What is the best approximation of the standard deviation of the measures \( -4,-3,0,8,9 \) ? 2 5 1 10

To find the best approximation of the standard deviation of the measures \( -4, -3, 0, 8, 9 \), we will follow these steps: 1. **Calculate the mean (\( \mu \))** of the data set. 2. **Calculate the variance (\( \sigma^2 \))** using the formula: \[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \] where \( N \) is the number of data points and \( x_i \) are the data points. 3. **Take the square root of the variance** to find the standard deviation (\( \sigma \)). Let's perform these calculations step by step. ### Step 1: Calculate the Mean The mean is calculated as follows: \[ \mu = \frac{-4 + (-3) + 0 + 8 + 9}{5} \] ### Step 2: Calculate the Variance Next, we will calculate the variance using the mean we just found. ### Step 3: Calculate the Standard Deviation Finally, we will take the square root of the variance to find the standard deviation. Let's perform these calculations. 1. Calculate the mean: \[ \mu = \frac{-4 - 3 + 0 + 8 + 9}{5} = \frac{10}{5} = 2 \] 2. Calculate the variance: \[ \sigma^2 = \frac{1}{5} \left[ (-4 - 2)^2 + (-3 - 2)^2 + (0 - 2)^2 + (8 - 2)^2 + (9 - 2)^2 \right] \] \[ = \frac{1}{5} \left[ (-6)^2 + (-5)^2 + (-2)^2 + (6)^2 + (7)^2 \right] \] \[ = \frac{1}{5} \left[ 36 + 25 + 4 + 36 + 49 \right] \] \[ = \frac{1}{5} \left[ 150 \right] = 30 \] 3. Calculate the standard deviation: \[ \sigma = \sqrt{30} \] Now, let's calculate \( \sqrt{30} \) to find the standard deviation. Calculating \( \sqrt{30} \) gives approximately \( 5.477 \). Now, we can compare this value with the options provided: 2, 5, 1, 10. The best approximation of the standard deviation from the given options is **5**.