If the sum of the squares of the deviations of 25 observations taken from the mean 40 is 900, then the coefficient of variation is

Options

A. `20%`
B. `12.5%`
C. `15%`
D. `18%`

Answer

To find the coefficient of variation given the sum of the squares of the deviations of 25 observations from the mean, we can follow these steps:

### Step 1: Understand the given information
We know:
- Number of observations (n) = 25
- Mean (x̄) = 40
- Sum of squares of deviations from the mean (Σ(xi - x̄)²) = 900

### Step 2: Calculate the standard deviation
The formula for standard deviation (σ) is:
\[ σ = \sqrt{\frac{Σ(xi - x̄)²}{n}} \]
Substituting the known values:
\[ σ = \sqrt{\frac{900}{25}} \]

### Step 3: Simplify the calculation
Calculating the fraction:
\[ \frac{900}{25} = 36 \]
Now, take the square root:
\[ σ = \sqrt{36} = 6 \]

### Step 4: Calculate the coefficient of variation
The coefficient of variation (CV) is given by the formula:
\[ CV = \frac{σ}{x̄} \times 100\% \]
Substituting the values we found:
\[ CV = \frac{6}{40} \times 100\% \]

### Step 5: Simplify the coefficient of variation
Calculating the fraction:
\[ \frac{6}{40} = 0.15 \]
Now, multiply by 100%:
\[ CV = 0.15 \times 100\% = 15\% \]

### Final Answer
The coefficient of variation is **15%**.

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