Question

find a. the mean; b. the deviation from the mean for each data item, and c. the sum of the deviations in part (b) for the following group of data items
28, 78, 53, 28, 103

a. the mean is 58. (simplify your answer.)
b. complete the following table by entering the deviations for each data value
(simplify your answers.)

Explanation:

Part b: Calculating Deviations from the Mean

To find the deviation from the mean for each data value, we use the formula:
Deviation = Data Value - Mean

Given the mean is \( 58 \), we calculate the deviation for each data point:

Step 1: Deviation for 28

Deviation = \( 28 - 58 = -30 \)

Step 2: Deviation for 78

Deviation = \( 78 - 58 = 20 \)

Step 3: Deviation for 53

Deviation = \( 53 - 58 = -5 \)

Step 4: Deviation for 28

Deviation = \( 28 - 58 = -30 \)

Step 5: Deviation for 103

Deviation = \( 103 - 58 = 45 \)

Part c: Sum of Deviations

To find the sum of the deviations, we add all the deviations calculated in part (b):

Step 1: List the deviations

\( -30, 20, -5, -30, 45 \)

Step 2: Sum the deviations

\( (-30) + 20 + (-5) + (-30) + 45 \)
\( = (-30 - 5 - 30) + (20 + 45) \)
\( = (-65) + 65 \)
\( = 0 \)

Final Answers

Part b (Table):

Data Value	Deviation
78	\( 20 \)
53	\( -5 \)
28	\( -30 \)
103	\( 45 \)

Part c:

The sum of the deviations is \( \boldsymbol{0} \).

Answer:

Part b: Calculating Deviations from the Mean

To find the deviation from the mean for each data value, we use the formula:
Deviation = Data Value - Mean

Given the mean is \( 58 \), we calculate the deviation for each data point:

Step 1: Deviation for 28

Deviation = \( 28 - 58 = -30 \)

Step 2: Deviation for 78

Deviation = \( 78 - 58 = 20 \)

Step 3: Deviation for 53

Deviation = \( 53 - 58 = -5 \)

Step 4: Deviation for 28

Deviation = \( 28 - 58 = -30 \)

Step 5: Deviation for 103

Deviation = \( 103 - 58 = 45 \)

Part c: Sum of Deviations

To find the sum of the deviations, we add all the deviations calculated in part (b):

Step 1: List the deviations

\( -30, 20, -5, -30, 45 \)

Step 2: Sum the deviations

\( (-30) + 20 + (-5) + (-30) + 45 \)
\( = (-30 - 5 - 30) + (20 + 45) \)
\( = (-65) + 65 \)
\( = 0 \)

Final Answers

Part b (Table):

Data Value	Deviation
78	\( 20 \)
53	\( -5 \)
28	\( -30 \)
103	\( 45 \)

Part c:

The sum of the deviations is \( \boldsymbol{0} \).