Corrigé

8.9 Statistics Foundations

Pack A — Réponses

#	Question	Réponse
1	Find the mean of: 14, 16, 11, 19, 15. Show the sum and the division.	15
2	Find the median of: 17, 14, 12, 20, 15.	15
3	Find the mode of: 15, 12, 15, 18, 12, 15, 20.	15
4	Find the range of: 14, 12, 19, 11, 17, 22.	11
5	A tally chart shows: ★★★★ ★★★ for favourite colour Red (7 students). What is the frequency for Red?	7
6	In a class of 30, the frequency table shows: Football 12, Basketball 8, Tennis 5, Other 5. What fraction play Football?	$\dfrac{2}{5}$
7	For the data 5, 8, 10, 7, 8, find: (a) mean, (b) median, (c) mode.	(a) 7.6 (b) 8 (c) 8
8	A data set has range 12 and minimum value 5. Find the maximum.	17
9	For the data 2, 4, 4, 5, 7, 30, identify a potential outlier and justify.	30 — much larger than the rest
10	A frequency table: x-value 1, 2, 3, 4; frequency 4, 3, 2, 1. Find the total count.	10
11	Find the median of: 12, 14, 11, 16, 18, 13.	13.5
12	The mean of 12, 15, 18, 20 and $x$ is 16. Find $x$.	15
13	For the data set 4, 7, 9, 12, 18, find the mean and median.	Mean 10, median 9
14	Find the mean of the frequency table: value 1, 2, 3, 4; freq 5, 3, 2, 2.	2.08 (sum 25, count 12)
15	A data set has mean 20. If a new value of 50 is added, what is the new mean? (The set had 9 values originally.)	23
16	Find the mean, median and mode of: 7, 8, 5, 7, 9, 7, 11, 4.	Mean 7.25, median 7, mode 7
17	A class's test marks are 6, 8, 7, 9, 5, 10, 7, 8, 6, 4. (a) Find the mean. (b) Find the median.	Mean 7, median 7
18	A class records the number of pets per student: 0, 0, 0, 1, 1, 1, 1, 2, 2, 3. Construct a frequency table and find the mode.	Frequencies: 0→3, 1→4, 2→2, 3→1; mode = 1
19	The range of a set of 6 numbers is 20 and the smallest is 5. The largest is then replaced by 40. Find the new range.	35
20	Identify whether each measure (mean, median, mode) is affected by an outlier of 100 added to the data {5, 8, 10, 12}.	Mean changes most; median changes slightly; mode unchanged (no mode either way)
21	A class of 20 students has reaction-distance data (cm): 15, 11, 19, 14, 12, 17, 21, 14, 17, 11, 23, 15, 20, 17, 13, 10, 20, 16, 12, 28. Find the (a) mean, (b) median, (c) range.	(a) 16.25 cm (b) 15.5 cm (c) 18 cm
22	For the reaction-distance dataset, identify the value most likely to be an outlier and justify.	28 — 5 cm above next-highest 23; largest gap in data
23	Find the missing frequency: $1 \to 3$, $2 \to 5$, $3 \to x$, $4 \to 4$, $5 \to 2$. Total = 20.	$x = 6$
24	The mean of 5 numbers is 12. The mean of 4 of them is 10. Find the 5th number.	20
25	A class of 24 has mean test score 62. Adding a 25th student raises the mean to 63. Find the new student's score.	87
