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Statistics question

2019 · 12 Jan · Shift 1 · Q23
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Statistics question

2019 · 12 Jan · Shift 1 · Q23

JEE MainMathematicsStatisticsMCQ+4 / −1
If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is :
  1. A
    31
  2. B
    50
  3. C
    51
  4. D
    30
View written solutionFree

Correct answer: A

  1. Let the 50 observations be x1,x2,…,x50x_1, x_2, \dots, x_{50}x1​,x2​,…,x50​.

  2. Given: the sum of deviations from 303030 is 505050.

    That means, ∑i=150(xi−30)=50\sum_{i=1}^{50} (x_i - 30) = 50∑i=150​(xi​−30)=50

  3. Expand the summation: ∑i=150xi−∑i=15030=50\sum_{i=1}^{50} x_i - \sum_{i=1}^{50} 30 = 50∑i=150​xi​−∑i=150​30=50

    Since there are 505050 observations, ∑i=15030=50⋅30=1500\sum_{i=1}^{50} 30 = 50 \cdot 30 = 1500∑i=150​30=50⋅30=1500

    So, ∑i=150xi−1500=50\sum_{i=1}^{50} x_i - 1500 = 50∑i=150​xi​−1500=50

  4. Hence, ∑i=150xi=1550\sum_{i=1}^{50} x_i = 1550∑i=150​xi​=1550

  5. Mean of the observations is: xˉ=∑xi50=155050=31\bar{x} = \frac{\sum x_i}{50} = \frac{1550}{50} = 31xˉ=50∑xi​​=501550​=31

  6. Therefore, the correct option is: A: 31\boxed{\text{A: }31}A: 31​

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