Question

What is the arithmetic mean of the following numbers? 1,2, 10, 6, 4,4,6,3,1,4

Answer 1

Answer:

The quadratic mean (rms) of a set of numbers is the square root of the sum of the squares of the numbers divided by the number of terms.











⎷

(

1

)

2

+

(

2

)

2

+

(

10

)

2

+

(

6

)

2

+

(

4

)

2

+

(

4

)

2

+

(

6

)

2

+

(

3

)

2

+

(

1

)

2

+

(

4

)

2

10

Step-by-step explanation:

One to any power is one.

√

1

+

(

2

)

2

+

(

10

)

2

+

(

6

)

2

+

(

4

)

2

+

(

4

)

2

+

(

6

)

2

+

(

3

)

2

+

(

1

)

2

+

(

4

)

2

10

Raise  

2

to the power of  

2

.

√

1

+

4

+

(

10

)

2

+

(

6

)

2

+

(

4

)

2

+

(

4

)

2

+

(

6

)

2

+

(

3

)

2

+

(

1

)

2

+

(

4

)

2

10

Raise  

10

to the power of  

2

.

√

1

+

4

+

100

+

(

6

)

2

+

(

4

)

2

+

(

4

)

2

+

(

6

)

2

+

(

3

)

2

+

(

1

)

2

+

(

4

)

2

10

Raise  

6

to the power of  

2

.

√

1

+

4

+

100

+

36

+

(

4

)

2

+

(

4

)

2

+

(

6

)

2

+

(

3

)

2

+

(

1

)

2

+

(

4

)

2

10

Raise  

4

to the power of  

2

.

√

1

+

4

+

100

+

36

+

16

+

(

4

)

2

+

(

6

)

2

+

(

3

)

2

+

(

1

)

2

+

(

4

)

2

10

Raise  

4

to the power of  

2

.

√

1

+

4

+

100

+

36

+

16

+

16

+

(

6

)

2

+

(

3

)

2

+

(

1

)

2

+

(

4

)

2

10

Raise  

6

to the power of  

2

.

√

1

+

4

+

100

+

36

+

16

+

16

+

36

+

(

3

)

2

+

(

1

)

2

+

(

4

)

2

10

Raise  

3

to the power of  

2

.

√

1

+

4

+

100

+

36

+

16

+

16

+

36

+

9

+

(

1

)

2

+

(

4

)

2

10

One to any power is one.

√

1

+

4

+

100

+

36

+

16

+

16

+

36

+

9

+

1

+

(

4

)

2

10

Raise  

4

to the power of  

2

.

√

1

+

4

+

100

+

36

+

16

+

16

+

36

+

9

+

1

+

16

10

Add  

1

and  

4

.

√

5

+

100

+

36

+

16

+

16

+

36

+

9

+

1

+

16

10

Add  

5

and  

100

.

√

105

+

36

+

16

+

16

+

36

+

9

+

1

+

16

10

Add  

105

and  

36

.

√

141

+

16

+

16

+

36

+

9

+

1

+

16

10

Add  

141

and  

16

.

√

157

+

16

+

36

+

9

+

1

+

16

10

Add  

157

and  

16

.

√

173

+

36

+

9

+

1

+

16

10

Add  

173

and  

36

.

√

209

+

9

+

1

+

16

10

Add  

209

and  

9

.

√

218

+

1

+

16

10

Add  

218

and  

1

.

√

219

+

16

10

Add  

219

and  

16

.

√

235

10

Cancel the common factor of  

235

and  

10

.

Tap for fewer steps...

Factor  

5

out of  

235

.

√

5

(

47

)

10

Cancel the common factors.

Tap for fewer steps...

Factor  

5

out of  

10

.

√

5

⋅

47

5

⋅

2

Cancel the common factor.

√

5

⋅

47

5

⋅

2

Rewrite the expression.

√

47

2

Rewrite  

√

47

2

as  

√

47

√

2

.

√

47

√

2

Multiply  

√

47

√

2

by  

√

2

√

2

.

√

47

√

2

⋅

√

2

√

2

Combine and simplify the denominator.

Tap for fewer steps...

Multiply  

√

47

√

2

and  

√

2

√

2

.

√

47

√

2

√

2

√

2

Raise  

√

2

to the power of  

1

.

√

47

√

2

√

2

1

√

2

Raise  

√

2

to the power of  

1

.

√

47

√

2

√

2

1

√

2

1

Use the power rule  

a

m

a

n

=

a

m

+

n

to combine exponents.

√

47

√

2

√

2

1

+

1

Add  

1

and  

1

.

√

47

√

2

√

2

2

Rewrite  

√

2

2

as  

2

.

Tap for fewer steps...

Use  

n

√

a

x

=

a

x

n

to rewrite  

√

2

as  

2

1

2

.

√

47

√

2

(

2

1

2

)

2

Apply the power rule and multiply exponents,  

(

a

m

)

n

=

a

m

n

.

√

47

√

2

2

1

2

⋅

2

Combine  

1

2

and  

2

.

√

47

√

2

2

2

2

Cancel the common factor of  

2

.

Tap for more steps...

√

47

√

2

2

1

Evaluate the exponent.

√

47

√

2

2

Simplify the numerator.

Tap for fewer steps...

Combine using the product rule for radicals.

√

47

⋅

2

2

Multiply  

47

by  

2

.

√

94

2

The result can be shown in multiple forms.

Exact Form:

√

94

2

Decimal Form:

4.84767985

…