Answer Key

Statistics

Pack A — Answers

#	Question	Answer
1	Find the mean of the data: $3, 5, 7, 9, 11.	7
2	Find the median of: $2, 5, 7, 9, 12.	7
3	Find the median of: $2, 4, 6, 8.	5
4	Find the mode of: $2, 3, 3, 4, 5, 3, 7.	3
5	Find the range of: $8, 12, 5, 15, 9, 7.	10
6	Find the mean of: $10, 14, 18, 22, 26.	18
7	Find the mode and the range of: $4, 5, 5, 6, 8.	mode = 5; range = 4
8	A boxplot has lower quartile 12, median 18, and upper quartile 24. State (a) the IQR and (b) the median.	(a) IQR = 12 (b) median = 18
9	A stem-and-leaf plot shows $1\,\|\,2\;5\;9$ and $2\,\|\,1\;4\;6\;7$. How many values are there?	7
10	In a frequency table, value $x$ has frequency 2; value 3 has frequency 5; value 5 has frequency 3. Total frequency 10. Mean: $x = ?$ given mean is 4. (Find $x$.)	$x = 5$
11	Find the mean of: $2(3), 4(5), 6(2) (where format is "value(frequency)").	3.8
12	A data set has 25 values. What is the median position?	13th value
13	For the ordered data $2, 4, 5, 7, 8, 10, 12, 14, 16, find $Q_1$ and $Q_3$.	$Q_1 = 4.5$, $Q_3 = 13$
14	A data set has $Q_1 = 14$ and $Q_3 = 28$. Find the IQR.	14
15	For the data $4, 6, 8, 10, 12, 14, 16, 18, 20, find min, $Q_1$, median, $Q_3$, max.	min=4, $Q_1$=7, med=12, $Q_3$=17, max=20
16	Two boxplots are shown. Boxplot A: median 18, IQR 10. Boxplot B: median 22, IQR 6. Which group has more consistent values?	Boxplot B (smaller IQR)
17	A test was sat by 20 students. Scores: 1(2), 2(5), 3(8), 4(3), 5(2). Find the mean.	2.9
18	From the frequency table, state the modal value: 1(4), 2(7), 3(10), 4(5), 5(4).	3
19	A data set $\{3, 5, 7, x, 11\}$ has mean $7$. Find $x$.	$x = 9$
20	A boxplot has whiskers from 0 to 50, $Q_1 = 10$, median 15, $Q_3 = 22$. Is the distribution symmetric or skewed?	Skewed (positively/right-skewed)
21	A cumulative frequency curve for 100 students has cf = 25 at score 12, cf = 50 at score 18, cf = 75 at score 23. State $Q_1$, median, $Q_3$.	$Q_1 = 12$, median = 18, $Q_3 = 23$
22	For the cumulative frequency data: $Q_1 = 14$, $Q_3 = 32$. Find the IQR.	18
23	A grouped frequency table has: $[0, 10)$ freq 4; $[10, 20)$ freq 8; $[20, 30)$ freq 6; $[30, 40)$ freq 2. Estimate the mean.	18
24	A boxplot has $Q_1 = 20$, $Q_3 = 40$. Using the 1.5×IQR rule, determine if 75 is an outlier.	Yes — 75 > 70 (upper fence)
25	Group A has median 25, IQR 10. Group B has median 25, IQR 20. Compare the spreads.	Same median; A more consistent (smaller IQR)
26	From the cumulative frequency curve: at score = 30, cf = 75; at score = 40, cf = 95. How many students scored between 30 and 40?	20
27	A grouped table has total frequency 60, with cumulative frequencies at class boundaries: 8, 22, 45, 60. The median is in which class?	Third class (covers cf 22 to 45; median position 30 lies here)
28	A data set has values 1, 2, 3, 4 with frequencies 4, $x$, 6, 3. The mean is 2.5. Find $x$.	$x = 3$
29	A boxplot shows min 5, $Q_1$ 12, median 20, $Q_3$ 28, max 35. What percentage of data lies above $Q_3$?	25%
30	A data set has $n = 30$ values with mean $14.5$. Find the sum.	435
31	Class A has 25 students with mean test score 60. Class B has 15 students with mean 72. Find the combined mean.	64.5
32	Six values are $\{12, 15, 17, x, 22, 25\}$. The median is 19. Find $x$.	$x = 21$
33	A cumulative frequency table on 200 values shows cf = 80 at score 50, cf = 120 at score 70. Estimate the score for the 60th percentile.	$\approx 70$ (cf = 120 is 60th percentile for 200)
34	Group A: min 5, $Q_1$ 12, median 18, $Q_3$ 25, max 35. Group B: min 8, $Q_1$ 18, median 22, $Q_3$ 28, max 38. Make two comparisons.	B has higher median and more concentrated middle (smaller IQR: 10 vs 13)
35	A data set has $Q_1 = 15$, $Q_3 = 35$. Identify values from $\{3, 10, 22, 50, 70\}$ that are outliers.	70 is an outlier
36	In a test, the mean of 20 students is 72. After re-marking, two scores increased by 5 each. What is the new mean?	72.5
37	Grouped data: $[0, 10)$ freq 5; $[10, 20)$ freq 12; $[20, 30)$ freq 8; $[30, 50)$ freq 5. State the modal class and explain why frequency density matters here.	$[10, 20)$; frequency density (freq/width) gives true comparison
38	Two data sets both have mean 10. Set A has range 4; Set B has range 20. Which is more spread out?	B
39	A class of $24$ students has mean test score $65$. A new student joins with score $90$. Find the new mean (1 d.p.).	66.0
40	Five integers have mean 5, median 5, and mode 7. State a possible set of integers.	$\{1, 3, 5, 7, 7\}$ (or similar)

