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For a frequency distribution the coefficient of skewness based upon the quartiles is 0.6. If the sum of the upper and lower quartiles is 100 and media
Question

For a frequency distribution the coefficient of skewness based upon the quartiles is 0.6. If the sum of the upper and lower quartiles is 100 and median is 38, What is the value of lower and upper quartiles :

A.

40 and 60

B.

20 and 80

C.

30 and 60

D.

30 and 70

Correct option is D

The correct answer is (d).

Introduction 

Bowley's coefficient of skewness provides a method to measure the asymmetry of a dataset relying on quartiles and the median.

By applying the given values of the sum of quartiles, the median, and the skewness coefficient into the formula, we can establish a system of linear equations. Solving these equations reveals the precise values for the lower and upper quartiles.

Information Booster
  • Bowley’s coefficient of skewness is defined as Sk=Q3+Q12Q2Q3Q1S_k = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}​​
  • Here, Q1Q_1​ is the lower quartile, Q3Q_3​ is the upper quartile, and Q2Q_2​ is the median.
  • We are given
  • Q3+Q1=100andQ2=38Q_3 + Q_1 = 100 and Q_2 = 38
  • This simplifies to
  • 0.6=10076Q3Q1=24Q3Q1 0.6 = \frac{100 - 76}{Q_3 - Q_1} = \frac{24}{Q_3 - Q_1}
  • Now, solving the system: Q3+Q1=100Q_3 + Q_1 = 100​ and Q3Q1=40Q_3 - Q_1 = 40​​
  • Adding the equations yields 2Q3=140=>Q3=702Q_3 = 140 \Rightarrow Q_3 = 70​, and consequently Q1=30Q_1 = 30​​
Additional Knowledge
  • Option A (40 and 60): If Q1=40andQ3=60Q_1=40 and Q_3=60​, their sum is 100, but Q3Q1=20Q_3 - Q_1 = 20​. The skewness would be 24/20=1.224/20 = 1.2​, which is incorrect.
  • Option B (20 and 80): If Q1=20andQ3=80Q_1=20 and Q_3=80​, their sum is 100, but Q3Q1=60Q_3 - Q_1 = 60​. The skewness would be 24/60=0.424/60 = 0.4​, which is incorrect.
  • Option C (30 and 60): The sum of these quartiles is 90, which violates the given condition that the sum of the quartiles must be exactly 100.

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