Let's go through the process of calculating the sample variance and standard deviation by hand for the given sample:

$6, 52, 13, 49, 35, 25, 31, 29, 31, 29$

Step 1: Find the Sample Mean

The first step is to calculate the mean of the sample. The formula for the sample mean ($\bar{x}$) is:

$\bar{x} = \frac{\sum x_i}{n}$

where $x_i$ are the sample values and $n$ is the sample size.

Here, $n = 10$, and the sum of the sample values is:

$\sum x_i = 6 + 52 + 13 + 49 + 35 + 25 + 31 + 29 + 31 + 29 = 300$

Now, calculate the sample mean:

$\bar{x} = \frac{300}{10} = 30$

Step 2: Calculate Each Deviation from the Mean

Next, subtract the sample mean from each data point to find the deviations:

$x_1 - \bar{x} = 6 - 30 = -24$ $x_2 - \bar{x} = 52 - 30 = 22$ $x_3 - \bar{x} = 13 - 30 = -17$ $x_4 - \bar{x} = 49 - 30 = 19$ $x_5 - \bar{x} = 35 - 30 = 5$ [ x_6 - \bar{x} = 25 -