Practice Maths

Comparing Data Sets — Solutions

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  1. Fluency

    (a) Mean (50) < Median (55) → negatively skewed (left tail). The mean is pulled down by low values.

    (b) Mean = Median = 70 → symmetric.

    (c) Mean (80) > Median (65) → positively skewed (right tail). The mean is pulled up by high values.

  2. Fluency

    Group X: Range = 55 − 10 = 45. IQR = 41 − 22 = 19.

    Group Y: Range = 60 − 15 = 45. IQR = 48 − 25 = 23.

    Both groups have the same total range (45). However, Group Y has a larger IQR (23 vs 19), meaning the middle 50% of Group Y data is more spread out. The upper half of Group Y extends higher (Q3 = 48 vs 41).

  3. Fluency

    Class A (reading stem leaves right-to-left): 65, 75, 76, 77, 82, 83, 91. 7 students.

    Class B (reading stem leaves left-to-right): 67, 68, 75, 76, 77, 78, 82, 83, 91. 9 students.

  4. Understanding

    Dataset 1 (ordered): 12, 15, 18, 20, 22, 25, 28. n = 7.
    Mean = 140/7 = 20. Median = 20 (4th value). Range = 16.
    Q1 = 15, Q3 = 25, IQR = 10.

    Dataset 2 (ordered): 5, 8, 20, 20, 20, 32, 35. n = 7.
    Mean = 140/7 = 20. Median = 20 (4th value). Range = 30.
    Q1 = 8, Q3 = 32, IQR = 24.

    Comparison: Both datasets have identical mean (20) and median (20). Dataset 2 has a much larger range (30 vs 16) and IQR (24 vs 10), indicating greater spread. Dataset 1 is more consistent around its centre. Dataset 2 has extreme low (5) and high (35) values, while Dataset 1 is more evenly spread. Dataset 2 has a clear mode of 20.

  5. Understanding

    Brisbane: Mean = (45+60+80+55+40+70)/6 = 350/6 ≈ 58.3 mm. Range = 80 − 40 = 40 mm.

    Cairns: Mean = (120+200+350+180+80+150)/6 = 1080/6 = 180 mm. Range = 350 − 80 = 270 mm.

    Comparison: Cairns has a much higher mean rainfall (180 mm vs 58.3 mm), receiving approximately 3 times as much rain on average. Cairns is also far more variable (range = 270 mm vs 40 mm), with monthly totals fluctuating dramatically. Brisbane has more stable and consistent rainfall across the 6 months.

  6. Understanding

    Males: Mean = (8+5+12+6+9+7+11+8+10+4)/10 = 80/10 = 8.0 hrs. Range = 12 − 4 = 8.

    Females: Mean = (10+12+8+14+11+9+13+7+12+14)/10 = 110/10 = 11.0 hrs. Range = 14 − 7 = 7.

    Comparison: Females study significantly more on average (11.0 hrs vs 8.0 hrs per week — a difference of 3 hrs). Females are also slightly more consistent (range 7 vs 8 for males), though the difference in spread is small. The centre difference is the more notable finding.

  7. Understanding

    (a) IQR = Q3 − Q1 = 26 − 12 = 14. Range = 34 − 5 = 29.

    (b) Lower fence = 12 − 1.5(14) = 12 − 21 = −9. Upper fence = 26 + 1.5(14) = 26 + 21 = 47.
    Min = 5 > −9 and Max = 34 < 47. No outliers.

    (c) Median (18) is closer to Q1 (12) than to Q3 (26): distance to Q1 = 6, distance to Q3 = 8. The box is slightly longer on the upper side, suggesting a slight positive skew (tail towards higher values).

  8. Problem Solving

    (a) Group 1 (sorted: 10.2, 10.5, 10.8, 11.0, 11.2, 11.5, 11.8):
    Min = 10.2, Q1 = 10.5, Median = 11.0, Q3 = 11.5, Max = 11.8

    Group 2 (sorted: 9.8, 10.1, 10.9, 11.1, 11.4, 12.0, 12.3):
    Min = 9.8, Q1 = 10.1, Median = 11.1, Q3 = 12.0, Max = 12.3

    (b) IQR Group 1 = 11.5 − 10.5 = 1.0 s. IQR Group 2 = 12.0 − 10.1 = 1.9 s.

    (c) Centre: Similar medians (11.0 vs 11.1 s) — roughly equal typical performance.
    Spread: Group 1 has a much smaller IQR (1.0 vs 1.9 s), making it more consistent.
    Shape: Group 1 appears roughly symmetric; Group 2 is slightly right-skewed (long upper whisker).
    Overall: Group 2 has a faster minimum time (9.8 s vs 10.2 s), suggesting some athletes in Group 2 are faster. However, Group 1 is more consistent and reliable across all athletes.

  9. Problem Solving

    Athletes (sorted: 52, 58, 60, 62, 65, 68, 70):
    Mean = 435/7 ≈ 62.1 bpm. Median = 62 bpm. Q1 = 58, Q3 = 68, IQR = 10.

    Non-athletes (sorted: 68, 72, 75, 78, 80, 82, 88):
    Mean = 543/7 ≈ 77.6 bpm. Median = 78 bpm. Q1 = 72, Q3 = 82, IQR = 10.

    Structured comparison:
    Shape: Both distributions appear roughly symmetric based on the median sitting approximately in the centre of each group’s range.
    Centre: Athletes have a lower median pulse rate (62 vs 78 bpm), consistent with the known physiological effect of aerobic fitness lowering resting heart rate.
    Spread: Both groups have equal IQR (10 bpm), indicating similar consistency within each group. Athletes’ range is slightly smaller (18 vs 20 bpm).
    Conclusion: Athletes have notably lower resting pulse rates (approx. 16 bpm lower on average) with similar variability to non-athletes.

  10. Problem Solving

    (a) “Do Year 11 girls spend more time on homework per night than boys, on average?”

    (b) Boys (sorted: 1.0, 1.5, 1.5, 1.5, 2.0, 2.0, 2.5, 3.0):
    Mean = 15.0/8 = 1.875 hrs. Median = (1.5+2.0)/2 = 1.75 hrs.

    Girls (sorted: 2.0, 2.0, 2.0, 2.5, 2.5, 2.5, 3.0, 3.5):
    Mean = 20.0/8 = 2.5 hrs. Median = (2.5+2.5)/2 = 2.5 hrs.

    (c) Comparison:
    Shape: Boys’ data is slightly positively skewed (mean 1.875 > median 1.75); girls’ is symmetric (mean = median = 2.5).
    Centre: Girls study more — mean 2.5 hrs vs 1.875 hrs for boys; medians 2.5 vs 1.75 hrs.
    Spread: Boys have a wider range (3.0−1.0 = 2.0 hrs) than girls (3.5−2.0 = 1.5 hrs), indicating boys’ homework time is more variable.

    (d) The data supports the claim that girls spend more time on homework. On average, girls spend 2.5 hours per night compared to 1.875 hours for boys, a difference of approximately 37 minutes. Girls are also slightly more consistent in their study habits. However, the sample is small (n = 8 per group) and results may not generalise to all students.