Corrigé
8.12 Analysing Data
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Define "spread" in statistics in one sentence. | The extent to which data values are scattered around a central value |
| 2 | For the data 4, 6, 8, 9, 12, find the (a) mean, (b) range. | (a) 7.8 (b) 8 |
| 3 | Describe the symmetry of a data set whose mean = median. | Roughly symmetric |
| 4 | Describe the skew of a distribution where mean > median. | Positive (right) skew |
| 5 | Describe the skew of a distribution where mean < median. | Negative (left) skew |
| 6 | Identify the outlier in the data 5, 6, 7, 7, 8, 9, 50. | 50 |
| 7 | A dataset has mean 10. Describe what each tail represents. | Left tail: values below the centre; right tail: values above |
| 8 | A uniform distribution has all values occurring at the same frequency. Sketch what this looks like. | Flat (all bars same height) |
| 9 | For the data 3, 4, 4, 5, 5, 5, 6, 7, identify the symmetry. | Symmetric (peak at 5) |
| 10 | For the data 2, 3, 5, 6, 9, find the mean absolute deviation (MAD). | 2 |
| 11 | For the data 4, 7, 8, 10, 15, find the (a) mean, (b) median, (c) MAD. | (a) 8.8 (b) 8 (c) 2.8 |
| 12 | Two classes' reaction times (ms) have means 200 and 210, ranges 50 and 30. Which class is faster on average and which more consistent? | Class 1 faster; Class 2 more consistent |
| 13 | A dataset has the mean = 50 and median = 45. State the skew direction. | Positive skew |
| 14 | For the data 5, 8, 9, 10, 12, 15, 30, find the median and explain whether 30 is likely an outlier. | Median 10. 30 likely an outlier (large gap from 15) |
| 15 | A test has marks 12, 14, 16, 18, 20 with mean 16. Compute the (a) MAD, (b) variance. | (a) 2.4 (b) 8 |
| 16 | Compare the spread of two data sets. Set A: 8, 9, 10, 11, 12 (range 4); Set B: 6, 8, 10, 12, 14 (range 8). Which has greater spread? | Set B (larger range) |
| 17 | For a uniform distribution of values 1, 2, 3, 4, 5, 6 (each appearing once), find the (a) mean, (b) MAD. | (a) 3.5 (b) 1.5 |
| 18 | Class A vs Class B mean test marks: A = 75, B = 80; standard deviations: A = 10, B = 4. Which class is more consistent? | Class B (smaller SD) |
| 19 | A teacher says "the class average is 72, so most students scored around 72." Critique this statement. | Wrong — mean alone doesn't imply concentration. Need spread to judge. |
| 20 | Two cricketers have batting averages: A 45, B 42. A's standard deviation is 25, B's is 8. Whose performance is more reliable? | B — much lower SD |
| 21 | For the reaction-distance dataset (sorted): 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 19, 20, 20, 21, 23, 28: find (a) Q1, (b) Q3, (c) IQR. | (a) 12.5 (b) 19.5 (c) 7 |
| 22 | Using Q1 = 12.5, Q3 = 19.5, find the lower and upper outlier fences ($Q_1 - 1.5 \cdot \text{IQR}$ and $Q_3 + 1.5 \cdot \text{IQR}$). Which values are outliers from a max 28 and min 10? | Fences 2 and 30; no formal outliers |
| 23 | Two boxplots: Class A min 8, Q1 12, median 15, Q3 18, max 25. Class B min 10, Q1 13, median 16, Q3 22, max 30. Which has (a) higher median, (b) larger IQR, (c) more skew? | (a) B (b) B (IQR 9 vs 6) (c) B (Q3 - median = 6 > median - Q1 = 3 → positive skew) |
| 24 | Suggest one inference about reaction times that can be drawn from a sample with mean 16 and median 16, vs one with mean 18 and median 14. | First: symmetric. Second: positive skew (long upper tail pulls mean above median). |
| 25 | A class of 30 has mean 75, range 40. A new student with score 100 joins. Find the new mean. | ≈ 75.8 |
| 26 | Mean salary in company A is £50k with SD £5k; in company B mean £45k with SD £20k. Compare typical salaries. | A: most employees within £45–55k; B: wider spread, fewer "typical" employees |
| 27 | A class's 5-number summary is 10, 15, 22, 28, 50. (a) Compute IQR. (b) Identify any outliers using 1.5×IQR rule. | (a) 13 (b) 50 is outlier (upper fence = 47.5) |
| 28 | In a survey of 1000 households, mean income $50k, median $40k. (a) State the skew. (b) Explain what this tells you about wealth distribution. | (a) Positive skew (b) A few very wealthy households pull the mean above the median |
