Multiple choice

A student scores the following marks in five test: $45, 54, 41, 57, 43$. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six test is

  1. $\dfrac{10}{\sqrt{3}}$
  2. $\dfrac{100}{\sqrt{3}}$
  3. $\dfrac{130}{3}$
  4. $\dfrac{10}{3}$
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A Correct answer
Explanation

Sum of 5 = 240. Mean of 6 = 48, so sum of 6 = 288. The 6th mark is 288 - 240 = 48. The marks are 45, 54, 41, 57, 43, 48. Mean = 48. Variance = [(45-48)^2 + (54-48)^2 + (41-48)^2 + (57-48)^2 + (43-48)^2 + (48-48)^2] / 6 = [9 + 36 + 49 + 81 + 25 + 0] / 6 = 200 / 6 = 100 / 3. Standard deviation = sqrt(100/3) = 10/sqrt(3).

AI explanation

By the mean formula, the sum of all six scores is 48 times 6 equals 288. Subtracting the five known scores gives the unknown sixth score: 288 minus 45 minus 54 minus 41 minus 57 minus 43 equals 48. The deviations from the mean are 3, 6, 7, 9 and 5, and the sum of their squares is 9 plus 36 plus 49 plus 81 plus 25 plus 0 equals 200. The variance is 200 divided by 6, making the standard deviation the square root of 100 divided by 3, which simplifies to 10 divided by the square root of 3.

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