Pack B — Answers

#	Question	Answer
1	Find the mean of the data: $4, 6, 8, 10, 12.	8
2	Find the median of: $3, 6, 10, 14, 18.	10
3	Find the median of: $5, 8, 12, 15.	10
4	Find the mode of: $5, 7, 7, 8, 9, 7, 10.	7
5	Find the range of: $14, 22, 8, 18, 25, 10.	17
6	Find the mean of: $6, 9, 12, 15, 18.	12
7	Find the mode and the range of: $7, 7, 9, 10, 14.	mode = 7; range = 7
8	A boxplot has lower quartile 12, median 18, and upper quartile 24. State (a) the IQR and (b) the median.	(a) IQR = 12 (b) median = 15
9	A stem-and-leaf plot shows $1\,\|\,2\;5\;9$ and $2\,\|\,1\;4\;6\;7$. How many values are there?	12 (smallest)
10	In a frequency table, value $x$ has frequency 2; value 3 has frequency 5; value 5 has frequency 3. Total frequency 10. Mean: $x = ?$ given mean is 4. (Find $x$.)	$x = 7$
11	Find the mean of: $1(4), 3(3), 5(3) (where format is "value(frequency)").	2.8
12	A data set has 25 values. What is the median position?	mean of 25th and 26th values
13	For the ordered data $3, 5, 8, 9, 11, 13, 15, 17, 20, find $Q_1$ and $Q_3$.	$Q_1 = 6.5$, $Q_3 = 16$
14	A data set has $Q_1 = 22$ and $Q_3 = 41$. Find the IQR.	19
15	For the data $2, 4, 5, 7, 9, 11, 12, 15, 18, find min, $Q_1$, median, $Q_3$, max.	min=2, $Q_1$=4.5, med=9, $Q_3$=13.5, max=18
16	Two boxplots are shown. Boxplot A: median 18, IQR 10. Boxplot B: median 22, IQR 6. Which group has more consistent values?	Boxplot B
17	A test was sat by 20 students. Scores: 1(2), 2(5), 3(8), 4(3), 5(2). Find the mean.	1.6
18	From the frequency table, state the modal value: 1(4), 2(7), 3(10), 4(5), 5(4).	30
19	A data set $\{3, 5, 7, x, 11\}$ has mean $8$. Find $x$.	$x = 14$
20	A boxplot has whiskers from 0 to 50, $Q_1 = 10$, median 15, $Q_3 = 22$. Is the distribution symmetric or skewed?	Positively (right) skewed
21	A cumulative frequency curve for 100 students has cf = 25 at score 12, cf = 50 at score 18, cf = 75 at score 23. State $Q_1$, median, $Q_3$.	$Q_1 = 5$, median = 11, $Q_3 = 17$
22	For the cumulative frequency data: $Q_1 = 25$, $Q_3 = 48$. Find the IQR.	23
23	A grouped frequency table has: $[0, 10)$ freq 4; $[10, 20)$ freq 8; $[20, 30)$ freq 6; $[30, 40)$ freq 2. Estimate the mean.	7.75
24	A boxplot has $Q_1 = 20$, $Q_3 = 40$. Using the 1.5×IQR rule, determine if 75 is an outlier.	No — 5 > $-20$ (lower fence)
25	Group A has median 25, IQR 10. Group B has median 25, IQR 20. Compare the spreads.	Same median; B more consistent
26	From the cumulative frequency curve: at score = 30, cf = 75; at score = 40, cf = 95. How many students scored between 30 and 40?	30
27	A grouped table has total frequency 60, with cumulative frequencies at class boundaries: 8, 22, 45, 60. The median is in which class?	Third class (cf 25 to 50; median at 40)
28	A data set has values 1, 2, 3, 4 with frequencies 4, $x$, 6, 3. The mean is 2.5. Find $x$.	$x = 3$
