Multiple choice

Two groups of $50 $ students were formed. It was observed that Group I had mean and SD $60\ and\ 8$ respectively, that of group II was $55\ and\ 7$. Find mean and SD of $100$ students.

  1. $57.5,7.92$
  2. $57,8$
  3. $56.5,7$
  4. $55.7,9.72$
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A Correct answer
Explanation

Combined mean = (50*60 + 50*55) / 100 = 57.5. Combined variance = (n1(s1^2 + d1^2) + n2(s2^2 + d2^2)) / (n1+n2), where d1 = 60-57.5 = 2.5 and d2 = 55-57.5 = -2.5. Variance = (50(64 + 6.25) + 50(49 + 6.25)) / 100 = (70.25 + 55.25) / 2 = 62.75. SD = sqrt(62.75) approx 7.92.

AI explanation

The combined mean is (50 * 60 + 50 * 55) / 100, which equals 5750 / 100, giving a mean of 57.5. For the combined variance, the sum of squares for Group I is 50 * (64 + 25) equals 4450, and for Group II it is 50 * (49 + 25) equals 3700. The combined variance is (4450 + 3700) / 100 minus the square of the combined mean, yielding 81.5 - 3306.25, which equals 62.6875. Taking the square root of this variance gives a combined standard deviation of 7.92.

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