Practice Maths

Measures of Centre and Spread Using Technology — Solutions

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  1. Mean, median, mode, range

    1. {4,7,9,9,11,14}:
    2. {3,5,5,8,10,12,12,15}:
    3. {20,25,25,30,35,40,40,40,50}:
    4. {6,8,10,10,12,14,16,20}:
    5. {15,18,21,21,24,27}:
    6. {2,4,4,6,8,8,10,12}:
    7. {50,55,60,65,70,70,80}:
    8. {100,105,110,110,115,120,120,125,130}:

  2. Q1, Q3, IQR

    1. {8,12,14,18,22,26,30}:
    2. {5,7,9,11,13,15,17,19}:
    3. {3,6,9,12,15,18,21,24}:
    4. {10,14,16,20,24,28,32,36}:
    5. {42,45,48,51,54,57,60}:
    6. {7,11,13,17,19,23,25,29,31}:
    7. {2,5,8,11,14,17,20,23}:
    8. {30,35,40,45,50,55,60,65,70}:

  3. Outlier detection

    1. Lower fence: 6−1.5×8 = −6; Upper fence: 14+1.5×8 = 26.
    2. Lower fence: 16−1.5×7 = 5.5; Upper fence: 23+1.5×7 = 33.5.
    3. Lower fence: 52−1.5×11.5 = 34.75; Upper fence: 63.5+1.5×11.5 = 80.75.
    4. Lower fence: 32−1.5×10 = 17; Upper fence: 42+1.5×10 = 57.
    5. Lower fence: 12−1.5×5 = 4.5; Upper fence: 17+1.5×5 = 24.5.
    6. Lower fence: 212.5−1.5×15 = 190; Upper fence: 227.5+1.5×15 = 250.

  4. Compare datasets

    1. Class A: mean=70, median=70, IQR=15. Class B: mean=70, median=70, IQR=30. Both classes have the same mean and median, but Class B has a much larger IQR (30 vs. 15), indicating scores are more spread out. Class A is more consistent.
    2. Athlete A: mean=13.975, median=13.5, IQR=0.5. Athlete B: mean=12.8, median=12.7, IQR=1.35. Athlete A has a smaller IQR (0.5 vs. 1.35), so Athlete A is more consistent, though Athlete B has a slightly lower mean time.
    3. With water: mean=23, median=23, IQR=6. Without water: mean=17, median=17, IQR=8. Plants with water grew taller on average (mean 23 cm vs. 17 cm) and were slightly more consistent (IQR 6 vs. 8). Watering appears to increase height and reduce variability.
    4. July: mean≈17.6, median=17, IQR=5. January: mean≈33.1, median=33, IQR=4. January temperatures are much higher (about 15–16 degrees higher) but both months have similar spread. January is slightly more consistent (IQR=4 vs. 5).
    5. Boys: Q1=158, Q3=170, IQR=12. Girls: Q1=155, Q3=165, IQR=10. Boys have a greater IQR (12 vs. 10), indicating more spread in heights among boys.
    6. Suburb A: median=505, IQR=60. Suburb B: median=512.5, IQR=97.5. Both suburbs have similar medians (about $505 000–$512 500), but Suburb B has a much larger IQR ($97 500 vs. $60 000), indicating greater variability in house prices in Suburb B.

  5. Spreadsheet formulas

    1. Mean:
    2. Median:
    3. Mode:
    4. Minimum:
    5. Maximum:
    6. Range:
    7. Q1:
    8. Q3:

  6. Extended problems

    1. Class test {58,62,65,68,70,72,75,78,80,92}:
      (i) Mean = 720÷10 = 72; Median = (70+72)÷2 = 71; Range = 92−58 = 34
      (ii) Q1 = 65, Q3 = 78, IQR = 13
      (iii) Lower fence: 65−1.5×13 = 45.5; Upper fence: 78+1.5×13 = 97.5. Since 92 < 97.5, 92 is NOT a potential outlier by this rule.
      (iv) Without 92: sum = 628, mean = 628÷9 ≈ 69.8. The mean decreases from 72 to approximately 69.8.
    2. Basketball players:
      Player A: Mean = 120÷8 = 15; Median = (14+16)÷2 = 15; Q1 = 11, Q3 = 19, IQR = 8
      Player B: Mean = 124÷8 = 15.5; Median = (15+15)÷2 = 15; Q1 = 11, Q3 = 20, IQR = 9
      Player A is more consistent — both players have a similar mean and median, but Player A has a smaller IQR (8 vs. 9), meaning the middle 50% of Player A’s scores are less spread out.
    3. Weather stations:
      Station 1: Median = 7, Q1 = 4, Q3 = 10, IQR = 6
      Station 2: Median = 7, Q1 = 3, Q3 = 11, IQR = 8
      Both stations have the same median (7 mm). Station 2 has a larger IQR (8 vs. 6), indicating more variable rainfall.
      Spreadsheet formulas: =MEDIAN(A1:A9), =QUARTILE(A1:A9,1), =QUARTILE(A1:A9,3), and IQR = QUARTILE(A1:A9,3)−QUARTILE(A1:A9,1).