29	A boxplot shows min 5, $Q_1$ 12, median 20, $Q_3$ 28, max 35. What percentage of data lies above $Q_3$?	25%
30	A data set has $n = 40$ values with mean $22.5$. Find the sum.	900
31	Class A has 25 students with mean test score 60. Class B has 15 students with mean 72. Find the combined mean.	60
32	Six values are $\{12, 15, 17, x, 22, 25\}$. The median is 19. Find $x$.	$x = 19$
33	A cumulative frequency table on 200 values shows cf = 80 at score 50, cf = 120 at score 70. Estimate the score for the 60th percentile.	$\approx 60$
34	Group A: min 5, $Q_1$ 12, median 18, $Q_3$ 25, max 35. Group B: min 8, $Q_1$ 18, median 22, $Q_3$ 28, max 38. Make two comparisons.	B has higher median; both have IQR 10
35	A data set has $Q_1 = 15$, $Q_3 = 35$. Identify values from $\{3, 10, 22, 50, 70\}$ that are outliers.	50 and 60 are outliers
36	In a test, the mean of 20 students is 72. After re-marking, two scores increased by 5 each. What is the new mean?	65.6
37	Grouped data: $[0, 10)$ freq 5; $[10, 20)$ freq 12; $[20, 30)$ freq 8; $[30, 50)$ freq 5. State the modal class and explain why frequency density matters here.	$[5, 10)$; with unequal widths use density
38	Two data sets both have mean 10. Set A has range 4; Set B has range 20. Which is more spread out?	B
39	A class of $19$ students has mean test score $70$. A new student joins with score $50$. Find the new mean (1 d.p.).	69.0
40	Five integers have mean 5, median 5, and mode 7. State a possible set of integers.	$\{1, 1, 3, 4, 11\}$ (or similar)

Problems — Worked Solutions

**Three averages.** Eight students scored the following marks out of 20: $$11,\ 14,\ 16,\ 13,\ 18,\ 12,\ 16,\ 10.$$ (a) Find the mean, median, mode, and range. (b) The teacher discovers that one mark was misread and should have been 19 instead of 11. Recompute the four statistics. (c) Which average is most affected by the correction? Why?

Answer

(a) mean 13.75, median 13.5, mode 16, range 8 (b) mean 14.75, median 15, mode 16, range 9 (c) See working

(a) Sort: 10, 11, 12, 13, 14, 16, 16, 18. - **Mean** = $(11+14+16+13+18+12+16+10)/8 = 110/8 = 13.75$. - **Median**: 8 values → mean of 4th and 5th = $(13+14)/2 = 13.5$. - **Mode** = 16 (appears twice). - **Range** = $18 - 10 = 8$. (b) Replace 11 with 19. New data sorted: 10, 12, 13, 14, 16, 16, 18, 19. - New sum = $110 - 11 + 19 = 118$. Mean = $118/8 = 14.75$. - Median = mean of 4th and 5th = $(14 + 16)/2 = 15$. - Mode still 16. - Range = $19 - 10 = 9$. (c) The **median** changed by 1.5 (from 13.5 to 15) while the mean changed by 1.0. Here, removing a low value (11) and adding a high value (19) shifted the central order so much that the median moved noticeably. In general, the mean is sensitive to changes in *any* value, while the median only changes when the central values shift.