| 29 | A teacher's annual class mean tests have decreased from 75 to 65 over 5 years. Does this prove the teacher's teaching is getting worse? List two factors to consider. | No — could be: changing student ability; different exam; smaller class with outliers |
| 30 | Two athletes record race times (s): A: 12.0, 12.2, 12.1, 12.3, 12.0. B: 11.8, 12.8, 11.9, 12.7, 12.0. Compare via mean and range. | A: mean 12.12, range 0.3 (consistent). B: mean 12.24, range 1.0 (variable). |
| 31 | For the reaction-distance dataset, compute the standard deviation (use the formula $\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}$). Mean = 16.25, n = 20. | SD ≈ 4.42 cm |
| 32 | A class's mean is 16.25 and median is 15.5. After removing the largest value (28), the new mean is 15.63 and new median is 15. Discuss the robustness of mean vs median. | Mean dropped 0.62; median dropped 0.5. Median is more robust to outliers. |
| 33 | Class A: mean 16.25 cm, sample of 20. Class B: mean 14.50 cm, sample of 30. Find the combined mean of all 50 students. | 15.2 cm |
| 34 | A linear model fits reaction time (ms) = $-8a + 270$ where $a$ is age, fitted to ages 8-16. (a) Predict age 12 reaction time. (b) Explain why the model gives -50 ms at age 40, and the kind of error involved. | (a) 174 ms (b) Cannot be negative — extrapolation error |
| 35 | Two boxplots: A (5-num: 10, 15, 20, 25, 30) and B (10, 12, 20, 28, 30). Both have median 20, range 20. Compare shapes. | A symmetric, B asymmetric. A IQR 10; B IQR 16. B has heavier upper-tail. |
| 36 | A teacher's test marks are: 65, 70, 72, 75, 80, 85, 90. Find Q1, Q3, IQR, and the 1.5×IQR fences. | Q1 = 70, Q3 = 85, IQR = 15. Fences: 47.5 and 107.5. |
| 37 | A reverse-percentage problem: in a survey, 65% of pupils said they "feel positive about school." 84 pupils said they did not feel positive. (a) How many surveyed? (b) How many felt positive? | (a) 240 (b) 156 |
| 38 | A class of $n$ students has mean test score 65. A new student with score 80 joins; new mean is 66.5. Find $n$. | $n = 9$ |
| 39 | A 2-way table: 130 study French, 90 study Spanish, 50 both. School of 200. Find (a) only French, (b) only Spanish, (c) neither. | (a) 80 (b) 40 (c) 30 |
| 40 | A histogram has unequal-width bars: width 4 → freq 12; width 4 → freq 20; width 8 → freq 24; width 4 → freq 4. Find the densities and the modal bar. | Densities 3, 5, 3, 1; modal bar = density 5 |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Define "spread" in statistics in one sentence. | Same. |
| 2 | For the data 4, 6, 8, 9, 12, find the (a) mean, (b) range. | (a) 10 (b) 10 |
| 3 | Describe the symmetry of a data set whose mean = median. | Same. |
| 4 | Describe the skew of a distribution where mean > median. | Same. |
| 5 | Describe the skew of a distribution where mean < median. | Same. |
| 6 | Identify the outlier in the data 5, 6, 7, 7, 8, 9, 50. | 80 |
| 7 | A dataset has mean 10. Describe what each tail represents. | Same. |
| 8 | A uniform distribution has all values occurring at the same frequency. Sketch what this looks like. | Same. |
| 9 | For the data 3, 4, 4, 5, 5, 5, 6, 7, identify the symmetry. | Symmetric (peak at 3) |
| 10 | For the data 2, 3, 5, 6, 9, find the mean absolute deviation (MAD). | 2.8 |
| 11 | For the data 4, 7, 8, 10, 15, find the (a) mean, (b) median, (c) MAD. | (a) 7 (b) 6 (c) 3.2 |
| 12 | Two classes' reaction times (ms) have means 200 and 210, ranges 50 and 30. Which class is faster on average and which more consistent? | Same. |
| 13 | A dataset has the mean = 50 and median = 45. State the skew direction. | Negative skew |
| 14 | For the data 5, 8, 9, 10, 12, 15, 30, find the median and explain whether 30 is likely an outlier. | Median 10. 25 likely an outlier (gap from 13) |
| 15 | A test has marks 12, 14, 16, 18, 20 with mean 16. Compute the (a) MAD, (b) variance. | (a) 2.4 (b) 8 |
| 16 | Compare the spread of two data sets. Set A: 8, 9, 10, 11, 12 (range 4); Set B: 6, 8, 10, 12, 14 (range 8). Which has greater spread? | Same. |
