Multiple choice

If $\displaystyle \sum_{i = 1}^{9}(x_{i} - 5) = 9$ and $\displaystyle \sum_{i = 1}^{9}(x_{i} - 5)^{2} = 45$, then the standard deviation of the $9$ items $x_{1}, x_{2}, ...., x_{9}$ is

  1. $2$
  2. $3$
  3. $9$
  4. $4$
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A Correct answer
Explanation

Standard deviation is the square root of the variance. Variance = (sum(xi-5)^2 / n) - (sum(xi-5) / n)^2. Here, variance = (45/9) - (9/9)^2 = 5 - 1 = 4. The standard deviation is sqrt(4) = 2.

AI explanation

The standard deviation can be found using the property that the variance of x is equal to the variance of (x minus c). First, find the mean of the 9 items (x minus 5) by dividing their sum, 9, by 9, giving a mean of 1. The variance of these 9 items is (the sum of their squares divided by n) minus the square of the mean, calculated as (45 divided by 9) minus 1 squared, resulting in 5 minus 1, which is 4. Therefore, the standard deviation is the square root of 4, which equals 2.

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