**Cumulative frequency analysis.** A class of 80 students sat a test marked out of 50. Here is a cumulative frequency table: | Mark | Cumulative frequency | |------|---------------------| | ≤ 10 | 8 | | ≤ 20 | 28 | | ≤ 30 | 52 | | ≤ 40 | 72 | | ≤ 50 | 80 | Use the curve to find: (a) the median; (b) the lower and upper quartiles, and hence the IQR; (c) the percentage of students scoring more than 35.

Answer

(a) ~25 (b) $Q_1 \approx 18$, $Q_3 \approx 33$, IQR ~ 15 (c) ~22%

For $n = 80$: $Q_1$ at cf 20; median at cf 40; $Q_3$ at cf 60. (a) Reading off: median at cf 40 → between $\leq 20$ (cf 28) and $\leq 30$ (cf 52). Linear interpolation: $20 + (40-28)/(52-28) \times 10 = 20 + 5 = 25$. **Median ≈ 25**. (b) $Q_1$ at cf 20: between cf 8 (mark 10) and 28 (mark 20). $10 + (20-8)/(28-8) \times 10 = 10 + 6 = 16$. $Q_1 \approx 16$. $Q_3$ at cf 60: between cf 52 (mark 30) and 72 (mark 40). $30 + (60-52)/(72-52) \times 10 = 30 + 4 = 34$. $Q_3 \approx 34$. **IQR ≈ 34 − 16 = 18**. (c) cf at mark 35: between cf 52 (mark 30) and 72 (mark 40). Linear: $52 + (35-30)/10 \times (72-52) = 52 + 10 = 62$. So 62 students scored ≤ 35, meaning $80 - 62 = 18$ students scored more than 35. Percentage: $18/80 \times 100 = \mathbf{22.5\%}$.

**Boxplot comparison.** Two football teams' goal totals over 20 matches are summarised: - **Team A:** min 0, $Q_1 = 1$, median 2, $Q_3 = 3$, max 5. - **Team B:** min 0, $Q_1 = 0$, median 1, $Q_3 = 4$, max 8. Compare the two teams. Which would you back to score more in the next match? Justify.

Answer

Team A: more consistent (lower IQR, higher median). Team B: more unpredictable but higher max

**Comparison.** - Median: A = 2 > B = 1 → Team A typically scores more per match. - IQR: A = $3 - 1 = 2$; B = $4 - 0 = 4$. Team A is more consistent. - Range: A = 5; B = 8. Team B more variable. **Backing decision.** Depends on goal: - Want a high-probability return → **Team A** (median 2 vs 1; more likely to score consistently around the median). - Want a chance of a big score → **Team B** (max 8, longer upper whisker). For a single match prediction, statistician would back **Team A** (higher median, more consistent middle 50% above zero — Team B has $Q_1 = 0$, meaning 25% of B's matches end in 0 goals).

**Missing values.** A teacher records the test marks for a class of 10 students. Nine of them scored: $$45,\ 67,\ 52,\ 78,\ 84,\ 56,\ 73,\ 65,\ 81.$$ The mean of all 10 students is 68. Find the score of the 10th student.

Answer

Mean × count = sum: $68 \times 10 = 680$. Sum of given 9 scores: $45 + 67 + 52 + 78 + 84 + 56 + 73 + 65 + 81$. $= 45 + 67 = 112$; $+52 = 164$; $+78 = 242$; $+84 = 326$; $+56 = 382$; $+73 = 455$; $+65 = 520$; $+81 = 601$. 10th score = $680 - 601 = \mathbf{79}$.

