Questions: Find the standard deviation for the group of data items. 9,10,11 ; 12,13 The standard deviation is (Round to two decimal places as needed.)

Transcript text: Find the standard deviation for the group of data items. \[ 9,10,11 ; 12,13 \] The standard deviation is $\square$ (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the Mean

To calculate the mean ($\mu$), sum all the data items and divide by the number of items ($n=5$). \[\mu = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{9 + 10 + 11 + 12 + 13}{5} = 11\]

Step 2: Calculate the Population Variance

Since we're treating the data as a population, divide by $n$. \[\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n} = \frac{(9 - 11)^2 + (10 - 11)^2 + (11 - 11)^2 + (12 - 11)^2 + (13 - 11)^2}{5} = 2\]

Step 3: Calculate the Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance. \[\sigma = \sqrt{\sigma^2} = \sqrt{2} = 1.41\]