| 17 | For a uniform distribution of values 1, 2, 3, 4, 5, 6 (each appearing once), find the (a) mean, (b) MAD. | (a) 7 (b) 3 |
| 18 | Class A vs Class B mean test marks: A = 75, B = 80; standard deviations: A = 10, B = 4. Which class is more consistent? | Same. |
| 19 | A teacher says "the class average is 72, so most students scored around 72." Critique this statement. | Same. |
| 20 | Two cricketers have batting averages: A 45, B 42. A's standard deviation is 25, B's is 8. Whose performance is more reliable? | Same. |
| 21 | For the reaction-distance dataset (sorted): 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 19, 20, 20, 21, 23, 28: find (a) Q1, (b) Q3, (c) IQR. | Same. |
| 22 | Using Q1 = 12.5, Q3 = 19.5, find the lower and upper outlier fences ($Q_1 - 1.5 \cdot \text{IQR}$ and $Q_3 + 1.5 \cdot \text{IQR}$). Which values are outliers from a max 28 and min 10? | Same. |
| 23 | Two boxplots: Class A min 8, Q1 12, median 15, Q3 18, max 25. Class B min 10, Q1 13, median 16, Q3 22, max 30. Which has (a) higher median, (b) larger IQR, (c) more skew? | Same. |
| 24 | Suggest one inference about reaction times that can be drawn from a sample with mean 16 and median 16, vs one with mean 18 and median 14. | Same. |
| 25 | A class of 30 has mean 75, range 40. A new student with score 100 joins. Find the new mean. | ≈ 66.2 |
| 26 | Mean salary in company A is £50k with SD £5k; in company B mean £45k with SD £20k. Compare typical salaries. | Same. |
| 27 | A class's 5-number summary is 10, 15, 22, 28, 50. (a) Compute IQR. (b) Identify any outliers using 1.5×IQR rule. | Same. |
| 28 | In a survey of 1000 households, mean income $50k, median $40k. (a) State the skew. (b) Explain what this tells you about wealth distribution. | Same. |
| 29 | A teacher's annual class mean tests have decreased from 75 to 65 over 5 years. Does this prove the teacher's teaching is getting worse? List two factors to consider. | Same. |
| 30 | Two athletes record race times (s): A: 12.0, 12.2, 12.1, 12.3, 12.0. B: 11.8, 12.8, 11.9, 12.7, 12.0. Compare via mean and range. | Same. |
| 31 | For the reaction-distance dataset, compute the standard deviation (use the formula $\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}$). Mean = 16.25, n = 20. | Same. |
| 32 | A class's mean is 16.25 and median is 15.5. After removing the largest value (28), the new mean is 15.63 and new median is 15. Discuss the robustness of mean vs median. | Same. |
| 33 | Class A: mean 16.25 cm, sample of 20. Class B: mean 14.50 cm, sample of 30. Find the combined mean of all 50 students. | Same. |
| 34 | A linear model fits reaction time (ms) = $-8a + 270$ where $a$ is age, fitted to ages 8-16. (a) Predict age 12 reaction time. (b) Explain why the model gives -50 ms at age 40, and the kind of error involved. | Same. |
| 35 | Two boxplots: A (5-num: 10, 15, 20, 25, 30) and B (10, 12, 20, 28, 30). Both have median 20, range 20. Compare shapes. | Same. |
| 36 | A teacher's test marks are: 65, 70, 72, 75, 80, 85, 90. Find Q1, Q3, IQR, and the 1.5×IQR fences. | Same. |
| 37 | A reverse-percentage problem: in a survey, 65% of pupils said they "feel positive about school." 84 pupils said they did not feel positive. (a) How many surveyed? (b) How many felt positive? | Same. |
| 38 | A class of $n$ students has mean test score 65. A new student with score 80 joins; new mean is 66.5. Find $n$. | Same. |
| 39 | A 2-way table: 130 study French, 90 study Spanish, 50 both. School of 200. Find (a) only French, (b) only Spanish, (c) neither. | Same. |
| 40 | A histogram has unequal-width bars: width 4 → freq 12; width 4 → freq 20; width 8 → freq 24; width 4 → freq 4. Find the densities and the modal bar. | Same. |
Problèmes — Solutions détaillées
**Reaction-distance dataset (department Numberphile).** A class of 20 Year 8 students measured reaction distance (cm): 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 19, 20, 20, 21, 23, 28. (a) Find the five-number summary: min, Q1, median, Q3, max. (b) Find the IQR. (c) Sketch a boxplot. (d) Using the 1.5×IQR rule, identify outliers.
