Year 10 Statistics Worksheets & Practice Tests

Worked examples

Year 10 statistics: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Finding quartiles and IQR

   Find the median, Q1, Q3 and interquartile range for the dataset: 12, 7, 15, 3, 9, 18, 11, 5, 14.

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   Key idea: Sort the data, find the median, then find Q1 and Q3 as the medians of the lower and upper halves.

   1. Sort the data: 3, 5, 7, 9, 11, 12, 14, 15, 18 (9 values).
   2. The median (Q2) is the 5th value: $Q2 = 11$.
   3. Lower half (values below the median): 3, 5, 7, 9. Q1 is the median of these four values: $Q1 = \frac{5 + 7}{2} = 6.$
   4. Upper half (values above the median): 12, 14, 15, 18. Q3 is the median of these four values: $Q3 = \frac{14 + 15}{2} = 14.5.$
   5. Calculate the IQR: $\text{IQR} = Q3 - Q1 = 14.5 - 6 = 8.5.$

   Answer: $Q2 = 11$, $Q1 = 6$, $Q3 = 14.5$, $\text{IQR} = 8.5$.

2. Example 2 · Identifying outliers using the 1.5 × IQR rule

   A dataset of ten reaction times (in milliseconds) is: 210, 230, 245, 250, 260, 275, 280, 290, 310, 480. For this dataset, $Q1 = 245$, $Q3 = 290$, and $\text{IQR} = 45$. Use the $1.5 \times \text{IQR}$ rule to determine whether any values are outliers.

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   Key idea: A value is an outlier if it lies below $Q1 - 1.5 \times \text{IQR}$ or above $Q3 + 1.5 \times \text{IQR}$.

   1. Calculate the lower fence: $Q1 - 1.5 \times \text{IQR} = 245 - 1.5 \times 45 = 245 - 67.5 = 177.5.$
   2. Calculate the upper fence: $Q3 + 1.5 \times \text{IQR} = 290 + 1.5 \times 45 = 290 + 67.5 = 357.5.$
   3. Any value below $177.5$ or above $357.5$ is an outlier. Check the extremes: the minimum is $210 > 177.5$ (not an outlier), and the maximum is $480 > 357.5$ (outlier).

   Answer: $480$ ms is an outlier. No values fall below the lower fence.

3. Example 3 · Comparing two boxplots

   Two classes each sat the same maths test scored out of 100. Their results are summarised below.

   	Class A	Class B
   Minimum	42	55
   Q1	58	62
   Median (Q2)	70	71
   Q3	80	76
   Maximum	95	88

   Compare the two distributions by commenting on centre, spread and skewness. Identify which class performed more consistently.

   Show worked solution

   Key idea: Compare boxplots by commenting on centre (median), spread (IQR and range) and skewness (symmetry of the box and whiskers).

   1. Centre: The medians are almost identical (Class A: $70$, Class B: $71$), so both classes performed similarly on average.
   2. Spread: $\text{IQR}_A = 80 - 58 = 22, \text{IQR}_B = 76 - 62 = 14.$ $\text{Range}_A = 95 - 42 = 53, \text{Range}_B = 88 - 55 = 33.$ Class B has a smaller IQR and smaller range, so Class B performed more consistently.
   3. Skewness: For Class A: $Q2 - Q1 = 12$ and $Q3 - Q2 = 10$, with a longer lower whisker ($70 - 42 = 28$) than upper whisker ($95 - 70 = 25$), indicating slight negative skew. For Class B: $Q2 - Q1 = 9$ and $Q3 - Q2 = 5$, with the upper half of the box compressed and whiskers roughly equal ($71 - 55 = 16$, $88 - 71 = 17$), indicating negative skew within the box (scores bunched toward the upper end).

   Answer: Both classes have nearly the same median (70 vs 71). Class B is more consistent (IQR of 14 vs 22, range of 33 vs 53). Both distributions are slightly negatively skewed, with Class B showing more compression in the upper half of the box.

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