  7. Which measure of centre is most appropriate?

    1. Median — the house price of $1 800 000 is an outlier that would greatly inflate the mean. The median is not affected by extreme values and better represents typical prices.
    2. Mode — shoe size 9 is the most common size and tells the shop owner which size to stock most. The mean shoe size would not correspond to a real size.
    3. Mean — the scores are fairly evenly spread with no extreme outliers, so the mean gives a good summary of the typical score.
    4. Mean — temperatures are consistent with no extreme outliers; the mean accurately represents the typical daily temperature.
    5. Median — the value of 8 goals is an outlier that would pull the mean upwards. The median of 1 goal better represents the typical match result.
    6. Mode — colour is a categorical variable, so mean and median have no meaning. The mode (blue) identifies the most common favourite colour.

  8. Effect of adding a value

    Original dataset: {10, 14, 16, 18, 20, 22}, n=6, sum=100, mean≈16.67, median=17

    1. Adding 16: new dataset {10, 14, 16, 16, 18, 20, 22}, n=7, sum=116, mean≈16.57, median=16. Mean barely changed; median decreased by 1.
    2. Adding 2: new dataset {2, 10, 14, 16, 18, 20, 22}, n=7, sum=102, mean≈14.57, median=16. Mean decreased by about 2.1; median unchanged.
    3. Adding 50: new dataset {10, 14, 16, 18, 20, 22, 50}, n=7, sum=150, mean≈21.43, median=18. Mean increased by about 4.8; median increased by 1.
    4. Adding 50 caused the biggest change in the mean (increased by ~4.8). Adding 16 caused the smallest change in the median (decreased by 1).
    5. The median was least affected by adding 50 (changed from 17 to 18, a change of 1). The mean changed by ~4.8. The median is resistant to extreme values because it only depends on the middle value(s), not the actual size of all values.

  9. Interpret IQR in context

    1. Sprinter A is more consistent. Despite having the same mean, Sprinter A’s IQR (0.2 s) is much smaller than Sprinter B’s (1.1 s), meaning A’s times vary less from race to race.
    2. IQR = 50 − 30 = 20. This means the middle 50% of the data values fall within a range of 20 units — half the values are between 30 and 50.
    3. Dataset B has more variable data. The larger IQR (25 vs. 5) means the middle 50% of Dataset B’s values are more spread out.
    4. Lower fence: 12 − 1.5 × 8 = 12 − 12 = 0. Upper fence: 20 + 1.5 × 8 = 20 + 12 = 32. Since 35 > 32, 35 is a potential outlier.
    5. The IQR of 6 minutes means the middle 50% of students had wait times within a range of 6 minutes. Half the students’ wait times were neither the very shortest nor the very longest, and those typical times varied by about 6 minutes.

  10. Full investigation

    1. Group A: sum=138, mean=13.8; ordered: {5,8,10,10,12,15,15,18,20,25}, median=(12+15)/2=13.5; range=25−5=20.
      Group B: sum=164, mean=16.4; ordered: {2,5,8,10,12,12,15,20,30,50}, median=(12+12)/2=12; range=50−2=48.
    2. Group A: Q1=10, Q3=18, IQR=8.
      Group B: Q1=8, Q3=20, IQR=12.
    3. Group B lower fence: 8−1.5×12=8−18=−10. Upper fence: 20+1.5×12=20+18=38. Since 50 > 38, 50 is a potential outlier in Group B.
    4. Group B has more spread: range = 48 vs. 20, and IQR = 12 vs. 8. Both measures indicate Group B’s pocket money is more variable.
    5. The median ($12) is a better representation of typical pocket money for Group B. The outlier ($50) inflates the mean to $16.40, making it appear students typically receive much more than most of them actually do. The median is not affected by the outlier.
    6. =AVERAGE(A1:A10) for mean; =MEDIAN(A1:A10) for median; =QUARTILE(A1:A10,1) for Q1; =QUARTILE(A1:A10,3) for Q3.