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} b. The sum of the deviations in part (a) is $\square$

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 the Deviations
2. Sum the Deviations

STEP 3

Alright, let's calculate the deviation for each data item.
Remember, the deviation is just the difference between the data item and the mean.
Our mean is $\text{Mean} = 84$ .

STEP 4

For $\text{Data Item} = 21$ , the deviation is $21 - 84 = \boldsymbol{-63}$ .
It's negative because $21$ is smaller than the mean.

STEP 5

For $\text{Data Item} = 35$ , the deviation is $35 - 84 = \boldsymbol{-49}$ .
Still negative!

STEP 6

For $\text{Data Item} = 49$ , the deviation is $49 - 84 = \boldsymbol{-35}$ .
Getting closer to zero!

STEP 7

For $\text{Data Item} = 84$ , the deviation is $84 - 84 = \boldsymbol{0}$ .
Makes sense, since $84$ is the mean!

STEP 8

For $\text{Data Item} = 126$ , the deviation is $126 - 84 = \boldsymbol{42}$ .
Now we're positive!

STEP 9

For $\text{Data Item} = 189$ , the deviation is $189 - 84 = \boldsymbol{105}$ .
Way bigger than the mean!

STEP 10

Now, let's add up all those deviations we just calculated.
This will tell us how spread out the data is overall.

STEP 11

$(-63) + (-49) + (-35) + 0 + 42 + 105$ .
Remember, adding a negative number is the same as subtracting its positive counterpart.

STEP 12

Let's group the negative numbers and the positive numbers: $(-63 - 49 - 35) + (42 + 105)$ .
This makes it easier to see what's going on.

STEP 13

$(-147) + 147 = 0$ .
Wow! 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 the Deviations
2. Sum the Deviations

STEP 3

Alright, let's calculate the deviation for each data item.
Remember, the deviation is just the difference between the data item and the mean.
Our mean is $\text{Mean} = 84$ .

STEP 4

For $\text{Data Item} = 21$ , the deviation is $21 - 84 = \boldsymbol{-63}$ .
It's negative because $21$ is smaller than the mean.

STEP 5

For $\text{Data Item} = 35$ , the deviation is $35 - 84 = \boldsymbol{-49}$ .
Still negative!

STEP 6

For $\text{Data Item} = 49$ , the deviation is $49 - 84 = \boldsymbol{-35}$ .
Getting closer to zero!

STEP 7

For $\text{Data Item} = 84$ , the deviation is $84 - 84 = \boldsymbol{0}$ .
Makes sense, since $84$ is the mean!

STEP 8

For $\text{Data Item} = 126$ , the deviation is $126 - 84 = \boldsymbol{42}$ .
Now we're positive!

STEP 9

For $\text{Data Item} = 189$ , the deviation is $189 - 84 = \boldsymbol{105}$ .
Way bigger than the mean!

STEP 10

Now, let's add up all those deviations we just calculated.
This will tell us how spread out the data is overall.

STEP 11

$(-63) + (-49) + (-35) + 0 + 42 + 105$ .
Remember, adding a negative number is the same as subtracting its positive counterpart.

STEP 12

Let's group the negative numbers and the positive numbers: $(-63 - 49 - 35) + (42 + 105)$ .
This makes it easier to see what's going on.

STEP 13

$(-147) + 147 = 0$ .
Wow! 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 the Deviations2. Sum the Deviations

STEP 3

Alright, let's **calculate the deviation** for each data item.Remember, the deviation is just the difference between the data item and the mean.Our **mean** is Mean=84\text{Mean} = 84Mean=84.

STEP 4

For Data Item=21\text{Data Item} = 21Data Item=21, the deviation is 21−84=−6321 - 84 = \boldsymbol{-63}21−84=−63.It's negative because 212121 is smaller than the mean.

STEP 5

For Data Item=35\text{Data Item} = 35Data Item=35, the deviation is 35−84=−4935 - 84 = \boldsymbol{-49}35−84=−49.Still negative!

STEP 6

For Data Item=49\text{Data Item} = 49Data Item=49, the deviation is 49−84=−3549 - 84 = \boldsymbol{-35}49−84=−35.Getting closer to zero!

STEP 7

For Data Item=84\text{Data Item} = 84Data Item=84, the deviation is 84−84=084 - 84 = \boldsymbol{0}84−84=0.Makes sense, since 848484 *is* the mean!

STEP 8

For Data Item=126\text{Data Item} = 126Data Item=126, the deviation is 126−84=42126 - 84 = \boldsymbol{42}126−84=42.Now we're positive!

STEP 9

For Data Item=189\text{Data Item} = 189Data Item=189, the deviation is 189−84=105189 - 84 = \boldsymbol{105}189−84=105.Way bigger than the mean!

STEP 10

Now, let's **add up all those deviations** we just calculated.This will tell us how spread out the data is overall.

STEP 11

(−63)+(−49)+(−35)+0+42+105(-63) + (-49) + (-35) + 0 + 42 + 105(−63)+(−49)+(−35)+0+42+105.Remember, adding a negative number is the same as subtracting its positive counterpart.

STEP 12

Let's group the negative numbers and the positive numbers: (−63−49−35)+(42+105)(-63 - 49 - 35) + (42 + 105)(−63−49−35)+(42+105).This makes it easier to see what's going on.

STEP 13

(−147)+147=0(-147) + 147 = 0(−147)+147=0.Wow! 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 000.

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1. Calculate the Deviations
2. Sum the Deviations

Alright, let's calculate the deviation for each data item.
Remember, the deviation is just the difference between the data item and the mean.
Our mean is $\text{Mean} = 84$ .

For $\text{Data Item} = 21$ , the deviation is $21 - 84 = \boldsymbol{-63}$ .
It's negative because $21$ is smaller than the mean.

For $\text{Data Item} = 35$ , the deviation is $35 - 84 = \boldsymbol{-49}$ .
Still negative!

For $\text{Data Item} = 49$ , the deviation is $49 - 84 = \boldsymbol{-35}$ .
Getting closer to zero!

For $\text{Data Item} = 84$ , the deviation is $84 - 84 = \boldsymbol{0}$ .
Makes sense, since $84$ is the mean!

For $\text{Data Item} = 126$ , the deviation is $126 - 84 = \boldsymbol{42}$ .
Now we're positive!

For $\text{Data Item} = 189$ , the deviation is $189 - 84 = \boldsymbol{105}$ .
Way bigger than the mean!

Now, let's add up all those deviations we just calculated.
This will tell us how spread out the data is overall.

$(-63) + (-49) + (-35) + 0 + 42 + 105$ .
Remember, adding a negative number is the same as subtracting its positive counterpart.

Let's group the negative numbers and the positive numbers: $(-63 - 49 - 35) + (42 + 105)$ .
This makes it easier to see what's going on.

$(-147) + 147 = 0$ .
Wow! The sum of the deviations is zero!

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