Using Measures of Dispersion (Edexcel GCSE Statistics)

Revision Note

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

    • standardised space score equals fraction numerator raw space value minus mean over denominator standard space deviation end fraction

      • 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 standardised space score equals fraction numerator x minus mu over denominator sigma end fraction

      • x is the raw data value, mu is the mean, sigma 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 (x, mu or sigma)

    • 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 fraction numerator x minus mu over denominator sigma end fraction to calculate the standardised scores for the maths and English exams

table row cell Maths space standardised space score end cell equals cell fraction numerator 80 minus 63.2 over denominator 11.9 end fraction equals 1.4117... end cell end table

table row cell English space standardised space score end cell equals cell fraction numerator 72 minus 56.1 over denominator 9.3 end fraction equals 1.7096... end cell end table

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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Roger B

Author: Roger B

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Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

Author: Dan Finlay

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.