(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)
**Outlier sensitivity.** The dataset has mean 16.25 and median 15.5. (a) Remove the largest value (28). Find the new mean and median. (b) Compare the changes. (c) Which measure is more robust to outliers? Explain.
(a) New mean ≈ 15.63, median = 15 (b) Mean dropped 0.62, median dropped 0.5 (c) Median — depends only on rank
**Comparing two boxplots.** Class A and Class B reaction distances: - Class A: min 8, Q1 12, median 15, Q3 18, max 25. - Class B: min 10, Q1 13, median 16, Q3 22, max 30. (a) Which class has a higher median? (b) Which class is more consistent (smaller IQR)? (c) Describe the skew of each. (d) Make a one-sentence comparison.
(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
**Combining two classes.** Class A: 20 students, mean 16.25. Class B: 30 students, mean 14.50. (a) Find the combined mean. (b) Why is this NOT the average of the two means?
(a) 15.2 (b) Weighted by class size — different counts
**Skew investigation.** A distribution of salaries in a small company has mean £45 000 and median £32 000. (a) Identify the skew direction. (b) What does this tell you about the salary distribution? (c) The CEO's salary is £200 000. Is this an outlier? Justify. (d) Which measure of central tendency is more representative of a typical employee?
(a) Positive (b) A few high salaries pull the mean above the median (c) Likely an outlier (d) Median
**Sampling-based inference.** A sample of 30 students has mean 75 and SD 10. The school of 600 students is the population. (a) Estimate the population mean. (b) Comment on the reliability with $n = 30$. (c) Estimate the percentage of students within ±10 marks of 75 (one SD). (d) Estimate the percentage between 65 and 85.
(a) 75 (b) Moderate — small sample (c) ~68% (d) ~68%
**Inference from a small sample.** A poll of 50 people finds 30 prefer brand A. Estimate the percentage preference in the population, and state your confidence.
60%; margin of error ≈ ±14%
**Comparing two data sets.** Class A and Class B test scores: A: 60, 70, 75, 80, 85. B: 50, 60, 75, 80, 100. (a) Find mean, median, range for each. (b) Which has higher mean? Higher median? (c) Which has more spread? (d) Comment on skew.
(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
**Inference validity.** A school claims its students have an average grade of 75. To verify: (a) Suggest two ways the school could test this claim. (b) State two issues that might cause the published average to be unreliable. (c) If sampling 50 students, what would be an acceptable margin of error?
(a) Stratified random sample / full census (b) Selective reporting / outliers / definition of "average" (c) ≈ ±5%
**MAD calculation.** Compute the Mean Absolute Deviation (MAD) of the data set 5, 8, 10, 12, 15. (a) Find the mean. (b) Find absolute deviations. (c) Find the MAD. (d) Interpret what MAD tells us.
(a) 10 (b) 5, 2, 0, 2, 5 (c) 2.8 (d) Average distance from the mean
**Box-and-whisker investigation.** 30 students' weekly reading hours have 5-number summary: 1, 3, 6, 10, 25. (a) Find IQR. (b) Identify any outliers using the 1.5×IQR rule. (c) Sketch the boxplot and describe the skew. (d) What might the value 25 represent?
(a) 7 (b) Upper fence 20.5; 25 > 20.5 → outlier (c) Right-skewed (long upper whisker) (d) An exceptional reader
**Validating an inference.** A teacher claims: "students who attend tutorials get higher marks". To support this: (a) Suggest a data-collection method. (b) State what statistical measures would help test the claim. (c) List two confounding factors that could mislead the conclusion. (d) Describe how a control group would help.
(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