Describing and comparing two distributions

Comparing two distributions

• Draw both box plots on the same scale, one above the other, so you can compare them directly.
• Work through four things every time: centre, spread, shape, and overlap — plus any unusual features.

Context: press-ups in one minute for a random sample from a Year 13 PE class (Sample A) and a Year 11 PE class (Sample B).

Compare the centre (median)

• Read each median and state which is higher and by how much, in context.
• Example: "The median for Sample A is 34 press-ups, which is 10 more than Sample B's median of 24."

Compare the spread (IQR)

• Compare the box widths (the IQRs). A wider box means the middle 50% is more spread out — less consistent.
• Example: "Sample A's middle 50% is a little more spread out than Sample B's, so the Year 13 results are slightly less consistent."

Compare the shape

• Describe skew for each group from the median's position in its box and the whisker lengths.
• Example: "Sample B is roughly symmetric; Sample A is skewed toward the higher values."

Overlap — do the boxes separate?

• Look at whether the boxes overlap or sit clearly apart.
• If the boxes overlap a lot, plenty of Sample B members score higher than plenty of Sample A members, even though A's median is higher — so the groups are not cleanly separated.
• Example: "The boxes overlap between about 26 and 42, so many Year 11 students did more press-ups than many Year 13 students."

Unusual features

• Point out outliers, gaps, or a very long whisker — anything that a single summary number would hide.
• These can make the median misleading, so they are worth a sentence.

Writing "I notice ..." statements in context

• Turn each comparison into a full sentence that names the variable, the groups, and a number:
  • "I notice the median number of press-ups for Sample A (34) is higher than for Sample B (24)."
  • "I notice Sample A's results are more spread out (wider box)."
  • "I notice the boxes overlap, so the groups are not clearly separated."