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Measures of Dispersion for Grouped Data · Form 5

Measures of Dispersion for Grouped Data: Practice Questions

Original SPM-style practice questions for Measures of Dispersion for Grouped Data, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Measures of Dispersion for Grouped Data, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

The class interval of a grouped data is 20 – 29. What is the midpoint of this class?

  1. A. 24
  2. B. 24.5
  3. C. 25
  4. D. 29

Question 2

A grouped data has the frequencies: 10–19 (5), 20–29 (8), 30–39 (12), 40–49 (7). Which is the modal class?

  1. A. 10 – 19
  2. B. 20 – 29
  3. C. 30 – 39
  4. D. 40 – 49

Question 3

For a grouped data the midpoints are 2, 5, 8 with frequencies 4, 6, 10 respectively. Estimate the mean.

  1. A. 5.0
  2. B. 5.5
  3. C. 5.9
  4. D. 6.0

Question 4

The frequencies of a grouped data are: 0–9 (5), 10–19 (9), 20–29 (11), 30–39 (7). In which class does the median lie?

  1. A. 0 – 9
  2. B. 10 – 19
  3. C. 20 – 29
  4. D. 30 – 39

Question 5

A set of 40 data is drawn as an ogive. At which cumulative frequency is the first quartile Q₁ read?

  1. A. 10
  2. B. 20
  3. C. 30
  4. D. 40

Question 6

The first quartile of a data is 24 and the third quartile is 41. Find the interquartile range.

  1. A. 17
  2. B. 32.5
  3. C. 41
  4. D. 65

Question 7

Set P has standard deviation 3.2 and Set Q has standard deviation 6.8. Which statement is correct?

  1. A. Set Q is more consistent than Set P
  2. B. Set P is more consistent than Set Q
  3. C. Both sets have equal dispersion
  4. D. Set P is more spread out than Set Q

Question 8

For a grouped data, N = 10, Σfx = 50 and Σfx² = 310. Find the standard deviation.

  1. A. √6 ≈ 2.45
  2. B. 6.00
  3. C. √31 ≈ 5.57
  4. D. 3.10

Question 9

What is the upper boundary of the class 30 – 39?

  1. A. 39
  2. B. 39.5
  3. C. 40
  4. D. 34.5

Question 10

A set of 60 data is displayed on an ogive. At which cumulative frequency is the third quartile Q₃ read?

  1. A. 15
  2. B. 30
  3. C. 45
  4. D. 60

Structured (Paper 2 style)

Question 1 (6 marks)

The table shows the time, in minutes, taken by 40 students to complete a task. Time (min): 10–19 (4), 20–29 (7), 30–39 (10), 40–49 (12), 50–59 (7).

(a) State the modal class. [1 mark] (b) Calculate the estimated mean time.

[3 marks] (c) Determine the class in which the median lies. [2 marks]

  1. (a) The modal class is the class with the highest frequency (12), which is 40–49.
  2. (b) The midpoints are 14.5, 24.5, 34.5, 44.5, 54.5.
  3. Σfx = 4×14.5 + 7×24.5 + 10×34.5 + 12×44.5 + 7×54.5 = 58 + 171.5 + 345 + 534 + 381.5 = 1490.
  4. Σf = 4 + 7 + 10 + 12 + 7 = 40, so estimated mean = 1490 ÷ 40 = 37.25 minutes.
  5. (c) Median position = N ÷ 2 = 40 ÷ 2 = 20. Cumulative frequencies: 4, 11, 21, 33, 40; position 20 falls in the class 30–39.

Question 2 (6 marks)

The table shows the scores of 20 players in a game. Score: 0–4 (3), 5–9 (8), 10–14 (6), 15–19 (3).

(a) Calculate the estimated mean score. [2 marks] (b) Calculate the variance.

[3 marks] (c) Hence, find the standard deviation. [1 mark]

  1. Midpoints x = 2, 7, 12, 17; frequencies f = 3, 8, 6, 3; Σf = 20.
  2. (a) Σfx = 3×2 + 8×7 + 6×12 + 3×17 = 6 + 56 + 72 + 51 = 185, so mean = 185 ÷ 20 = 9.25.
  3. (b) Σfx² = 3×2² + 8×7² + 6×12² + 3×17² = 12 + 392 + 864 + 867 = 2135.
  4. Variance = Σfx² ÷ N − mean² = 2135 ÷ 20 − 9.25² = 106.75 − 85.5625 = 21.1875.
  5. (c) Standard deviation = √21.1875 ≈ 4.60.

Question 3 (6 marks)

The table shows the masses, in kg, of 50 parcels. Mass (kg): 1–5 (6), 6–10 (10), 11–15 (14), 16–20 (12), 21–25 (8).

(a) State the cumulative frequency at each upper boundary. [2 marks] (b) Using the cumulative frequencies (ogive), estimate the first quartile Q₁ and the third quartile Q₃.

[3 marks] (c) Hence, find the interquartile range. [1 mark]

  1. (a) Upper boundaries: 5.5, 10.5, 15.5, 20.5, 25.5; cumulative frequencies: 6, 16, 30, 42, 50.
  2. (b) Q₁ is at position N ÷ 4 = 50 ÷ 4 = 12.5, which lies in the class 6–10 (boundary 5.5–10.5).
  3. Q₁ = 5.5 + ((12.5 − 6) ÷ 10) × 5 = 5.5 + 3.25 = 8.75 kg.
  4. Q₃ is at position 3N ÷ 4 = 3 × 50 ÷ 4 = 37.5, which lies in the class 16–20 (boundary 15.5–20.5).
  5. Q₃ = 15.5 + ((37.5 − 30) ÷ 12) × 5 = 15.5 + 3.125 = 18.625 kg.
  6. (c) Interquartile range = Q₃ − Q₁ = 18.625 − 8.75 = 9.875 kg.

Question 4 (4 marks)

A grouped data has classes with midpoints 3, 8, 13, 18 and frequencies 4, k, 6, 2. The estimated mean is 9.

(a) Write an equation in terms of k. [2 marks] (b) Hence, find the value of k.

[2 marks]

  1. Σf = 4 + k + 6 + 2 = 12 + k; Σfx = 4×3 + k×8 + 6×13 + 2×18 = 12 + 8k + 78 + 36 = 126 + 8k.
  2. (a) Mean = Σfx ÷ Σf gives (126 + 8k) ÷ (12 + k) = 9.
  3. (b) 126 + 8k = 9(12 + k) = 108 + 9k.
  4. 126 − 108 = 9k − 8k, so k = 18. Check: Σf = 30, Σfx = 270, mean = 270 ÷ 30 = 9.

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

What's included in this grouped dispersion practice set?

It includes Paper 1 questions on reading variance and standard deviation from a frequency table, and Paper 2 structured questions building a full table before calculating measures of dispersion for grouped data. All questions are original practice for this site, not past SPM papers.

Why attempt the question before checking the worked solution?

Building the frequency table and doing the calculation yourself shows exactly which step, such as finding midpoints or squaring deviations, you find hardest. Looking at the solution first only shows you the final numbers, not where your own method needs correcting.

What's a common mistake with grouped dispersion questions?

Students often use the class boundary instead of the midpoint when building the frequency table, which throws off every later calculation. Set up your table carefully first, check each column against the worked solution, and only then move on to variance or standard deviation.

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