Question

eight adults are surveyed and asked how many credit cards they possess. their responses are 4, 0, 3, 1, 5, 2, 2, and 3. complete the table below.

$x$	$\bar{x}$	$x - \bar{x}$	$(x - \bar{x})^2$
0	2.5
3	2.5
1	2.5
5	2.5
2	2.5
2	2.5
3	2.5

using the formula for standard deviation
$sigma = \sqrt{\frac{\sum \left(x - \bar{x}\
ight)^2}{n}}$

Explanation:

Step1: Calculate \( x - \bar{x} \) and \( (x - \bar{x})^2 \) for each \( x \)

For \( x = 4 \):
\( x - \bar{x} = 4 - 2.5 = 1.5 \)
\( (x - \bar{x})^2 = (1.5)^2 = 2.25 \)

For \( x = 0 \):
\( x - \bar{x} = 0 - 2.5 = -2.5 \)
\( (x - \bar{x})^2 = (-2.5)^2 = 6.25 \)

For \( x = 3 \):
\( x - \bar{x} = 3 - 2.5 = 0.5 \)
\( (x - \bar{x})^2 = (0.5)^2 = 0.25 \)

For \( x = 1 \):
\( x - \bar{x} = 1 - 2.5 = -1.5 \)
\( (x - \bar{x})^2 = (-1.5)^2 = 2.25 \)

For \( x = 5 \):
\( x - \bar{x} = 5 - 2.5 = 2.5 \)
\( (x - \bar{x})^2 = (2.5)^2 = 6.25 \)

For \( x = 2 \):
\( x - \bar{x} = 2 - 2.5 = -0.5 \)
\( (x - \bar{x})^2 = (-0.5)^2 = 0.25 \)

For \( x = 2 \):
\( x - \bar{x} = 2 - 2.5 = -0.5 \)
\( (x - \bar{x})^2 = (-0.5)^2 = 0.25 \)

For \( x = 3 \):
\( x - \bar{x} = 3 - 2.5 = 0.5 \)
\( (x - \bar{x})^2 = (0.5)^2 = 0.25 \)

Step2: Sum the \( (x - \bar{x})^2 \) values

\( \sum (x - \bar{x})^2 = 2.25 + 6.25 + 0.25 + 2.25 + 6.25 + 0.25 + 0.25 + 0.25 \)
\( = (2.25 + 2.25) + (6.25 + 6.25) + (0.25 + 0.25 + 0.25 + 0.25) \)
\( = 4.5 + 12.5 + 1 \)
\( = 18 \)

Step3: Calculate the standard deviation \( \sigma \)

Given \( n = 8 \),
\( \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}} = \sqrt{\frac{18}{8}} = \sqrt{2.25} = 1.5 \)

Answer:

The completed table is:

\( x \)	\( \bar{x} \)	\( x - \bar{x} \)	\( (x - \bar{x})^2 \)
0	2.5	-2.5	6.25
3	2.5	0.5	0.25
1	2.5	-1.5	2.25
5	2.5	2.5	6.25
2	2.5	-0.5	0.25
2	2.5	-0.5	0.25
3	2.5	0.5	0.25

The standard deviation \( \sigma = 1.5 \)