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Solutions — Full Answer Key
MathematicsYear 8 · 8.12 Analysing Data
Solutions · Full Answer Key
Pack A answers · Pack B answers · Problem-solving worked solutions
Pack A — Answers
Bronze
1.The extent to which data values are scattered around a central value
2.(a) 7.8 (b) 8
3.Roughly symmetric
4.Positive (right) skew
5.Negative (left) skew
6.50
7.Left tail: values below the centre; right tail: values above
8.Flat (all bars same height)
9.Symmetric (peak at 5)
10.2
Silver
11.(a) 8.8 (b) 8 (c) 2.8
12.Class 1 faster; Class 2 more consistent
13.Positive skew
14.Median 10. 30 likely an outlier (large gap from 15)
15.(a) 2.4 (b) 8
16.Set B (larger range)
17.(a) 3.5 (b) 1.5
18.Class B (smaller SD)
19.Wrong — mean alone doesn't imply concentration. Need spread to judge.
20.B — much lower SD
Gold
21.(a) 12.5 (b) 19.5 (c) 7
22.Fences 2 and 30; no formal outliers
23.(a) B (b) B (IQR 9 vs 6) (c) B (Q3 - median = 6 > median - Q1 = 3 → positive skew)
24.First: symmetric. Second: positive skew (long upper tail pulls mean above median).
25.≈ 75.8
26.A: most employees within £45–55k; B: wider spread, fewer "typical" employees
27.(a) 13 (b) 50 is outlier (upper fence = 47.5)
28.(a) Positive skew (b) A few very wealthy households pull the mean above the median
29.No — could be: changing student ability; different exam; smaller class with outliers
30.A: mean 12.12, range 0.3 (consistent). B: mean 12.24, range 1.0 (variable).
Platinum
31.SD ≈ 4.42 cm
32.Mean dropped 0.62; median dropped 0.5. Median is more robust to outliers.
33.15.2 cm
34.(a) 174 ms (b) Cannot be negative — extrapolation error
35.A symmetric, B asymmetric. A IQR 10; B IQR 16. B has heavier upper-tail.
36.Q1 = 70, Q3 = 85, IQR = 15. Fences: 47.5 and 107.5.
37.(a) 240 (b) 156
38.
39.(a) 80 (b) 40 (c) 30
40.Densities 3, 5, 3, 1; modal bar = density 5
Pack B — Answers
Bronze
1.Same.
2.(a) 10 (b) 10
3.Same.
4.Same.
5.Same.
6.80
7.Same.
8.Same.
9.Symmetric (peak at 3)
10.2.8
Silver
11.(a) 7 (b) 6 (c) 3.2
12.Same.
13.Negative skew
14.Median 10. 25 likely an outlier (gap from 13)
15.(a) 2.4 (b) 8
16.Same.
17.(a) 7 (b) 3
18.Same.
19.Same.
20.Same.
Gold
21.Same.
22.Same.
23.Same.
24.Same.
25.≈ 66.2
26.Same.
27.Same.
28.Same.
29.Same.
30.Same.
Platinum
31.Same.
32.Same.
33.Same.
34.Same.
35.Same.
36.Same.
37.Same.
38.Same.
39.Same.
40.Same.
Problem-solving — Worked Solutions
1Problem 1
Answer
(a) 10, 12.5, 15.5, 19.5, 28 (b) 7 (c) Box 12.5–19.5, median 15.5, whiskers to 10 and 28 (d) No formal outliers (fences 2 and 30; max 28 < 30)
Full working
(a) Min 10, max 28. Q1 = mean of 5th and 6th = (12 + 13)/2 = 12.5. Median = (15 + 16)/2 = 15.5. Q3 = (19 + 20)/2 = 19.5.
(b) IQR = 19.5 - 12.5 = 7.
(c) Box from 12.5 to 19.5, with vertical at 15.5 (median); whiskers to 10 (left) and 28 (right).
(d) Lower fence = 12.5 - 10.5 = 2. Upper = 19.5 + 10.5 = 30. Max 28 < 30 → no outlier. 28 is borderline (just below the fence).
(b) IQR = 19.5 - 12.5 = 7.
