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Fluency · Pack B

8.12 Analysing Data

Answer each question. Show working where needed.

Bronze

  1. Define "spread" in statistics in one sentence.

  2. For the data 4, 6, 8, 9, 12, find the (a) mean, (b) range.

  3. Describe the symmetry of a data set whose mean = median.

  4. Describe the skew of a distribution where mean > median.

  5. Describe the skew of a distribution where mean < median.

  6. Identify the outlier in the data 5, 6, 7, 7, 8, 9, 50.

  7. A dataset has mean 10. Describe what each tail represents.

  8. A uniform distribution has all values occurring at the same frequency. Sketch what this looks like.

  9. For the data 3, 4, 4, 5, 5, 5, 6, 7, identify the symmetry.

  10. For the data 2, 3, 5, 6, 9, find the mean absolute deviation (MAD).

Silver

  1. For the data 4, 7, 8, 10, 15, find the (a) mean, (b) median, (c) MAD.

  2. Two classes' reaction times (ms) have means 200 and 210, ranges 50 and 30. Which class is faster on average and which more consistent?

  3. A dataset has the mean = 50 and median = 45. State the skew direction.

  4. For the data 5, 8, 9, 10, 12, 15, 30, find the median and explain whether 30 is likely an outlier.

  5. A test has marks 12, 14, 16, 18, 20 with mean 16. Compute the (a) MAD, (b) variance.

  6. 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?

  7. For a uniform distribution of values 1, 2, 3, 4, 5, 6 (each appearing once), find the (a) mean, (b) MAD.

  8. Class A vs Class B mean test marks: A = 75, B = 80; standard deviations: A = 10, B = 4. Which class is more consistent?

  9. A teacher says "the class average is 72, so most students scored around 72." Critique this statement.

  10. Two cricketers have batting averages: A 45, B 42. A's standard deviation is 25, B's is 8. Whose performance is more reliable?

Gold

  1. 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.

  2. 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?

  3. 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?

  4. 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.

  5. A class of 30 has mean 75, range 40. A new student with score 100 joins. Find the new mean.

  6. Mean salary in company A is £50k with SD £5k; in company B mean £45k with SD £20k. Compare typical salaries.

  7. A class's 5-number summary is 10, 15, 22, 28, 50. (a) Compute IQR. (b) Identify any outliers using 1.5×IQR rule.

  8. In a survey of 1000 households, mean income $50k, median $40k. (a) State the skew. (b) Explain what this tells you about wealth distribution.

  9. 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.

  10. 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.

Platinum

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. A teacher's test marks are: 65, 70, 72, 75, 80, 85, 90. Find Q1, Q3, IQR, and the 1.5×IQR fences.

  7. 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?

  8. A class of $n$ students has mean test score 65. A new student with score 80 joins; new mean is 66.5. Find $n$.

  9. A 2-way table: 130 study French, 90 study Spanish, 50 both. School of 200. Find (a) only French, (b) only Spanish, (c) neither.

  10. 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.