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Mathematics

If the mean of x - 3, x - 1, 7, x, 2x - 1 and 3x - 5 is 3.5, then the median is :

  1. 2.5

  2. 3.5

  3. 3.8

  4. 3.9

Statistics

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Answer

Given,

Mean = 3.5

Total number of observations = 6

Mean = Sum of ObservationsTotal number of Observations\dfrac{\text{Sum of Observations}}{\text{Total number of Observations}}

⇒ 3.5 = x−3+x−1+7+x+2x−1+3x−56\dfrac{x - 3 + x - 1 + 7 + x + 2x - 1 + 3x - 5}{6}

⇒ 3.5 = 8x−36\dfrac{8x - 3}{6}

⇒ 21 = 8x - 3

⇒ 8x = 24

⇒ x = 3.

∴ The observations are :

0, 2, 7, 3, 5, 4

By arranging the data in ascending order, we get :

0, 2, 3, 4, 5, 7

Number of observations (n) = 6, which is even.

By formula,

⇒Median=(n2)thterm+(n2+1)thterm2⇒Median=(62)thterm+(62+1)thterm2⇒Median=3rd term+4th term2⇒Median=3+42⇒Median=72\Rightarrow \text{Median} = \dfrac{\left(\dfrac{n}{2}\right)^{\text{th}} \text{term} + \left(\dfrac{n}{2} + 1\right)^{\text{th}} \text{term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\left(\dfrac{6}{2}\right)^{\text{th}} \text{term} + \left(\dfrac{6}{2} + 1\right)^{\text{th}} \text{term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\text{3rd term} + \text{4th term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\text{3} + \text{4}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{7}{2} \\[1em]

∴ Median = 3.5.

Hence, option 2 is the correct option.

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