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GMAT Exam  >  GMAT Questions  >  Arrange the following sets in the order of th... Start Learning for Free
Arrange the following sets in the order of the increasing magnitude of their mean to standard deviation ratio.
I. {50, 60, 70, 80}
II. {35, 40, 45, 50}
III. {90, 110, 130, 150}
  • a)
    I < II < III
  • b)
    II < III < I
  • c)
    II < I < III
  • d)
    III < I < II
  • e)
    III < II < I
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Arrange the following sets in the order of the increasing magnitude of...
Given:
  • {50, 60, 70, 80}
  • {35, 40, 45, 50}
  • {90, 110, 130, 150}
To Find: Arrange the sets in increasing order of the (Mean / Standard deviation) ratio.
Approach:
  1. Let’s assume the mean of set-I be m and standard deviation be d.
  2. We would try to express the terms of the other two sets in terms of set-I for establishing a relation between the standard deviations and the means of the set
  3. Also, we will use the following properties of standard deviation to calculate the standard deviation of the other sets in terms of d:
    • If all the numbers of a set are multiplied or divided by the same constant x, the standard deviation is also multiplied or divided by |x| respectively.
    • If all the numbers of a set are increased or decreased by the same constant x, the standard deviation of the set does not change.
Working out:
  1. Set-I: {50, 60, 70, 80}
    • Assuming the mean of set {50, 60, 70, 80} as m and standard deviation be d
    • Ratio = m/d………(1)
  2. Set-II: {35, 40, 45, 50}
    • {50, 60, 70, 80} → Dividing all the terms of the set by 2, results in the standard deviation of d/2 and mean of m/2 . The set becomes = {25, 30, 35, 40}
    • Adding 10 to all the numbers of the set {25, 30, 35, 40} does not change its standard deviation but increases its mean by 10. The set becomes = {35, 40, 45, 50}
    • Hence, the set {35, 40, 45, 50} has a standard deviation of d/2  and mean of  
  3. Set-III: {90, 110, 130, 150}
    • ​​{50, 60, 70, 80}→ Multiplying all the terms of the set by 2, results in standard deviation of 2d and mean of 2m. The set becomes = { 100, 120, 140, 160}
    • Subtracting 10 from all the terms of the set {100, 120, 140, 160} does not change its standard deviation but decreases the mean by 10. The set becomes = {90, 110, 130, 150}
    • Hence, the set {90, 110, 130, 150} has a standard deviation of 2d and mean of 2m – 10
  4. Comparing (1), (2) and (3), we can arrange the sets in the ratio of mean to standard deviation as
    a. III < I < II
Answer : D
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Most Upvoted Answer
Arrange the following sets in the order of the increasing magnitude of...
The mean to standard deviation ratio measures the spread or variability of the data relative to the average value. A higher ratio indicates a larger spread relative to the mean, while a lower ratio indicates a smaller spread relative to the mean.

To find the mean to standard deviation ratio, we first need to calculate the mean and standard deviation for each set.

I. {50, 60, 70, 80}
Mean: (50 + 60 + 70 + 80)/4 = 65
Standard deviation: √[((50-65)² + (60-65)² + (70-65)² + (80-65)²)/4] = √[(225 + 25 + 25 + 225)/4]
= √[500/4] = √125 = 11.18
Mean to standard deviation ratio: 65/11.18 = 5.82

II. {35, 40, 45, 50}
Mean: (35 + 40 + 45 + 50)/4 = 42.5
Standard deviation: √[((35-42.5)² + (40-42.5)² + (45-42.5)² + (50-42.5)²)/4] = √[(56.25 + 6.25 + 6.25 + 56.25)/4]
= √[125/4] = √31.25 = 5.59
Mean to standard deviation ratio: 42.5/5.59 = 7.61

III. {90, 110, 130, 150}
Mean: (90 + 110 + 130 + 150)/4 = 120
Standard deviation: √[((90-120)² + (110-120)² + (130-120)² + (150-120)²)/4] = √[(900 + 100 + 100 + 900)/4]
= √[2000/4] = √500 = 22.36
Mean to standard deviation ratio: 120/22.36 = 5.36

Therefore, the sets in increasing order of the magnitude of their mean to standard deviation ratio are:
III. {90, 110, 130, 150} (ratio = 5.36)
I. {50, 60, 70, 80} (ratio = 5.82)
II. {35, 40, 45, 50} (ratio = 7.61)
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