Using Measures of Dispersion (Edexcel GCSE Statistics)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
Comparing data sets
How do I compare two data sets?
You may be given two sets of data that relate to a context
To compare data sets, you need to
compare an average (measure of central tendency)
Mode, median or mean
AND compare a measure of spread (measure of dispersion)
Range, interquartile range (IQR), interpercentile or interdecile range, standard deviation
You need to use the same average and the same measure of spread for both data sets
You may need to decide which average should be used
See the 'Using Measures of Central Tendency' revision note
Which measure of spread to use depends on which average is used
If you compare the modes of the data sets
then use the range (quantitative data only)
If you compare the medians of the data sets
then use the range, interquartile range, interpercentile range or interdecile range
interquartile range is the most common choice
standard deviation should not be used with median
If you compare the means of the data sets
then use the range or standard deviation
standard deviation is the most common choice, if it is available
interquartile range should not be used with mean
How do I write a conclusion when comparing two data sets?
When comparing averages and spreads, you need to
compare numbers
describe what this means in the context of the question ('in real life')
Copy the exact wording from the question in your answer
There should be four parts to your conclusion
For example:
"The median score of class A (45) is higher than the median score of class B (32)."
"This means that, on average, class A performed better than class B in the test."
"The range of class A (5) is lower than the range of class B (12)."
"This means the scores in class A were less spread out than scores in class B."
Other good phrases for lower ranges include:
"scores are closer together"
"scores are more consistent"
"there is less variation in the scores"
What restrictions are there when drawing conclusions?
The data set may be too small to be truly representative
Measuring the heights of only 5 pupils in a whole school is not enough to talk about averages and spreads
The data set may be biased
Measuring the heights of just the older year groups in a school will make the average appear too high
The conclusions might be influenced by who is presenting them
A politician might choose to compare a different type of average if it helps to strengthen their argument!
What else could I be asked?
You may need to think from the point of view of another person
A teacher might not want a large spread of marks
It might show that they haven't taught the topic very well!
An examiner might want a large spread of marks
It makes it clearer when assigning grade boundaries, A, B, C, D, E, ...
You may be asked to compare data from a sample with data from the population as a whole
For example, to determine how representative the sample is of the population
Examiner Tips and Tricks
To get full marks when when comparing data sets in the exam, you must
be sure to use appropriate averages and measures of spread
compare the numbers
say what the numbers mean in the context of the question
Worked Example
Julie collects data showing the distances travelled by snails and slugs during a ten-minute interval. She records a summary of her findings, as shown in the table below.
| Median | Interquartile Range |
Snails | 7.1 cm | 3.1 cm |
Slugs | 9.7 cm | 4.5 cm |
Compare the distances travelled by snails and slugs during the ten-minute interval.
Compare the numerical values of the median (an average)
Describe what this means in the context of the question
Slugs have a higher median than snails (9.7 cm > 7.1 cm)
This suggests that, on average, slugs travel further than snails
Compare the numerical values of the interquartile range (the spread)
Describe what this means in the context of the question
Snails have a lower range than slugs (3.1 cm < 4.5 cm)
This suggests that there is less variation in the distances travelled by snails
Standardising Data
What do we mean by standardising data?
It is possible to standardise the data collected in two samples
This makes it easier to compare data values in the two samples
e.g. Michelle scores an 80 on a maths exam and a 72 on an English exam
The two exams are quite different
So which of those is really the 'better' score?
How do I standardise data?
Each data value is converted into a standardised score using the formula
the raw value is the original data value from the data set
the mean is the mean of the data set the raw value belongs to
the standard deviation is the standard deviation of the data set the raw value belongs to
The formula calculates how many standard deviations the raw value is away from the mean
This can also be written as
is the raw data value, is the mean, is the standard deviation
The formula is not on the exam formula sheet, so you need to remember it
If the raw value is greater than the mean then the standardised score will be positive
The more positive the standardised score is, the higher the raw value is compared with the average for the data set
If the raw value is less than the mean then the standardised score will be negative
The more negative the standardised score is, the lower the raw value is compared with the average for the data set
Assuming that a higher raw value is a good thing, then
A higher standardised score means a better result than a lower standardised score
A more positive standardised score means a better result than a less positive one
A more negative standardised score means a worse result than a less negative one
You may be given the standardised score and asked to find one of the other values (, or )
If you know any three values in the formula you can find the fourth
Substitute in the values you know
And solve for the one you want to know
Examiner Tips and Tricks
You may need to calculate the standard deviation for a data set before standardising scores
Remember that the formulas for standard deviation are on the exam formula sheet
Worked Example
The table shows the mean and standard deviation for the scores on a maths exam and an English exam for all the students who sat the exams.
Mean | Standard deviation | |
---|---|---|
Maths | 63.2 | 11.9 |
English | 56.1 | 9.3 |
Michelle received a score of 80 on the maths exam, and a score of 72 on the English exam.
Use standardised scores for this information to compare Michelle’s performance on the maths exam with her performance on the English exam.
Explain how you reach your conclusion.
Use to calculate the standardised scores for the maths and English exams
Michelle's raw score is higher on the maths exam
But the standardised scores show that compared to all students who sat the exams she actually performed better on the English exam
Michelle's standardised score for the English exam is higher than her standardised score for the maths exam. This means that compared to other students who sat the exams Michelle performed better on the English exam than she did on the maths exam.
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