NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Data displays and distributions

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Required skills (click to reveal)

A group of 18 students in Sydney recorded the mass (in kilograms) of their backpacks. The recorded masses are: 5.2, 5.4, 5.6, 5.7, 5.8, 5.9, 6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 7.1, 7.3.

Construct a stem-and-leaf plot using the whole number part as the stem and the decimal part as the leaf.

What is the exact value of the median mass of the backpacks?

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 data set contains 18 backpack masses in kilograms, recorded in Sydney. The values are: 5.2, 5.4, 5.6, 5.7, 5.8, 5.9, 6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 7.1, 7.3.
  2. To construct a stem-and-leaf plot, the stem is the whole number part (kg), and the leaf is the decimal part (tenths of a kg). The stems range from 5 to 7.
  3. List the data in ascending order (already done). For stem 5: leaves are 2, 4, 6, 7, 8, 9 (from 5.2 to 5.9).
  4. For stem 6: leaves are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (from 6.0 to 6.9).
  5. For stem 7: leaves are 1, 3 (from 7.1 and 7.3).
  6. The stem-and-leaf plot is now complete: 5 | 2 4 6 7 8 9; 6 | 0 1 2 3 4 5 6 7 8 9; 7 | 1 3.
  7. To find the median, note that there are 18 data points (even number), so the median is the average of the 9th and 10th values in the ordered list.
  8. Count the values: 1st to 6th are in stem 5 (5.2 to 5.9). 7th to 16th are in stem 6 (6.0 to 6.9). The 9th value is the 3rd in stem 6: 6.2. The 10th value is the 4th in stem 6: 6.3.
  9. Calculate the average of the 9th and 10th values: (6.2 + 6.3) / 2 = 12.5 / 2 = 6.25.
  10. The median mass is 6.25 kg. This is exact because 6.2 and 6.3 are both measured to the nearest 0.1 kg, and their average is a terminating decimal.
  11. Verify: the 9th and 10th values are correctly identified as 6.2 and 6.3. Their sum is 12.5, and dividing by 2 gives 6.25.
  12. The median mass of the backpacks is 6.25 kg.

Reference answer: 6.25 kg

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