Question

Find the standard deviation of the numbers: -3, 0, 3, 8.

• A. 4.45
• B. 3.08
• C. 3.56
• D. 4.06

Hint:

To calculate the standard deviation, first find the mean, then the squared differences from the mean, and finally compute the square root of the average of those squared differences.

Answer 1

The Correct Option is D

The formula for the standard deviation (S.D) of a set of values is: \[ S.D = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \] Where: - \( N \) is the number of values, - \( x_i \) are the individual values, - \( \mu \) is the mean of the values. ### Step 1: Find the Mean (\(\mu\)) The mean is given by the formula: \[ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i \] Substitute the given values: \[ \mu = \frac{-3 + 0 + 3 + 8}{4} = \frac{8}{4} = 2 \] ### Step 2: Calculate the squared differences from the mean Now, calculate the squared differences from the mean for each value: - For \( x_1 = -3 \): \( (-3 - 2)^2 = (-5)^2 = 25 \) - For \( x_2 = 0 \): \( (0 - 2)^2 = (-2)^2 = 4 \) - For \( x_3 = 3 \): \( (3 - 2)^2 = (1)^2 = 1 \) - For \( x_4 = 8 \): \( (8 - 2)^2 = (6)^2 = 36 \) ### Step 3: Calculate the variance The variance is the average of the squared differences: \[ \text{Variance} = \frac{25 + 4 + 1 + 36}{4} = \frac{66}{4} = 16.5 \] ### Step 4: Find the Standard Deviation Finally, take the square root of the variance: \[ S.D = \sqrt{16.5} \approx
4.06 \] Thus, the standard deviation is approximately \(
4.06 \).