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Chapter 18: Problem 694

The sum of the squares of deviation for 10 observations taken from their mean 30 is 90 . The coefficient of variation is (a) \(20 \%\) (b) \(10 \%\) (c) \(11 \%\) (d) \(12 \%\)

Short Answer

Expert verified
The coefficient of variation in this problem is found by first calculating the standard deviation, which is \(\sigma = \sqrt{\frac{90}{10}} = 3\). Then, the coefficient of variation is calculated as \(CV = \frac{3}{30} \times 100 = 10\%\). Therefore, the correct answer is (b) \(10\%\).

Step by step solution

01

Utilize the formula for standard deviation

The sum of the squares of deviation is used to find the standard deviation of a set of observations, and is found using the following formula: \[ \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{N}} \] where \(sigma\) (\(\sigma\)) is the standard deviation, \(x\) represents individual data points, \(\bar{x}\) is the mean of the observations, and \(N\) is the number of observations.
02

Calculate standard deviation from given information

In this exercise, we are given the sum of the squares of deviation (\(90\)), the mean (\(30\)), and the number of observations (\(10\)). We can now use the formula from Step 1 to find the standard deviation (\(\sigma\)): \[\sigma = \sqrt{\frac{90}{10}} \] Upon simplifying this expression, we get: \[\sigma = \sqrt{9} \] Therefore, the standard deviation is \(\sigma = 3\).
03

Calculate the coefficient of variation

Now that we have found the standard deviation, we can calculate the coefficient of variation. Recall that the coefficient of variation is the ratio of the standard deviation to the mean, multiplied by 100. Using the mean of 30 and the standard deviation of 3, we get: \[CV = \frac{3}{30} \times 100\] Further simplifying, we see that the coefficient of variation is \(10\%\). So the correct answer is (b) \(10\%\).

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Most popular questions from this chapter

\(A\) and \(B\) are events of same experiments with \(P(A)=0.2\) \(P(B)=0.5\) Maximum value of \(P\left(A^{\prime} \cap B\right)=\) (a) \(0.2\) (b) \(0.5\) (c) \(0.1\) (d) \(0.4\)

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