(c) Box from 12.5 to 19.5, with vertical at 15.5 (median); whiskers to 10 (left) and 28 (right).
(d) Lower fence = 12.5 - 10.5 = 2. Upper = 19.5 + 10.5 = 30. Max 28 < 30 → no outlier. 28 is borderline (just below the fence).
2Problem 2
Answer
(a) New mean ≈ 15.63, median = 15 (b) Mean dropped 0.62, median dropped 0.5 (c) Median — depends only on rank
Full working
(a) New sum = 325 - 28 = 297. New count = 19. New mean = 297/19 ≈ 15.63. New median: 19 values, 10th = 15.
(b) Mean change ≈ 0.62; median change = 0.5.
(c) Median is more robust because it depends only on the position of the middle value, not on the magnitudes of extreme values.
(b) Mean change ≈ 0.62; median change = 0.5.
(c) Median is more robust because it depends only on the position of the middle value, not on the magnitudes of extreme values.
3Problem 3
Answer
(a) B (16 > 15) (b) A (IQR 6 < 9) (c) A roughly symmetric, B positive skew (d) A is more consistent but B has slightly higher centre with more variability
Full working
(a) Median: A 15, B 16 → B higher.
(b) IQR(A) = 6; IQR(B) = 9. A more consistent.
(c) A: med - Q1 = 3, Q3 - med = 3 → symmetric. B: med - Q1 = 3, Q3 - med = 6 → positive skew (upper tail stretched).
(d) "Class A had a slightly lower and more consistent reaction distance, while Class B had a higher centre with a longer upper tail."
(b) IQR(A) = 6; IQR(B) = 9. A more consistent.
(c) A: med - Q1 = 3, Q3 - med = 3 → symmetric. B: med - Q1 = 3, Q3 - med = 6 → positive skew (upper tail stretched).
(d) "Class A had a slightly lower and more consistent reaction distance, while Class B had a higher centre with a longer upper tail."
4Problem 4
Answer
(a) 15.2 (b) Weighted by class size — different counts
Full working
(a) Sum A = . Sum B = . Combined sum = 760. Combined n = 50. Combined mean = 760/50 = 15.2.
(b) The average of the means ignores the fact that Class B has more students. Use the **weighted** mean: .
(b) The average of the means ignores the fact that Class B has more students. Use the **weighted** mean: .
5Problem 5
Answer
(a) Positive (b) A few high salaries pull the mean above the median (c) Likely an outlier (d) Median
Full working
(a) Mean > median → positive (right) skew.
(b) The bulk of employees earn around £32k (the median), but a small number of high earners raise the mean to £45k.
(c) £200 000 is far above the median (£32k) — very likely an outlier.
(d) The median (£32k) better represents a typical employee; the mean is distorted by extreme values.
(b) The bulk of employees earn around £32k (the median), but a small number of high earners raise the mean to £45k.
(c) £200 000 is far above the median (£32k) — very likely an outlier.
(d) The median (£32k) better represents a typical employee; the mean is distorted by extreme values.
6Problem 6
Answer
(a) 75 (b) Moderate — small sample (c) ~68% (d) ~68%
Full working
(a) Sample mean = 75 is the best point estimate.
(b) is small but acceptable. Standard error ≈ SD/ = 10/ ≈ 1.8 → estimate is 75 ± 3.6 (95% CI).
(c) For roughly bell-shaped: ~68% within ±1 SD.
(d) Same range (65 to 85): ~68%.
(b) is small but acceptable. Standard error ≈ SD/ = 10/ ≈ 1.8 → estimate is 75 ± 3.6 (95% CI).
(c) For roughly bell-shaped: ~68% within ±1 SD.
(d) Same range (65 to 85): ~68%.
7Problem 7
Answer
60%; margin of error ≈ ±14%
Full working
Sample proportion = 30/50 = 60%. MoE ≈ = 14%. So true preference likely in 46–74%. Cannot confidently say A > B (50% threshold).
8Problem 8
Answer
(a) A: 74, 75, 25; B: 73, 75, 50 (b) A: higher mean; tied median (c) B (range 50 > 25) (d) B positively skewed; A roughly symmetric
Full working
(a) A: sum 370, mean 74, median 75 (middle), range 25. B: sum 365, mean 73, median 75, range 50.