**Mean from grouped table.** A survey of times taken to complete a puzzle (minutes): | Time (min) | Frequency | |-----------|-----------| | [0, 5) | 8 | | [5, 10) | 14 | | [10, 15) | 22 | | [15, 20) | 12 | | [20, 25) | 4 | (a) Estimate the mean time taken. (b) State the modal class. (c) State the median class.

Answer

(a) ≈ 11.67 min (b) [10, 15) (c) [10, 15)

Total frequency: $8 + 14 + 22 + 12 + 4 = 60$. (a) Use midpoints: 2.5, 7.5, 12.5, 17.5, 22.5. Sum: $2.5(8) + 7.5(14) + 12.5(22) + 17.5(12) + 22.5(4)$ $= 20 + 105 + 275 + 210 + 90 = 700$. Mean: $700/60 \approx \mathbf{11.67}$ min. (b) Highest frequency 22 → **modal class is [10, 15)**. (c) Median position = 30 (between 30th and 31st). Cumulative: 8, 22, 44, 56, 60. 30 lies between 22 and 44 → **median class is [10, 15)**.

**Standard deviation (informal).** Two sets of test scores: - Set A: 70, 70, 70, 70, 70 - Set B: 50, 60, 70, 80, 90 (a) Find the mean of each set. (b) Find the range of each set. (c) Which has higher variability? How can you tell visually?

Answer

(a) Both 70 (b) A: 0; B: 40 (c) B much more variable

(a) Set A: all 70, mean = 70. Set B: $(50+60+70+80+90)/5 = 350/5 = 70$. Both means are 70. (b) Set A range: $70 - 70 = 0$. Set B range: $90 - 50 = 40$. (c) Set B has much higher variability (spread). The same mean does **not** mean the same shape — Set A has zero spread (all identical), Set B varies from 50 to 90. On a dot plot Set A is one cluster at 70; Set B is spread out.

**Skewness.** A boxplot has min 10, $Q_1 = 14$, median 18, $Q_3 = 25$, max 50. (a) Find the IQR. (b) Determine whether the distribution is symmetric, positively skewed, or negatively skewed. (c) Are there any outliers using the $1.5 \times$ IQR rule?

Answer

(a) 11 (b) Positively skewed (c) Yes — values above 41.5 are outliers

(a) IQR = $Q_3 - Q_1 = 25 - 14 = 11$. (b) Whisker lengths: lower = $14 - 10 = 4$; upper = $50 - 25 = 25$. Upper >> lower → **positive (right) skew**. Also: median (18) closer to $Q_1$ (14) than to $Q_3$ (25), confirming right skew. (c) Fences: lower = $14 - 1.5(11) = 14 - 16.5 = -2.5$; upper = $25 + 16.5 = 41.5$. Max = 50 > 41.5 → **outlier at the high end**. The exact value isn't shown by the boxplot, but the maximum 50 lies in outlier territory.

**Combined groups.** Class A has 24 students with mean test score 68. Class B has 16 students with mean 76. (a) Find the combined mean of all 40 students. (b) Is the combined mean the simple average of 68 and 76? Why or why not? (c) If the combined mean were exactly 72, how would the class sizes need to compare?

Answer

(a) 71.2 (b) No — weighted average (c) Equal sizes (24 each)

(a) Sum A: $24 \times 68 = 1632$. Sum B: $16 \times 76 = 1216$. Combined sum: 2848. Combined mean: $2848/40 = \mathbf{71.2}$. (b) Simple average $(68+76)/2 = 72$. The combined mean (71.2) is **not** the simple average — it is closer to 68 because Class A has more students. The combined mean is a **weighted average**, where each class is weighted by its size. (c) For combined mean = 72 (the simple average of 68 and 76), the two classes would need to have **equal weights** = equal sizes. So 24 in each, for example. (Or any other equal-size pair.)

**Misleading statistic.** A property agent advertises: > "The average price of a flat in our area is £350,000." Sample of 10 recently sold flats: £180k, £200k, £220k, £230k, £240k, £250k, £270k, £280k, £290k, £1,300k. (a) Calculate the mean and median. (b) Comment on which is a fairer summary of "typical" flat prices. (c) What is the role of the outlier here?

