Questions: Look at the following data and complete the table to find the sample standard deviation 17,8,12,26,17,29,10

Solution Steps

To find the sample standard deviation, we need to follow these steps:

1. Calculate the mean (average) of the data set.
2. Subtract the mean from each data point and square the result.
3. Sum all the squared results.
4. Divide this sum by the number of data points minus one (n-1) to get the variance.
5. Take the square root of the variance to get the sample standard deviation.

The mean (\(\mu\)) of the data set is calculated as follows: \[ \mu = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{17 + 8 + 12 + 26 + 17 + 29 + 10}{7} = 17.0 \]

Subtract the mean from each data point and square the result: \[ \begin{align_} (17 - 17.0)^2 &= 0.0 \\ (8 - 17.0)^2 &= 81.0 \\ (12 - 17.0)^2 &= 25.0 \\ (26 - 17.0)^2 &= 81.0 \\ (17 - 17.0)^2 &= 0.0 \\ (29 - 17.0)^2 &= 144.0 \\ (10 - 17.0)^2 &= 49.0 \\ \end{align_} \]

Sum all the squared differences: \[ \sum (x_i - \mu)^2 = 0.0 + 81.0 + 25.0 + 81.0 + 0.0 + 144.0 + 49.0 = 380.0 \]

Divide the sum of squared differences by \(n-1\) (where \(n\) is the number of data points): \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = \frac{380.0}{7-1} = 63.3333 \]

Take the square root of the variance to get the sample standard deviation: \[ \sigma = \sqrt{\sigma^2} = \sqrt{63.3333} \approx 7.958 \]

Final Answer