How to Compare Distributions Using Box Plots at GCSE

What does a box plot show?

A box plot (also called a box-and-whisker diagram) displays five key values for a data set:

• Minimum — the lowest value (left whisker end)
• Lower quartile (Q1) — the value 25% of the way through the ordered data
• Median (Q2) — the middle value
• Upper quartile (Q3) — the value 75% of the way through the ordered data
• Maximum — the highest value (right whisker end)

From these five values you can immediately read off:

• Interquartile range (IQR) = Q3 − Q1 (the width of the box)
• Range = Maximum − Minimum (the total width of the diagram)
• Skewness — indicated by where the median sits within the box

How do you compare two distributions using box plots?

Follow these steps when answering a "compare" question in the exam:

1. Read off the key values. For both distributions, note the median, Q1, Q3, minimum and maximum. Calculate the IQR and range for each.
2. Compare the medians. State which distribution has the higher (or lower) median and say what this means in the context of the question.
3. Compare the spread. Use the IQR (preferred) or the range to say which distribution is more spread out or more consistent. State which measure you are using.
4. Comment on skewness if asked. If the median is closer to Q1 the distribution is positively skewed; if closer to Q3 it is negatively skewed; if central it is roughly symmetric.
5. Write your comparison in context. Every comparison sentence must include the context — the name of the group, the subject being measured, and what a higher or lower value actually means.

Sample comparison table: two exam classes

The following table shows box plot values for the maths exam scores of two classes.

A complete comparison answer for these two distributions:

"Class B had a higher median score (62 marks) than Class A (58 marks), suggesting Class B performed better overall. However, Class A had a larger IQR (32 marks compared to 18 marks), indicating that Class A's scores were more spread out and less consistent."

Notice: two sentences, each containing a comparison and a contextual interpretation. This is the structure GCSE mark schemes expect.

What does skewness tell you in a box plot?

• Positive skew: The median is closer to Q1, and the right whisker is longer. Most values cluster at the lower end, with a few high outliers pulling the distribution to the right.
• Negative skew: The median is closer to Q3, and the left whisker is longer. Most values cluster at the higher end, with a few low values pulling the distribution to the left.
• Symmetric: The median sits in the middle of the box and the whiskers are roughly equal in length.

Skewness affects how you interpret the median — in a positively skewed distribution the mean is pulled higher than the median by the extreme values.

How to read a GCSE exam question on box plots

A typical GCSE exam question reads:

"The box plots show the waiting times, in minutes, at two hospitals. Compare the distributions of waiting times."

This is your signal to:

• Compare the medians (which hospital has shorter typical waiting times?).
• Compare the IQRs (which hospital is more consistent?).
• Relate each comparison back to the context (hospitals, waiting times).

If the question says "compare fully" or gives three marks, also comment on skewness or the range. If it gives two marks, one comparison with context is sufficient for each mark.

Common mistakes when comparing box plots

• Making comparisons without context. Writing "Class A has a higher median" earns no marks; "Class A had a higher median score, suggesting higher typical performance" earns the mark.
• Confusing IQR and range. The IQR is the better measure of spread because it ignores extreme values. Always state which measure you are using.
• Saying "more data" when you mean "more spread." A box plot does not show the number of data values — it shows their distribution. Never write "Class A has more data than Class B."
• Forgetting to state the actual values. Quoting specific figures (e.g. "median of 62") strengthens your answer and earns method marks even if you misread a scale.

Frequently asked questions

Q: Do I always need to use the median when comparing box plots?

Yes — the median is the appropriate measure of location when using a box plot, because the box plot is built around quartiles rather than the mean. Always compare medians, not means, unless you are told additional information that allows you to calculate the mean.

Q: What if the box plots overlap significantly?

Overlapping box plots mean the distributions are similar in range or IQR. Focus your comparison on the medians and state that the overall spread is comparable. You can still comment on which group has a higher central tendency even if the distributions overlap.

Q: How many comparisons do I need to make?

As a rule of thumb: one mark per distinct, contextualised comparison. A two-mark question needs two separate points (typically: one about location, one about spread). A three-mark question needs three points — location, spread, and either skewness or a comment about range alongside IQR.

Q: Can I use the range instead of the IQR for spread?

Yes, but the IQR is preferred because it is not affected by outliers. In an exam, state which measure you are using. If the question mentions outliers in the scenario, using the IQR shows better statistical judgement and is more likely to receive full marks.

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