(b) A's mean 74 vs B's 73; both have median 75.
(c) B has wider range (50 > 25).
(d) A is roughly symmetric (mean ≈ median). B has lower mean than median due to the outlier-like 50 pulling it left; but 100 is also extreme. Spread is what differentiates them.
(b) A's mean 74 vs B's 73; both have median 75.
(c) B has wider range (50 > 25).
(d) A is roughly symmetric (mean ≈ median). B has lower mean than median due to the outlier-like 50 pulling it left; but 100 is also extreme. Spread is what differentiates them.
9Problem 9
Answer
(a) Stratified random sample / full census (b) Selective reporting / outliers / definition of "average" (c) ≈ ±5%
Full working
(a) (i) Full census: average over all students. (ii) Stratified random sample of, say, 100 students across years and abilities.
(b) (i) Selective reporting: may exclude weak students. (ii) Definition: is it mean, median, mode? (iii) Time of year: changing student body.
(c) For n=50, MoE ≈ . To get ±5% need n ≈ 400.
(b) (i) Selective reporting: may exclude weak students. (ii) Definition: is it mean, median, mode? (iii) Time of year: changing student body.
(c) For n=50, MoE ≈ . To get ±5% need n ≈ 400.
10Problem 10
Answer
(a) 10 (b) 5, 2, 0, 2, 5 (c) 2.8 (d) Average distance from the mean
Full working
(a) Sum 50, mean 10.
(b) Absolute deviations: = 5, 2, 0, 2, 5.
(c) Sum 14, divide by 5: MAD = 2.8.
(d) MAD says that on average, values are 2.8 units away from the mean. Smaller MAD = more clustered data.
(b) Absolute deviations: = 5, 2, 0, 2, 5.
(c) Sum 14, divide by 5: MAD = 2.8.
(d) MAD says that on average, values are 2.8 units away from the mean. Smaller MAD = more clustered data.
11Problem 11
Answer
(a) 7 (b) Upper fence 20.5; 25 > 20.5 → outlier (c) Right-skewed (long upper whisker) (d) An exceptional reader
Full working
(a) IQR = 10 - 3 = 7.
(b) Lower fence = 3 - 10.5 = -7.5. Upper = 10 + 10.5 = 20.5. Max 25 > 20.5 → outlier.
(c) Boxplot has box from 3 to 10, median 6, with a long upper whisker to 25 → right-skewed.
(d) A student who reads exceptionally much — perhaps an avid reader (or a measurement error worth checking).
(b) Lower fence = 3 - 10.5 = -7.5. Upper = 10 + 10.5 = 20.5. Max 25 > 20.5 → outlier.
(c) Boxplot has box from 3 to 10, median 6, with a long upper whisker to 25 → right-skewed.
(d) A student who reads exceptionally much — perhaps an avid reader (or a measurement error worth checking).
12Problem 12
Answer
(a) Record marks for tutorial vs non-tutorial attendees (b) Compare means + spreads / boxplots (c) Self-selection: motivated students attend; harder topics drive both (d) Random assignment to tutorial vs control
Full working
(a) Track exam marks for two groups: students who attended ≥ 3 tutorials, and those who attended fewer.
(b) Compute mean and median for each group. Compare via boxplots. Note IQR and overlaps.
(c) **Self-selection**: motivated/anxious students attend tutorials — their higher marks may reflect motivation, not the tutorials. **Topic difficulty**: tutorials may be scheduled for tough topics → both attendance and marks affected.
(d) Random assignment (e.g. some students mandated to attend, others not) eliminates self-selection. Compare group means with a fair design.
(b) Compute mean and median for each group. Compare via boxplots. Note IQR and overlaps.
(c) **Self-selection**: motivated/anxious students attend tutorials — their higher marks may reflect motivation, not the tutorials. **Topic difficulty**: tutorials may be scheduled for tough topics → both attendance and marks affected.
(d) Random assignment (e.g. some students mandated to attend, others not) eliminates self-selection. Compare group means with a fair design.