Questions: For the following set of data, find the sample standard deviation, to the nearest hundredth. 19,20,30,27,17,22,25,30

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: $$\sum x_i = 19 + 20 + 30 + 27 + 17 + 22 + 25 + 30 = 190$$ Thus, the mean is: $$\mu = \frac{190}{8} = 23.75$$

The variance $\sigma^2$ is calculated using the formula: $$\sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1}$$ First, we find $(x_i - \mu)^2$ for each data point:

• $(19 - 23.75)^2 = 22.5625$
• $(20 - 23.75)^2 = 14.0625$
• $(30 - 23.75)^2 = 38.0625$
• $(27 - 23.75)^2 = 10.5625$
• $(17 - 23.75)^2 = 45.5625$
• $(22 - 23.75)^2 = 3.0625$
• $(25 - 23.75)^2 = 1.5625$
• $(30 - 23.75)^2 = 38.0625$

Now, summing these values: $$\sum (x_i - \mu)^2 = 22.5625 + 14.0625 + 38.0625 + 10.5625 + 45.5625 + 3.0625 + 1.5625 + 38.0625 = 190.5$$ Now, we can calculate the variance: $$\sigma^2 = \frac{190.5}{8-1} = \frac{190.5}{7} \approx 25.07$$

The standard deviation $\sigma$ is the square root of the variance: $$\sigma = \sqrt{\sigma^2} = \sqrt{25.07} \approx 5.01$$

Thus, the sample standard deviation is $5.01$.

Final Answer