Median, Mode and Range — Solutions

1. Find Median, Mode and Range

   1. {4, 8, 3, 6, 5}: Median = 5, no mode, Range = 5 ▶ View Solution
   2. {12, 7, 12, 9, 8, 12, 5}: Median = 9, Mode = 12, Range = 7 ▶ View Solution
   3. {3, 3, 5, 7, 9, 11, 11}: Median = 7, Mode = 3 and 11 (bimodal), Range = 8 ▶ View Solution
   4. {2, 4, 6, 8, 10}: Median = 6, no mode, Range = 8 ▶ View Solution
   5. {15, 8, 22, 8, 17, 8}: Median = 11.5, Mode = 8, Range = 14 ▶ View Solution
   6. {1, 2, 3, 4, 5, 6, 7, 8, 9}: Median = 5, no mode, Range = 8 ▶ View Solution
   7. {30, 25, 30, 40, 35, 25}: Median = 30, Mode = 25 and 30 (bimodal), Range = 15 ▶ View Solution
   8. {7, 9, 11, 13, 15, 17}: Median = 12, no mode, Range = 10 ▶ View Solution

2. Which Measure to Use?

   1. Shoe shop — mode: Mode — most popular size to stock; mean/median may not correspond to any real shoe size ▶ View Solution
   2. Company salary — median: Median — $280 000 outlier distorts mean ($86 167 vs median $49 000); median better represents typical employee ▶ View Solution
   3. Mean ≈6.1, median = 6: Yes — differ by only 0.1; data is roughly symmetric with no major outliers ▶ View Solution
   4. When mean and median differ greatly: When data contains outliers — extreme values pull mean but leave median near the centre ▶ View Solution

3. Outliers and Their Effect

   1. Outlier in {8, 9, 10, 10, 11, 12, 45}: 45; mean with outlier = 15, without = 10 ▶ View Solution
   2. Median with/without outlier: Median unchanged at 10 both times — outlier does not shift the middle position ▶ View Solution
   3. Why median is “resistant”: Depends on position not size; one extreme value can’t shift the middle position no matter how large ▶ View Solution
   4. Test results {55, 60, 62, 65, 68, 70, 72, 98}: Mean ≈69, Median = 66.5; median more representative — 98 pulls the mean above most students’ scores ▶ View Solution

4. Median & Range in Context

   1. Street prices: Street A: Median $480k, Range $110k  |  Street B: Median $480k, Range $410k — same median but B far more variable ▶ View Solution
   2. Shoe sizes: Mode = 8, Median = 8, Range = 4 ▶ View Solution
   3. Cricket batters: Batter A: Median 30, Range 47  |  Batter B: Median 31, Range 7 — B far more consistent ▶ View Solution
   4. Seventh value between 11 and 13: 12 ▶ View Solution

5. Calculate from a Data Table

   1. Ordered temperatures: 19, 19, 22, 24, 25, 28, 31 ▶ View Solution
   2. Median: 24°C ▶ View Solution
   3. Mode: 19°C ▶ View Solution
   4. Range: 12°C ▶ View Solution
   5. Day of median: Sunday ▶ View Solution

6. True or False?

   1. Dataset can have more than one mode: True — e.g. {3, 3, 5, 5} has two modes (bimodal) ▶ View Solution
   2. Median of {3, 5, 7, 9} is 7: False — (5 + 7) ÷ 2 = 6 ▶ View Solution
   3. All same values gives range of 0: True ▶ View Solution
   4. Large value always changes median significantly: False — median is resistant to outliers; shifts by at most one position ▶ View Solution
   5. No repeated values means no mode: True ▶ View Solution
   6. Range measures spread not centre: True ▶ View Solution

7. Spot the Error

   1. Step skipped: Data not ordered first — took 3rd value from unordered list ▶ View Solution
   2. Correct working: Ordered: 1, 3, 5, 7, 9; Median = 5 ▶ View Solution
   3. Why order first: Median is the middle value by size, not by position in the original list ▶ View Solution

8. Compare Two Teams

   1. Median and range: Sharks: Median 3, Range 4  |  Eagles: Median 2, Range 7 ▶ View Solution
   2. Mode: Sharks: 2 and 3 (bimodal)  |  Eagles: 0 ▶ View Solution
   3. More consistent: Sharks — range of 4 vs Eagles’ range of 7 ▶ View Solution
   4. Which to watch: Accept any reasoned answer comparing consistency vs excitement ▶ View Solution

9. Which is Correct?

   1. Correct student: Student A — averaged the two middle values for an even count ▶ View Solution
   2. Student B’s mistake: Took only the 3rd value; with 6 values must average positions 3 and 4 ▶ View Solution
   3. Even number of values: Average the two middle values (positions n÷2 and n÷2 + 1) ▶ View Solution

10. Extended Investigation

    1. Median before: 7.5 ▶ View Solution
    2. Median after: 3.5 ▶ View Solution
    3. Range before/after: Before: 4  |  After: 3 ▶ View Solution
    4. Summary: Median dropped from 7.5 to 3.5 — treatment substantially reduced pain; range slightly narrower (4 to 3) ▶ View Solution