Using Measures of Dispersion (Edexcel GCSE Statistics)

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

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