Math / Data & Statistics

QuestionA group of data items and their mean are given. $21,35,49,84,126,189 ; \text { Mean }=84$ a. Find the deviation from the mean for each of the data items. b. Find the sum of the deviations in part (a). a. Type the deviation from the mean for each of the data items. \begin{tabular}{|c|c|c|c|c|c|} \hline 21 & 35 & 49 & 84 & 126 & 189 \\ \hline & & & & & \\ \hline \end{tabular}

Studdy Solution

STEP 1

What is this asking? We're given a set of numbers and their average, and we need to find how far each number is from the average, and then add up all those distances! Watch out! Remember that deviations can be positive or negative!
Don't mix up the mean with other statistical measures like median or mode.

STEP 2

1. Calculate Deviations
2. Sum the Deviations

STEP 3

Alright, let's calculate the deviation for each data point!
Remember, the deviation is just the difference between the data point and the mean, which is 84.

STEP 4

For $\$21$ , the deviation is $21 - 84 = -63$ .
It's negative because $\$21$ is smaller than the mean.

STEP 5

For $\$35$ , the deviation is $35 - 84 = -49$ .
Still negative!

STEP 6

For $\$49$ , the deviation is $49 - 84 = -35$ .
Getting closer to zero!

STEP 7

For $\$84$ , the deviation is $84 - 84 = 0$ .
Makes sense, since it's the mean itself!

STEP 8

For $\$126$ , the deviation is $126 - 84 = 42$ .
Now we're positive!

STEP 9

For $\$189$ , the deviation is $189 - 84 = 105$ .
Way up there!

STEP 10

Now, let's add up all those deviations: $-63 + (-49) + (-35) + 0 + 42 + 105$ .

STEP 11

Combining the negative deviations, we get $-63 - 49 - 35 = -147$ .

STEP 12

Adding the positive deviations, we get $42 + 105 = 147$ .

STEP 13

Finally, $-147 + 147 = 0$ .
The sum of the deviations is zero!

STEP 14

a. The deviations are: $-63, -49, -35, 0, 42, 105$ . b. The sum of the deviations is $\$0$ .

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Studdy Solution

STEP 1

STEP 2

1. Calculate Deviations
2. Sum the Deviations

STEP 3

Alright, let's calculate the deviation for each data point!
Remember, the deviation is just the difference between the data point and the mean, which is 84.

STEP 4

For $\$21$ , the deviation is $21 - 84 = -63$ .
It's negative because $\$21$ is smaller than the mean.

STEP 5

For $\$35$ , the deviation is $35 - 84 = -49$ .
Still negative!

STEP 6

For $\$49$ , the deviation is $49 - 84 = -35$ .
Getting closer to zero!

STEP 7

For $\$84$ , the deviation is $84 - 84 = 0$ .
Makes sense, since it's the mean itself!

STEP 8

For $\$126$ , the deviation is $126 - 84 = 42$ .
Now we're positive!

STEP 9

For $\$189$ , the deviation is $189 - 84 = 105$ .
Way up there!

STEP 10

Now, let's add up all those deviations: $-63 + (-49) + (-35) + 0 + 42 + 105$ .

STEP 11

Combining the negative deviations, we get $-63 - 49 - 35 = -147$ .

STEP 12

Adding the positive deviations, we get $42 + 105 = 147$ .

STEP 13

Finally, $-147 + 147 = 0$ .
The sum of the deviations is zero!

STEP 14

a. The deviations are: $-63, -49, -35, 0, 42, 105$ . b. The sum of the deviations is $\$0$ .

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Studdy Solution

STEP 1

STEP 2

1. Calculate Deviations2. Sum the Deviations

STEP 3

Alright, let's **calculate the deviation** for each data point!Remember, the deviation is just the difference between the data point and the **mean**, which is **84**.

STEP 4

For $21\$21$21, the deviation is 21−84=−6321 - 84 = -6321−84=−63.It's negative because $21\$21$21 is smaller than the **mean**.

STEP 5

For $35\$35$35, the deviation is 35−84=−4935 - 84 = -4935−84=−49.Still negative!

STEP 6

For $49\$49$49, the deviation is 49−84=−3549 - 84 = -3549−84=−35.Getting closer to zero!

STEP 7

For $84\$84$84, the deviation is 84−84=084 - 84 = 084−84=0.Makes sense, since it's the **mean** itself!

STEP 8

For $126\$126$126, the deviation is 126−84=42126 - 84 = 42126−84=42.Now we're positive!

STEP 9

For $189\$189$189, the deviation is 189−84=105189 - 84 = 105189−84=105.Way up there!

STEP 10

Now, let's **add up all those deviations**: −63+(−49)+(−35)+0+42+105-63 + (-49) + (-35) + 0 + 42 + 105−63+(−49)+(−35)+0+42+105.

STEP 11

Combining the negative deviations, we get −63−49−35=−147-63 - 49 - 35 = -147−63−49−35=−147.

STEP 12

Adding the positive deviations, we get 42+105=14742 + 105 = 14742+105=147.

STEP 13

Finally, −147+147=0-147 + 147 = 0−147+147=0.The sum of the deviations is **zero**!

STEP 14

a. The deviations are: −63,−49,−35,0,42,105-63, -49, -35, 0, 42, 105−63,−49,−35,0,42,105. b. The sum of the deviations is $0\$0$0.

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1. Calculate Deviations
2. Sum the Deviations

Alright, let's calculate the deviation for each data point!
Remember, the deviation is just the difference between the data point and the mean, which is 84.

For $\$21$ , the deviation is $21 - 84 = -63$ .
It's negative because $\$21$ is smaller than the mean.

For $\$35$ , the deviation is $35 - 84 = -49$ .
Still negative!

For $\$49$ , the deviation is $49 - 84 = -35$ .
Getting closer to zero!

For $\$84$ , the deviation is $84 - 84 = 0$ .
Makes sense, since it's the mean itself!

For $\$126$ , the deviation is $126 - 84 = 42$ .
Now we're positive!

For $\$189$ , the deviation is $189 - 84 = 105$ .
Way up there!

Now, let's add up all those deviations: $-63 + (-49) + (-35) + 0 + 42 + 105$ .

Combining the negative deviations, we get $-63 - 49 - 35 = -147$ .

Adding the positive deviations, we get $42 + 105 = 147$ .

Finally, $-147 + 147 = 0$ .
The sum of the deviations is zero!

a. The deviations are: $-63, -49, -35, 0, 42, 105$ . b. The sum of the deviations is $\$0$ .