Question

1. fill in the table below: fill in the differences of each data value from the mean, then the squared differences.

$x$	$x - \bar{x}$	$(x - \bar{x})^2$
11
14
6
8
	$\sum (x - \bar{x})^2 =$

1. calculate the sample standard deviation ($s$).

$s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} = $ (please round your answer to two decimal places)

note: $n$ is the sample size or the number of data value in the sample.

calculate the standard deviation and variance of the sample quantitative data shown, to two decimal places.

Explanation:

Part 2: Filling the Table

First, we need to find the mean ($\bar{x}$) of the data set \( \{2, 11, 14, 6, 8\} \).

Step 1: Calculate the mean ($\bar{x}$)

The formula for the mean is \( \bar{x} = \frac{\sum x}{n} \), where \( n \) is the number of data points.
\[
\sum x = 2 + 11 + 14 + 6 + 8 = 41
\]
\[
n = 5
\]
\[
\bar{x} = \frac{41}{5} = 8.2
\]

Step 2: Calculate \( x - \bar{x} \) and \( (x - \bar{x})^2 \) for each data point

• For \( x = 2 \):
• \( x - \bar{x} = 2 - 8.2 = -6.2 \)
• \( (x - \bar{x})^2 = (-6.2)^2 = 38.44 \)
• For \( x = 11 \):
• \( x - \bar{x} = 11 - 8.2 = 2.8 \)
• \( (x - \bar{x})^2 = (2.8)^2 = 7.84 \)
• For \( x = 14 \):
• \( x - \bar{x} = 14 - 8.2 = 5.8 \)
• \( (x - \bar{x})^2 = (5.8)^2 = 33.64 \)
• For \( x = 6 \):
• \( x - \bar{x} = 6 - 8.2 = -2.2 \)
• \( (x - \bar{x})^2 = (-2.2)^2 = 4.84 \)
• For \( x = 8 \):
• \( x - \bar{x} = 8 - 8.2 = -0.2 \)
• \( (x - \bar{x})^2 = (-0.2)^2 = 0.04 \)

Step 3: Calculate \( \sum (x - \bar{x})^2 \)

\[
\sum (x - \bar{x})^2 = 38.44 + 7.84 + 33.64 + 4.84 + 0.04 = 84.8
\]

Filled Table:

\( x \)	\( x - \bar{x} \)	\( (x - \bar{x})^2 \)
11	2.8	7.84
14	5.8	33.64
6	-2.2	4.84
8	-0.2	0.04
		\( \sum (x - \bar{x})^2 = 84.8 \)

Part 3: Calculate the sample standard deviation (\( s \))

The formula for the sample standard deviation is \( s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} \)

Step 1: Identify \( \sum (x - \bar{x})^2 \) and \( n \)

We know \( \sum (x - \bar{x})^2 = 84.8 \) and \( n = 5 \), so \( n - 1 = 4 \).

Step 2: Substitute into the formula

\[
s = \sqrt{\frac{84.8}{4}} = \sqrt{21.2} \approx 4.60
\]

Part 2 Table Answers:

\( x \)	\( x - \bar{x} \)	\( (x - \bar{x})^2 \)
11	2.8	7.84
14	5.8	33.64
6	-2.2	4.84
8	-0.2	0.04
		\( \sum (x - \bar{x})^2 = 84.8 \)

Part 3 Answer:

The sample standard deviation \( s \approx \boxed{4.60} \)

Answer:

Part 2: Filling the Table

First, we need to find the mean ($\bar{x}$) of the data set \( \{2, 11, 14, 6, 8\} \).

Step 1: Calculate the mean ($\bar{x}$)

The formula for the mean is \( \bar{x} = \frac{\sum x}{n} \), where \( n \) is the number of data points.
\[
\sum x = 2 + 11 + 14 + 6 + 8 = 41
\]
\[
n = 5
\]
\[
\bar{x} = \frac{41}{5} = 8.2
\]

Step 2: Calculate \( x - \bar{x} \) and \( (x - \bar{x})^2 \) for each data point

• For \( x = 2 \):
• \( x - \bar{x} = 2 - 8.2 = -6.2 \)
• \( (x - \bar{x})^2 = (-6.2)^2 = 38.44 \)
• For \( x = 11 \):
• \( x - \bar{x} = 11 - 8.2 = 2.8 \)
• \( (x - \bar{x})^2 = (2.8)^2 = 7.84 \)
• For \( x = 14 \):
• \( x - \bar{x} = 14 - 8.2 = 5.8 \)
• \( (x - \bar{x})^2 = (5.8)^2 = 33.64 \)
• For \( x = 6 \):
• \( x - \bar{x} = 6 - 8.2 = -2.2 \)
• \( (x - \bar{x})^2 = (-2.2)^2 = 4.84 \)
• For \( x = 8 \):
• \( x - \bar{x} = 8 - 8.2 = -0.2 \)
• \( (x - \bar{x})^2 = (-0.2)^2 = 0.04 \)

Step 3: Calculate \( \sum (x - \bar{x})^2 \)

\[
\sum (x - \bar{x})^2 = 38.44 + 7.84 + 33.64 + 4.84 + 0.04 = 84.8
\]

Filled Table:

\( x \)	\( x - \bar{x} \)	\( (x - \bar{x})^2 \)
11	2.8	7.84
14	5.8	33.64
6	-2.2	4.84
8	-0.2	0.04
		\( \sum (x - \bar{x})^2 = 84.8 \)

Part 3: Calculate the sample standard deviation (\( s \))

The formula for the sample standard deviation is \( s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} \)

Step 1: Identify \( \sum (x - \bar{x})^2 \) and \( n \)

We know \( \sum (x - \bar{x})^2 = 84.8 \) and \( n = 5 \), so \( n - 1 = 4 \).

Step 2: Substitute into the formula

\[
s = \sqrt{\frac{84.8}{4}} = \sqrt{21.2} \approx 4.60
\]

Part 2 Table Answers:

\( x \)	\( x - \bar{x} \)	\( (x - \bar{x})^2 \)
11	2.8	7.84
14	5.8	33.64
6	-2.2	4.84
8	-0.2	0.04
		\( \sum (x - \bar{x})^2 = 84.8 \)

Part 3 Answer:

The sample standard deviation \( s \approx \boxed{4.60} \)