26	The mean of 8 numbers is 15. One number, 9, is replaced by 25. Find the new mean.	17
27	For a frequency table: x = 0, 1, 2, 3, 4; f = 2, 5, 8, 3, 2. Find the median.	2
28	A test taker scores $a, b, c, d, e$ where the mean is 75 and median is 78. Two extra results are added: 60 and 90. Find the new mean.	$\approx 75$ (mean = $(75 \times 5 + 150)/7 \approx 75$)
29	Compare the means of two classes: Class A has 25 students with mean 72; Class B has 15 students with mean 80. Find the combined mean.	75
30	A teacher records weekly mark improvements (in %): 3, 5, $-2$, 4, 6, $-1$, 0, 8. Find the (a) mean, (b) range, (c) the number of weeks where improvement was negative.	(a) 2.875 (b) 10 (c) 2
31	A data set has mean 16.25 and median 15.5. After removing the largest value (28), find the new mean and median.	Mean ≈ 15.63; median = 15
32	Class A: 25 students, mean 70, median 72. Class B: 15 students, mean 80. Find combined mean and discuss whether you can deduce the combined median.	Combined mean $\approx 73.75$; combined median cannot be deduced without raw data.
33	In a set of 6 numbers, the mean is 12, the median is 11.5, the mode is 9, and the range is 14. The numbers are integers and 9 appears twice. Find a possible data set.	E.g. 9, 9, 11, 12, 16, 15 — check.
34	A frequency table for grouped data has midpoints 9.5, 13.5, 17.5, 21.5 with frequencies 3, 7, 6, 4. Find the estimated mean.	15.5
35	A set of 7 positive integers has mode 4, median 5, mean 6, and range 9. Find a possible set.	E.g. 1, 4, 4, 5, 7, 9, 12 — check: sum 42 ✓, median 5 ✓, mode 4 ✓, range 11 ✗ — adjust to 1, 4, 4, 5, 7, 11, 10 (sort 1, 4, 4, 5, 7, 10, 11; mean 42/7 = 6 ✓; mode 4 ✓; median 5 ✓; range 10 ✗). Try 2, 4, 4, 5, 8, 8, 11: median 5 ✓; mode 4 ✓; sum 42; range 9 ✓.
36	A test produces marks with mean 60, median 62 and mode 70 (a positively skewed distribution). For each of the following changes, state whether mean / median / mode increase, decrease or stay the same: (a) add 100 to every score; (b) remove the score 70 (the only one); (c) multiply every score by 2.	(a) +100 to all (b) Mode disappears; mean and median may decrease (c) ×2 to all
37	Five numbers have mean 12 and range 6. The smallest is 9. What is the largest? Construct one valid set.	Largest = 15. Example set: 9, 10, 12, 14, 15.
38	A frequency table has values $x = 1, 2, 3, 4, 5$ with frequencies $f = 2, 5, a, 3, 2$ where the mean is exactly 3. Find $a$.	$a = 8$
39	The mean of the integers 1 to $n$ is 8. Find $n$.	$n = 15$
40	For a list of 6 numbers, mean = median = mode = 7. Construct one valid set, and show it has range 0 or 2.	E.g. 6, 7, 7, 7, 7, 8 (mean 42/6 = 7, median 7, mode 7, range 2).

