NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Comparing data distributions

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The weekly electricity consumption (in kWh) for 12 households in Melbourne and 12 households in Brisbane is recorded. The Melbourne data set is: 240, 250, 255, 260, 265, 270, 275, 280, 285, 290, 295, 300.

The Brisbane data set is: 230, 240, 245, 250, 255, 260, 265, 270, 275, 280, 285, 310. Both data sets are ordered.

Calculate the exact interquartile range (IQR) for the Brisbane data set. The IQR is the difference between the upper quartile (Q3) and the lower quartile (Q1).

For a data set with 12 values, Q1 is the average of the 3rd and 4th values, and Q3 is the average of the 9th and 10th values.

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Worked solution

  1. The Brisbane data set has 12 values, which are already ordered: 230, 240, 245, 250, 255, 260, 265, 270, 275, 280, 285, 310.
  2. To find the lower quartile (Q1), calculate the average of the 3rd and 4th values in the ordered data set.
  3. The 3rd value is 245 and the 4th value is 250. So, Q1 = (245 + 250) / 2.
  4. Calculate: 245 + 250 = 495. Then, 495 / 2 = 247.5. Therefore, Q1 = 247.5 kWh.
  5. To find the upper quartile (Q3), calculate the average of the 9th and 10th values in the ordered data set.
  6. The 9th value is 275 and the 10th value is 280. So, Q3 = (275 + 280) / 2.
  7. Calculate: 275 + 280 = 555. Then, 555 / 2 = 277.5. Therefore, Q3 = 277.5 kWh.
  8. The interquartile range (IQR) is the difference between Q3 and Q1: IQR = Q3 - Q1.
  9. Substitute the values: IQR = 277.5 - 247.5.
  10. Calculate: 277.5 - 247.5 = 30.0.
  11. The IQR is 30.0 kWh. This represents the spread of the middle 50% of the Brisbane electricity consumption data.
  12. The final answer is 30.0 kWh, which is exact and matches the required precision.

Reference answer: 30.0

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