Answer

(a) Mean = £346k; median = £245k (b) Median fairer (c) £1.3M flat skews the mean upward

(a) Sum: $180+200+220+230+240+250+270+280+290+1300 = 3460$ (thousand). Mean = $3460/10 = 346$ (thousand) = **£346,000** — consistent with the advertised £350k (rounded). Median: 10 values → mean of 5th and 6th sorted = $(240 + 250)/2 = 245$ (thousand) = **£245,000**. (b) The **median is fairer** for "typical": 9 out of 10 flats are below £300k. The mean is pulled upward by the single £1.3M outlier. (c) The outlier (£1.3M flat) drags the mean from ~£240k upwards to £346k — a 44% inflation. The median is immune. The agent has chosen the statistic that suits their advertising.

**Reverse problem.** The mean of 5 distinct positive integers is 10, the median is 9, and the mode is 7. (a) Find a possible set of 5 integers. (b) How many different sets are possible? Find all.

Answer

(a) e.g. $\{7, 7, 9, 11, 16\}$ (b) Multiple — list in working

Constraints: mean 10 → sum 50. Median 9 → 3rd value (sorted) = 9. Mode 7 → 7 appears more than any other value. With 5 distinct positive ints — wait, **distinct** contradicts mode 7 (which requires repetition). **Re-read the problem:** "5 distinct positive integers" is incompatible with a mode. So we interpret the question as "5 positive integers (not necessarily distinct) with mode 7" — i.e. 7 must appear at least twice (so it's the unique mode). Sort: $a, b, 9, d, e$ with sum 50, $a + b + d + e = 41$. Mode 7 means 7 must appear ≥ 2 times. Case 1: $a = b = 7$. Then $d + e = 41 - 14 = 27$, with $d \geq 9$, $e \geq d$, and no other value repeated. Options: $d = 9$ (but then 9 appears twice — equals mode 7's count of 2: not unique mode unless we exclude this). $d = 10, e = 17$; $d = 11, e = 16$; $d = 12, e = 15$; $d = 13, e = 14$. So sets: $\{7,7,9,10,17\}, \{7,7,9,11,16\}, \{7,7,9,12,15\}, \{7,7,9,13,14\}$. Case 2: One of $\{a, b\}$ = 7, and one of $\{d, e\}$ = 7? No — $d \geq 9$, so 7 can't appear above the median. **Possible sets**: at least 4, as listed above.

**Effect of transformation.** A data set has mean 60 and IQR 12. (a) Each value is increased by 5. What is the new mean and IQR? (b) Each value is doubled. What is the new mean and IQR? (c) Each value is increased by 5 and then doubled. What is the new mean and IQR?

Answer

(a) mean 65, IQR 12 (b) mean 120, IQR 24 (c) mean 130, IQR 24

(a) Adding a constant to all values shifts the mean by the same amount but doesn't change the spread: - New mean = $60 + 5 = 65$. - IQR unchanged = 12. (b) Multiplying all values by 2 scales both the centre and spread by 2: - New mean = $60 \times 2 = 120$. - New IQR = $12 \times 2 = 24$. (c) Add 5, then double. New value = $2(x + 5) = 2x + 10$. - New mean = $2(60) + 10 = 130$. - New IQR: only the $2x$ contributes to spread (the +10 is constant). New IQR = $2 \times 12 = 24$.

**Survey design.** A council wants to know the average household income in a town. They survey 50 randomly selected households. (a) Why might the mean of the 50 incomes not equal the true population mean? (b) Why might the median be a better statistic than the mean? (c) How could they improve the estimate?

Answer

(a) Sampling variability (b) Robust to high earners (c) Larger sample / stratified sampling

(a) **Sampling variability** — any random sample is just one possible draw from the population. The sample mean fluctuates around the true mean. With $n = 50$ the estimate may be off by several percent of the true value. (b) **Income distributions are typically right-skewed** (a few very high earners). The mean is pulled upward by these high values, giving a misleadingly high "typical" income. The median is robust to outliers and better reflects the typical household. (c) **Improvements:** - Increase sample size (smaller sampling error). - **Stratified sampling** by neighbourhood, age, or housing type, so each sub-group is represented proportionally. - Combine mean and median — report both, along with IQR or range, for a fuller picture.