NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Statistical investigations

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A statistical investigation was conducted in Adelaide to examine the effect of a new teaching method on student performance in a Year 7 mathematics test. The test scores (out of 100) for 12 students who used the new method were: 78, 82, 76, 80, 84, 79, 81, 83, 77, 82, 80, 85.

Calculate the exact mean score. Express your answer as a simplified fraction.

Concept illustrations

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Median, mode and range in sorted data. The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.23389↑ medianmode 3 range 7
The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.
Comparing three category counts. Illustrative category counts: A = 2, B = 4, C = 9. Separate bars compare categories; the vertical scale starts at zero.05102A4B9C5
Illustrative category counts: A = 2, B = 4, C = 9. Separate bars compare categories; the vertical scale starts at zero.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The given data consists of 12 test scores: 78, 82, 76, 80, 84, 79, 81, 83, 77, 82, 80, 85. These are discrete numerical values representing student performance out of 100.
  2. To find the mean, sum all the scores and divide by the number of students, which is 12.
  3. Add the scores step by step: 78 + 82 = 160; 160 + 76 = 236; 236 + 80 = 316; 316 + 84 = 400; 400 + 79 = 479; 479 + 81 = 560; 560 + 83 = 643; 643 + 77 = 720; 720 + 82 = 802; 802 + 80 = 882; 882 + 85 = 967.
  4. The total sum of the 12 scores is 967.
  5. Divide the sum by the number of students: 967 ÷ 12.
  6. Perform the division: 967 ÷ 12. 12 × 80 = 960, so 967 − 960 = 7. Therefore, 967 ÷ 12 = 80 + 7/12.
  7. The result is 80 and 7/12. As an improper fraction: (80 × 12 + 7) / 12 = (960 + 7) / 12 = 967/12.
  8. Check if 967/12 can be simplified. The greatest common divisor of 967 and 12 is 1, since 967 is a prime number (verified by testing divisibility by primes up to √967 ≈ 31.1).
  9. Therefore, 967/12 is already in simplest form.
  10. The mean score is 967/12.
  11. Verify by multiplying the mean by 12: (967/12) × 12 = 967, which matches the original sum.
  12. The exact mean score is 967/12.

Reference answer: 967/12

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