Questions: Find the standard deviation for the group of data items.

Solution Steps

To find the mean \( \mu \) of the dataset, we use the formula: \[ \mu = \frac{\sum x_i}{n} \] where \( \sum x_i \) is the sum of all data points and \( n \) is the number of data points. For our dataset: \[ \mu = \frac{38 + 45 + 48 + 55 + 58}{5} = \frac{244}{5} = 48.8 \]

The variance \( \sigma^2 \) is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \] We first find \( (x_i - \mu)^2 \) for each data point:

• For \( x_1 = 38 \): \( (38 - 48.8)^2 = 116.64 \)
• For \( x_2 = 45 \): \( (45 - 48.8)^2 = 14.44 \)
• For \( x_3 = 48 \): \( (48 - 48.8)^2 = 0.64 \)
• For \( x_4 = 55 \): \( (55 - 48.8)^2 = 38.44 \)
• For \( x_5 = 58 \): \( (58 - 48.8)^2 = 83.64 \)

Now, summing these squared differences: \[ \sum (x_i - \mu)^2 = 116.64 + 14.44 + 0.64 + 38.44 + 83.64 = 253.8 \]

Now, we can calculate the variance: \[ \sigma^2 = \frac{253.8}{5} = 50.96 \]

The standard deviation \( \sigma \) is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} = \sqrt{50.96} \] Thus, the standard deviation is: \[ \sigma \approx 7.14 \]

Final Answer