Multiple choice

The marks of a student in $6$ tests are $41, 45, 54, 57, 43$ and x. If the mean marks of these tests is $48$, then standard deviation of these tests is?

  1. $\dfrac{10}{\sqrt{3}}$
  2. $\dfrac{10}{\sqrt{2}}$
  3. $\dfrac{10}{3}$
  4. $\dfrac{20}{3}$
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A Correct answer
Explanation

Mean = (41+45+54+57+43+x)/6 = 48. 240+x = 288, so x = 48. Data: 41, 45, 54, 57, 43, 48. Deviations from 48: -7, -3, 6, 9, -5, 0. Squares: 49, 9, 36, 81, 25, 0. Sum = 200. Variance = 200/6 = 100/3. Standard deviation = sqrt(100/3) = 10/sqrt(3).

AI explanation

Using the mean formula, we first find the sum of the marks by multiplying the mean by the number of tests: 48 times 6 = 288. Subtracting the five known marks gives the sixth score: 288 minus 41 minus 45 minus 54 minus 57 minus 43 equals x equals 48. The deviations of all six marks from the mean of 48 are 7, 3, 6, 9, 5 and 0, so the sum of their squares is 49 plus 9 plus 36 plus 81 plus 25 plus 0 equals 200. We divide this sum by the number of tests to find the variance, which is 200 divided by 6, and taking the square root gives the standard deviation of 10 divided by the square root of 3.

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