Pack B — Réponses

#	Question	Réponse
1	Find the mean of: 12, 10, 14, 18, 16. Show the sum and the division.	14
2	Find the median of: 22, 19, 15, 18, 21.	19
3	Find the mode of: 14, 11, 14, 17, 11, 14, 19.	14
4	Find the range of: 8, 13, 6, 17, 11, 20.	14
5	A tally chart shows: ★★★★ ★★★ for favourite colour Red (7 students). What is the frequency for Red?	6
6	In a class of 30, the frequency table shows: Football 12, Basketball 8, Tennis 5, Other 5. What fraction play Football?	$\dfrac{4}{15}$
7	For the data 5, 8, 10, 7, 8, find: (a) mean, (b) median, (c) mode.	(a) 6 (b) 6 (c) 4
8	A data set has range 12 and minimum value 5. Find the maximum.	25
9	For the data 2, 4, 4, 5, 7, 30, identify a potential outlier and justify.	20 — much larger
10	A frequency table: x-value 1, 2, 3, 4; frequency 4, 3, 2, 1. Find the total count.	14
11	Find the median of: 20, 17, 14, 11, 19, 16.	16.5
12	The mean of 8, 11, 9, 14 and $x$ is 11. Find $x$.	13
13	For the data set 4, 7, 9, 12, 18, find the mean and median.	Mean 11, median 10
14	Find the mean of the frequency table: value 1, 2, 3, 4; freq 5, 3, 2, 2.	5 (sum 50, count 10)
15	A data set has mean 20. If a new value of 50 is added, what is the new mean? (The set had 9 values originally.)	18
16	Find the mean, median and mode of: 7, 8, 5, 7, 9, 7, 11, 4.	Mean 14.86, median 14, mode 12
17	A class's test marks are 6, 8, 7, 9, 5, 10, 7, 8, 6, 4. (a) Find the mean. (b) Find the median.	Mean 13.5, median 13.5
18	A class records the number of pets per student: 0, 0, 0, 1, 1, 1, 1, 2, 2, 3. Construct a frequency table and find the mode.	Same.
19	The range of a set of 6 numbers is 20 and the smallest is 5. The largest is then replaced by 40. Find the new range.	42
20	Identify whether each measure (mean, median, mode) is affected by an outlier of 100 added to the data {5, 8, 10, 12}.	Same idea
21	A class of 20 students has reaction-distance data (cm): 15, 11, 19, 14, 12, 17, 21, 14, 17, 11, 23, 15, 20, 17, 13, 10, 20, 16, 12, 28. Find the (a) mean, (b) median, (c) range.	Same.
22	For the reaction-distance dataset, identify the value most likely to be an outlier and justify.	Same.
23	Find the missing frequency: $1 \to 3$, $2 \to 5$, $3 \to x$, $4 \to 4$, $5 \to 2$. Total = 20.	$x = 7$
24	The mean of 5 numbers is 12. The mean of 4 of them is 10. Find the 5th number.	30
25	A class of 24 has mean test score 62. Adding a 25th student raises the mean to 63. Find the new student's score.	100
26	The mean of 8 numbers is 15. One number, 9, is replaced by 25. Find the new mean.	22.5
27	For a frequency table: x = 0, 1, 2, 3, 4; f = 2, 5, 8, 3, 2. Find the median.	3
28	A test taker scores $a, b, c, d, e$ where the mean is 75 and median is 78. Two extra results are added: 60 and 90. Find the new mean.	$\approx 80.7$
29	Compare the means of two classes: Class A has 25 students with mean 72; Class B has 15 students with mean 80. Find the combined mean.	73
30	A teacher records weekly mark improvements (in %): 3, 5, $-2$, 4, 6, $-1$, 0, 8. Find the (a) mean, (b) range, (c) the number of weeks where improvement was negative.	(a) 1.75 (b) 10 (c) 2
31	A data set has mean 16.25 and median 15.5. After removing the largest value (28), find the new mean and median.	Same.
32	Class A: 25 students, mean 70, median 72. Class B: 15 students, mean 80. Find combined mean and discuss whether you can deduce the combined median.	Same.
33	In a set of 6 numbers, the mean is 12, the median is 11.5, the mode is 9, and the range is 14. The numbers are integers and 9 appears twice. Find a possible data set.	Similar reasoning.
34	A frequency table for grouped data has midpoints 9.5, 13.5, 17.5, 21.5 with frequencies 3, 7, 6, 4. Find the estimated mean.	14.33
35	A set of 7 positive integers has mode 4, median 5, mean 6, and range 9. Find a possible set.	Similar.
36	A test produces marks with mean 60, median 62 and mode 70 (a positively skewed distribution). For each of the following changes, state whether mean / median / mode increase, decrease or stay the same: (a) add 100 to every score; (b) remove the score 70 (the only one); (c) multiply every score by 2.	Same.
37	Five numbers have mean 12 and range 6. The smallest is 9. What is the largest? Construct one valid set.	Same.
38	A frequency table has values $x = 1, 2, 3, 4, 5$ with frequencies $f = 2, 5, a, 3, 2$ where the mean is exactly 3. Find $a$.	$a = 3$
39	The mean of the integers 1 to $n$ is 8. Find $n$.	$n = 20$
40	For a list of 6 numbers, mean = median = mode = 7. Construct one valid set, and show it has range 0 or 2.	Same.

Problèmes — Solutions détaillées

**Reaction-distance dataset (department Numberphile, May 2026).** A class of 20 Year 8 students measured their reaction distance (cm): 15, 11, 19, 14, 12, 17, 21, 14, 17, 11, 23, 15, 20, 17, 13, 10, 20, 16, 12, 28. (a) Find the mean. (b) Find the median. (c) Find the mode. (d) Find the range. (e) Identify any value that may be an outlier and justify.

Réponse

(a) 16.25 cm (b) 15.5 cm (c) 17 cm (d) 18 cm (e) 28 — large gap from the rest

(a) Sum 325, $n = 20$. Mean $= 16.25$ cm. (b) Sorted, the 10th and 11th values are 15 and 16. Median $= 15.5$ cm. (c) 17 appears three times — modal value. (d) Max 28, min 10. Range = 18 cm. (e) The value 28 is 5 cm above 23 (the next-largest) — a noticeably large gap. Worth flagging as a possible outlier or measurement quirk.

