Find the standard deviation for the group of data items. \[ 7,7,7,10,13,13,13 \]

To find the standard deviation of the given data set \( 7, 7, 7, 10, 13, 13, 13 \), we can follow these steps: 1. **Calculate the mean (average)** of the data set. 2. **Calculate the variance** by finding the average of the squared differences from the mean. 3. **Take the square root of the variance** to get the standard deviation. ### Step 1: Calculate the Mean The mean \( \mu \) is calculated as follows: \[ \mu = \frac{\text{sum of all data items}}{\text{number of data items}} = \frac{7 + 7 + 7 + 10 + 13 + 13 + 13}{7} \] Calculating the sum: \[ 7 + 7 + 7 + 10 + 13 + 13 + 13 = 70 \] Now, divide by the number of data items (which is 7): \[ \mu = \frac{70}{7} = 10 \] ### Step 2: Calculate the Variance Next, we calculate the variance \( \sigma^2 \) using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \] where \( x_i \) are the data items, \( \mu \) is the mean, and \( N \) is the number of data items. Now, we calculate each squared difference from the mean: - For \( 7 \): \( (7 - 10)^2 = (-3)^2 = 9 \) - For \( 7 \): \( (7 - 10)^2 = (-3)^2 = 9 \) - For \( 7 \): \( (7 - 10)^2 = (-3)^2 = 9 \) - For \( 10 \): \( (10 - 10)^2 = (0)^2 = 0 \) - For \( 13 \): \( (13 - 10)^2 = (3)^2 = 9 \) - For \( 13 \): \( (13 - 10)^2 = (3)^2 = 9 \) - For \( 13 \): \( (13 - 10)^2 = (3)^2 = 9 \) Now, sum these squared differences: \[ 9 + 9 + 9 + 0 + 9 + 9 + 9 = 54 \] Now, divide by the number of data items (7): \[ \sigma^2 = \frac{54}{7} \approx 7.714285714285714 \] ### Step 3: Calculate the Standard Deviation Finally, take the square root of the variance to find the standard deviation \( \sigma \): \[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{54}{7}} \approx \sqrt{7.714285714285714} \approx 2.776 \] Thus, the standard deviation of the data set \( 7, 7, 7, 10, 13, 13, 13 \) is approximately: \[ \boxed{2.776} \]