Interpreting Data (CIE A Level Maths: Probability & Statistics 1)

You may be asked to comment on how statistics could affect data or how removing or adding a new piece of data could change statistics you have calculated. You may also be asked to compare two data sets of a similar context.

Analysing statistics calculated from a data set:

• You could be asked to use some statistics you have calculated, such as the median or interquartile range, to make decisions about the set of data you were analysing
  • Writing about the mean or median gives information about where the data is located
  • Writing about the range, interquartile range, standard deviation or variance gives information about how spread out the data is
    • A lower value means the data set is more consistent
    • A greater value means the data set is more spread out, or varied
• When commenting on a data set you should usually write about both a measure of location and a measure of spread
• You should pair the median with the range or interquartile range and the mean with the standard deviation or variance
  • Use the mean and standard deviation/variance when the data is roughly symmetrical and does not contain outliers
  • Use the median and interquartile range when the data contains outliers
• You should always write your analysis in the context of the data
  • Sometimes a lower mean/median is better: time taken to complete a puzzle
  • Sometimes a higher mean/median is better: score on a test

Changing a data set:

• Sometimes it might be discovered that a value was omitted from a data set and needs to be added in
• Similarly a data value could be found to be an error and will need to be removed (cleaned) from the data set
• It is important to be aware of how the measures of location and spread may change when the data is added or removed
  • Adding or removing a data value to the set could change the mean, median and quartiles
  • How each statistics changes will depend on where the data value lies within the data set
    • Adding a data point below the mean, or removing one from above will cause the mean to decrease
    • Adding a data point above the mean, or removing one from below will cause the mean to increase
    • The median and quartiles may or may not change depending on the data value, you should always check these cases individually
• Adding or removing extreme values will change the value of the mean by a lot but affect the median in the same way as any other value

Comparing two data sets

• When comparing two data sets you must comment on both:
  • A measure of location such as the mean, median or mode and
  • A measure of spread such as the range, interquartile range, variance or standard deviation
    • If you comment on the mean as the measure of location you should use the standard deviation or variance as the measure of spread
    • If you comment on the median as the measure of location you should use the interquartile range as the measure of spread
• You should use information about the data to decide which measure of location and measure of spread is the best to use
  • If data contains extreme values then it is best to use the median and interquartile range to compare the data sets
  • Extreme values can cause the mean to be an unreliable statistic
• It is common to be asked to compare two data sets that are represented as two box plots on the same scale
  • You should write about both the median and the interquartile range
• When writing a comparison about the data you should always write in context