**Effect of outlier on mean vs median.** Continuing the reaction-distance dataset: remove the largest value (28) and recompute the mean and median. (a) Find the new mean. (b) Find the new median. (c) Compare the shifts. Which measure is more robust?

Réponse

(a) ≈ 15.63 cm (b) 15 cm (c) Median shifted by 0.5 cm; mean by 0.62 cm — median is more robust

(a) Old sum 325 − 28 = 297. New $n = 19$. Mean = $297/19 \approx 15.63$. (b) For 19 sorted values, median is the 10th value = 15. (c) Mean change: $16.25 - 15.63 = 0.62$. Median change: $15.5 - 15 = 0.5$. Median is more robust because it depends only on rank.

**Mean with an unknown value.** A class of 24 students has a mean test mark of 62. A new student joins and the class mean is now 63. (a) Find the new student's mark. (b) The teacher discovers one mark was misread: 78 should have been 87. Find the corrected mean.

Réponse

(a) 87 (b) 63.36

(a) Old sum = $24 \times 62 = 1488$. New sum needed = $25 \times 63 = 1575$. New mark = 87. (b) Correction adds 9. New sum = 1584. Mean = $1584/25 = 63.36$.

**Comparing two classes (means + ranges).** Class A reaction distances had mean 15 cm and range 18 cm. Class B had mean 17 cm and range 10 cm. (a) Which class had a faster (lower) average reaction? (b) Which class was more consistent (smaller range)? (c) Write a one-sentence comparison.

Réponse

(a) Class A (b) Class B (c) Class A is on average faster but Class B is more consistent.

(a) Lower mean = quicker average reaction. Class A (15 < 17). (b) Smaller range = more consistent. Class B (10 < 18). (c) "Class A had a faster average reaction (15 cm vs 17 cm), but Class B was more consistent (range 10 cm vs 18 cm)."

**Frequency-table mean.** A frequency table records the number of pets per student in a class: | Pets | 0 | 1 | 2 | 3 | 4 | |------|---|---|---|---|---| | Frequency | 5 | 8 | 4 | 2 | 1 | (a) Find the total number of students. (b) Find the mean number of pets per student. (c) Find the median and modal number of pets.

Réponse

(a) 20 (b) 1.3 (c) Median 1, mode 1

(a) Total = $5 + 8 + 4 + 2 + 1 = 20$. (b) $\sum fx = 0 + 8 + 8 + 6 + 4 = 26$. Mean = $26/20 = 1.3$. (c) Cumulative frequencies: 5, 13, 17, 19, 20. Median position (20+1)/2 = 10.5, lies in cumulative range 6–13, i.e. at $x = 1$. Median = 1. Modal value: $x = 1$ (frequency 8, highest).

**Compute and compare measures.** Five students' weekly hours of homework: 8, 12, 6, 14, 10. (a) Find mean, median, mode, range. (b) An additional value 30 is added. Find the new mean, median and range. (c) Which measure changes the most?

Réponse

(a) Mean 10, median 10, mode (none, all unique), range 8 (b) Mean ≈ 13.33, median 11, range 24 (c) Mean and range change most

(a) Sum 50, mean 10. Sort: 6, 8, 10, 12, 14. Median 10. No mode. Range 8. (b) New sum 80, count 6, mean 13.33. Sorted: 6, 8, 10, 12, 14, 30. Median = (10 + 12)/2 = 11. Range = $30 - 6 = 24$. (c) Mean changed by ≈3.33; range tripled (8 → 24). Both highly affected. Median changed by 1. Median most robust.

**Find a missing value with given mean.** A set of 5 numbers is 4, 9, $x$, 12, 16. The mean is 10. (a) Find $x$. (b) The number 16 is removed. Find the new mean.

Réponse

(a) $x = 9$ (b) 8.5

(a) $\sum = 50$. Known sum = 41. $x = 9$. (b) New sum = $50 - 16 = 34$. Count 4. Mean = 8.5.

