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

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Required skills (click to reveal)

A statistical investigation was conducted in Perth to examine the effect of a new study technique on the number of correct answers in a 20-question quiz. A group of 12 students used the technique, and their scores were: 14, 16, 15, 17, 14, 16, 15, 18, 14, 16, 15, 17.

Calculate the exact interquartile range (IQR) of the scores. Express your answer as a simplified fraction or integer.

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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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. List the 12 scores in ascending order: 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18.
  2. The data set has 12 values, which is even, so the median (Q2) is the average of the 6th and 7th values.
  3. The 6th value is 15, and the 7th value is 16. So, Q2 = (15 + 16) / 2 = 15.5.
  4. To find Q1 (first quartile), consider the lower half of the data: the first 6 values: 14, 14, 14, 15, 15, 15.
  5. The lower half has 6 values, so Q1 is the average of the 3rd and 4th values in this subset.
  6. The 3rd value is 14, and the 4th value is 15. So, Q1 = (14 + 15) / 2 = 14.5.
  7. To find Q3 (third quartile), consider the upper half of the data: the last 6 values: 16, 16, 16, 17, 17, 18.
  8. The upper half has 6 values, so Q3 is the average of the 3rd and 4th values in this subset.
  9. The 3rd value is 16, and the 4th value is 17. So, Q3 = (16 + 17) / 2 = 16.5.
  10. The interquartile range (IQR) is calculated as IQR = Q3 - Q1.
  11. Substitute values: IQR = 16.5 - 14.5 = 2.0.
  12. The IQR is exactly 2, which is an integer and does not need to be expressed as a fraction.

Reference answer: 2

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