Multiple choice

If $x_1, x_2,......x_{10}$ are $10$ observations, in which mean of $x_1, x_2, x_3, x_4$ is $11$ while mean of $x_5, x_6, ...x_{10}$ is $16$. Also $x_1^2+x^2_2+....x^2_{10}=2000$ then value of standard deviation is?

  1. $1.5$
  2. $3$
  3. $2.5$
  4. $2$
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D Correct answer
Explanation

Mean of x1..x4 = 11, sum = 44. Mean of x5..x10 = 16, sum = 96. Total sum = 140. Mean = 14. Variance = (Sum x^2)/n - (Mean)^2 = 2000/10 - 14^2 = 200 - 196 = 4. Standard deviation = sqrt(4) = 2.

AI explanation

Using the property of combined mean, the overall mean is calculated as (4 * 11 + 6 * 16) / 10, which equals 140 / 10 or 14. Applying the direct method for standard deviation, the variance is (2000 / 10) minus the square of the mean (14 squared), giving 200 minus 196, which is 4. The standard deviation is the square root of the variance, resulting in 2.

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