and 6), which all have an equal chance. So the probability of rolling a six is
It is important to note that probabilities are numbers between zero and one, expressed as fractions, decimals or percentages. Probabilities close to zero are unlikely to occur and probabilities close to one are more likely.
If is the probability of something happening, and is the probability of that it not happening, then:
This is because 1 represents the total probability of all possible outcomes, and in this case, there are two outcomes: that it happens or not.
Sometimes, you can use this to calculate a probability. For example, the probability that a fair die lands on six is What is the probability that it does not land on six?
If you toss a coin 10 times, it may land on heads exactly 5 times, because for each toss, the probability of heads is However, it could instead land heads 4 times, or 7 times, or even not at all, because the process is random, and the tosses are independent.
In both scientific contexts and everyday situations, calculations often involve extremely large or small numbers.
You must take extra care with the order of magnitude, especially when estimating.
For example, it is simple to calculate but more challenging to find
There are many shortcuts that can help you, including:
- using English words like ‘million’ and facts such as that
- writing out the numbers in full and counting the number of zeroes,
- writing the calculation in standard form, and
- round the values to very easy numbers for the arithmetic, so it is easier to focus on the correct order of magnitude.
Question walkthrough
Make an order of magnitude calculation
Rounds given values to one significant figure, converts to consistent SI units, then multiplies rainfall height by island area to find volume and by water’s density to estimate the mass of rainwater to an order of magnitude.
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. It is called absolute uncertainty, which 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 because you cannot give a measurement with greater accuracy than 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.
When you add or subtract measurements, you must add the absolute uncertainties in the measurements together:
Often, the quantity measured is the difference between two measurements, and the absolute uncertainties of the two measurements are equal. In such cases, the percentage uncertainty becomes:
It is useful to know that ‘quantity measured’ in the equation is the sum or difference of the quantities measured.
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.
Question walkthrough
Calculate absolute and relative uncertainty in a sum of measurements
Converts all component masses to the same units before summing, then finds the absolute uncertainty (from four combined reading uncertainties) and the relative uncertainty in the total mass.
When calculating the percentage or relative uncertainty, ensure that the absolute uncertainty and measurement are expressed in the same units.
For example, a length is measured as using a ruler with millimetre divisions. The absolute uncertainty is:
As a distance measurement comes from two readings, in this case at and the combined absolute uncertainty is:
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.
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.
When you multiply or divide measurements, first calculate the percentage uncertainty in each measurement. Then, add the percentage uncertainties together to work out the total uncertainty.
When you raise a measurement to a power, you multiply the percentage uncertainty by that power.
For example, when you square a measurement, you multiply the percentage uncertainty by
Question walkthrough
Calculate relative uncertainty when two measurements are multiplied or divided
Doubles the ruler’s reading uncertainty to account for two readings per length measurement, finds the percentage uncertainty in each side length, then sums them to find the percentage uncertainty in the block’s face area.
A student measures the potential difference across a resistor as and the current through it as . The absolute uncertainties are and respectively.
The student uses the values to calculate resistance using the equation:
The student then calculates the percentage uncertainty in the resistance.