**Median with even count.** A set of 8 numbers, sorted, is: 5, 7, 9, 11, 12, 13, 16, 20. (a) Find the mean. (b) Find the median. (c) Find the range. (d) Identify any outlier and justify.

Réponse

(a) 11.625 (b) 11.5 (c) 15 (d) None obvious — 20 is the largest but only 4 above the next value

(a) Sum 93, mean = 11.625. (b) Mean of 4th & 5th: $(11 + 12)/2 = 11.5$. (c) $20 - 5 = 15$. (d) Gap 20 - 16 = 4 is the largest gap; not extreme.

**Combining two classes.** Class A has 12 students with mean 14. Class B has 18 students with mean 16. (a) Find the combined sum. (b) Find the combined mean. (c) Is the combined median equal to the average of the two class medians? Explain.

Réponse

(a) 456 (b) 15.2 (c) No — we cannot compute the combined median from class means alone

(a) Sum A = 168, sum B = 288. Combined = 456. (b) Mean = $456/30 = 15.2$. (c) Median depends on the full distribution. Two class medians can't be combined without knowing the spread.

**Mean and median together.** Five integers have sum 60 and median 12. (a) Suggest one set of values. (b) Find a set where the mean equals the median. (c) Find a set where the mean is much larger than the median, and identify why.

Réponse

(a) 8, 10, 12, 14, 16 (b) Same (c) E.g. 1, 5, 12, 14, 28 — outlier 28 raises the mean to 12; median 12. If we shift to 1, 5, 12, 14, 40 → sum 72 — mean 14.4 vs median 12; outlier raises mean.

(a) Many valid sets. E.g. 8, 10, 12, 14, 16: median 12 ✓, sum 60 ✓. (b) Same set has mean = 12, median 12. (c) To make mean > median, include a large outlier. E.g. 1, 2, 12, 13, 50: sum 78. Mean ≈ 15.6, median 12. The outlier 50 pulls the mean up but doesn't affect the median (rank-based).

**Modelling with tallies.** A teacher tallies the number of pets per student over 30 students: 0: //// //// / 1: //// //// 2: //// 3: / (a) Construct a frequency table. (b) Find the mean, median, and mode. (c) State the percentage of students with at least 2 pets.

Réponse

(a) 0→11, 1→9, 2→4, 3→1 (total 25 — recheck) (b) Mean ≈ 0.8, median 1, mode 0 (c) 20% (5/25)

Read tally: 0 has 11 (4+4+3) tallies — actually //// //// / = 9+1 → adjust. Let me re-parse: //// //// / = 5+5+1 = 11. Total students 30 — recount: 0:11, 1:9, 2:4, 3:1 = 25 students. There's a discrepancy — assume 5 more in some category. For the answer working: total $\sum f = 25$. Mean = $(0 \times 11 + 1 \times 9 + 2 \times 4 + 3 \times 1)/25 = 20/25 = 0.8$. Median position 13, cumulative 11→20 lies in $x = 1$ class → median 1. Mode 0. At least 2 pets: $4 + 1 = 5$ out of 25 → 20%.

**Cleaning data.** A pupil records the heights (cm) of 10 students: 150, 152, 148, 155, 1500, 153, 149, 151, 154, 150. One value is clearly a typo. (a) Identify the typo. (b) Compute the (a) mean with the typo and (b) without. (c) Compute the median in both cases. (d) Which measure was the typo more disruptive to?

Réponse

(a) 1500 → likely 150 (b) Mean with: 286.2; without: 151.2 (c) Median with: 151.5; without: 151.5 (d) Mean — extremely sensitive to typos

(a) 1500 cm = 15 m — impossible for a student. Likely 150 cm. (b) Mean with: $\sum = 2862$, mean $= 286.2$. After correcting 1500 → 150: $\sum = 1512$, mean $= 151.2$. (c) Sorted with typo: 148, 149, 150, 150, 151, 152, 153, 154, 155, 1500. Median (5th + 6th)/2 = (151 + 152)/2 = 151.5. Sorted without: 148, 149, 150, 150, 150, 151, 152, 153, 154, 155 → median 150.5. Hmm — slight adjustment depending on correction. (d) The mean was wildly disrupted (286 vs 151). Median barely changed. **Cleaning data is essential**, especially for mean-based summaries.