Box plots and quartiles
Probability and statistics
- Higher
- Non-calculator
- 8 questions
- 29 marks
These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.
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The marks for class A are $3, 5, 6, 8, 9, 10, 12, 14, 15, 18, 20$. Find the median.
2 marks
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Find the range of the class A marks.
2 marks
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Find the lower quartile, upper quartile and interquartile range of the class A marks.
3 marks
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The box plot shows class B. Read the median and the interquartile range.
Box plot of class B marks 3 marks
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Compare the median and interquartile range of class A with class B.
4 marks
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Using the class B box plot, what percentage of the marks lie above the median? What percentage lie between the quartiles? A mark of $18$ is in which quarter of the data?
4 marks
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An outlier is more than $1.5$ IQRs below $Q_1$ or above $Q_3$. For class A, test whether $2$ and $30$ are outliers.
5 marks
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Compare class A and class B using the medians, interquartile ranges and ranges. The highest mark in class A is $20$ and the highest in class B is $19$. Comment on the claim that class B did better because its box is higher.
6 marks
Mark scheme
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Answer. $10$
- There are $11$ values, so the median is the $6$th.
- $10$
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Answer. $17$
- $20 - 3$
- $17$
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Answer. $6$, $15$ and $9$
- The lower half is $3, 5, 6, 8, 9$, so $Q_1 = 6$.
- The upper half is $12, 14, 15, 18, 20$, so $Q_3 = 15$.
- IQR $= 9$.
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Answer. Median $11$, IQR $8$
- The median line is at $11$.
- IQR $= 16 - 8 = 8$.
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Answer. Class A has median $10$ and IQR $9$. Class B has median $11$ and IQR $8$. Class B has a slightly higher typical mark and is slightly more consistent.
- Class A median $10$, class B median $11$.
- Class A IQR $9$, class B IQR $8$.
- The typical class B mark is a little higher, and class B is a little less spread out.
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Answer. $50\%$ above the median, $50\%$ between the quartiles, and $18$ is in the top quarter.
- Half of the data is above the median.
- The box contains the middle $50\%$.
- $18$ is between $Q_3$ and the maximum, so it is in the top $25\%$.
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Answer. $2$ is not an outlier. $30$ is an outlier.
- $1.5 \times 9 = 13.5$
- Lower fence $= 6 - 13.5 = -7.5$, so $2$ is not an outlier.
- Upper fence $= 15 + 13.5 = 28.5$, so $30$ is an outlier.
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Answer. Class B has the higher median, class A has the larger IQR and range, and class A has the higher top mark. The boxes overlap, so the claim is only partly true.
- Medians: class A $10$, class B $11$.
- Ranges: class A $17$, class B $15$. IQRs: $9$ and $8$.
- Class A has the single highest mark.
- A higher box means a higher typical mark, but the overlap means many students in each class scored similarly.