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Mathematics

There are 50 numbers. Each number is subtracted from 53 and the mean of the numbers so obtained is found to be -3.5. The mean of the given numbers is :

  1. 49.5

  2. 53

  3. 46.5

  4. 56.5

Statistics

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Answer

Let the original numbers be :

x1, x2, x3, ….., x50

And their mean be 'a'

So,

⇒a=x1+x2+⋯+x5050⇒x1+x2+⋯+x50=50a\Rightarrow a = \dfrac{x1 + x2 + \dots + x{50}}{50} \\[1em] \Rightarrow x1 + x2 + \dots + x{50} = 50a \\[1em]

Each number is subtracted from 53.

∴ The new numbers are :

(53−x1),(53−x2),(53−x3),…,(53−x50)(53 - x1), (53 - x2), (53 - x3), \dots, (53 - x{50})

Sum of new numbers is :

⇒ (53−x1)+(53−x2)+(53−x3)+⋯+(53−x50)(53 - x1) + (53 - x2) + (53 - x3) + \dots + (53 - x{50})

⇒ 53 + 53 + ….. + 53 - (x1 + x2 + x3 + ….. + x50)

⇒ 50 × 53 - 50a

⇒ 2650 - 50a

Mean = Total SumTotal observations\dfrac{\text{Total Sum}}{\text{Total observations}}

⇒ -3.5 = 2650−50a50\dfrac{2650 - 50a}{50}

⇒ -175 = 2650 - 50a

⇒ 50a = 2650 + 175

⇒ a = 282550\dfrac{2825}{50}

⇒ a = 56.5

∴ Mean of original numbers = 56.5.

Hence, option 4 is the correct option.

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