Statistics: •• including box plots •• including quartiles and inter-quartile range (Higher tier only)

The idea

Higher tier extends the study of univariate distributions to box plots and to quartiles and the interquartile range. A box plot shows the minimum, lower quartile, median, upper quartile and maximum on a scale, so the box holds the middle half of the data. The lower quartile is the value one quarter of the way through the ordered data and the upper quartile is three quarters of the way through. The interquartile range is upper quartile minus lower quartile and measures the spread of the middle half, so it is less affected by outliers than the range. Use box plots to compare distributions by matching medians and interquartile ranges.

What earns the marks

• Identifies the five values needed for a box plot: minimum, lower quartile, median, upper quartile and maximum.
• Finds the lower quartile as the value one quarter of the way through the ordered data and the upper quartile as the value three quarters of the way through.
• Calculates the interquartile range as upper quartile minus lower quartile.
• Explains that the interquartile range measures the spread of the middle half of the data and is less affected by outliers than the range.
• Compares two box plots by referring to the medians and the interquartile ranges, and comments on the spread and skew suggested by the boxes.

Common mistakes and how to put them right

• Calculating the interquartile range as lower quartile minus upper quartile.

  Put it right: Subtract the lower quartile from the upper quartile so the result is positive.

• Treating the median as the mean.

  Put it right: Remember the median is the middle value of the ordered data, while the mean is the total divided by the count.

• Reading the box plot as showing the range only.

  Put it right: Read the five labelled values and use the box to describe the middle half of the data.

• Assuming the interquartile range is affected by outliers in the same way as the range.

  Put it right: State that the interquartile range uses the middle half and so is more resistant to extreme values.

Exam tips

• Order the data first and count carefully before locating quartiles, especially when the number of values is not a multiple of four.
• On a box plot, check the scale and label each of the five values so the examiner can see your reading.
• When comparing box plots, quote the median and interquartile range for each distribution and state what the difference means in context.