Statistics Examples

$2$ , $4$ , $6$ , $8$ , $10$ , $12$

Step 1

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.

$\sqrt{\frac{{(2)}^{2} + {(4)}^{2} + {(6)}^{2} + {(8)}^{2} + {(10)}^{2} + {(12)}^{2}}{6}}$

Step 2

Simplify the result.

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Step 2.1

Simplify the expression.

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Step 2.1.1

Raise $2$ to the power of $2$ .

$\sqrt{\frac{4 + {(4)}^{2} + {(6)}^{2} + {(8)}^{2} + {(10)}^{2} + {(12)}^{2}}{6}}$

Step 2.1.2

Raise $4$ to the power of $2$ .

$\sqrt{\frac{4 + 16 + {(6)}^{2} + {(8)}^{2} + {(10)}^{2} + {(12)}^{2}}{6}}$

Step 2.1.3

Raise $6$ to the power of $2$ .

$\sqrt{\frac{4 + 16 + 36 + {(8)}^{2} + {(10)}^{2} + {(12)}^{2}}{6}}$

Step 2.1.4

Raise $8$ to the power of $2$ .

$\sqrt{\frac{4 + 16 + 36 + 64 + {(10)}^{2} + {(12)}^{2}}{6}}$

Step 2.1.5

Raise $10$ to the power of $2$ .

$\sqrt{\frac{4 + 16 + 36 + 64 + 100 + {(12)}^{2}}{6}}$

Step 2.1.6

Raise $12$ to the power of $2$ .

$\sqrt{\frac{4 + 16 + 36 + 64 + 100 + 144}{6}}$

Step 2.1.7

Add $4$ and $16$ .

$\sqrt{\frac{20 + 36 + 64 + 100 + 144}{6}}$

Step 2.1.8

Add $20$ and $36$ .

$\sqrt{\frac{56 + 64 + 100 + 144}{6}}$

Step 2.1.9

Add $56$ and $64$ .

$\sqrt{\frac{120 + 100 + 144}{6}}$

Step 2.1.10

Add $120$ and $100$ .

$\sqrt{\frac{220 + 144}{6}}$

Step 2.1.11

Add $220$ and $144$ .

$\sqrt{\frac{364}{6}}$

Step 2.2

Cancel the common factor of $364$ and $6$ .

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Step 2.2.1

Factor $2$ out of $364$ .

$\sqrt{\frac{2 (182)}{6}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $2$ out of $6$ .

$\sqrt{\frac{2 \cdot 182}{2 \cdot 3}}$

Step 2.2.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 182}{2 \cdot 3}}$

Step 2.2.2.3

Rewrite the expression.

$\sqrt{\frac{182}{3}}$

Step 2.3

Rewrite $\sqrt{\frac{182}{3}}$ as $\frac{\sqrt{182}}{\sqrt{3}}$ .

$\frac{\sqrt{182}}{\sqrt{3}}$

Step 2.4

Multiply $\frac{\sqrt{182}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{182}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{\sqrt{182}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{182} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{182} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{182} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{182} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.5.5

Add $1$ and $1$ .

$\frac{\sqrt{182} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{182} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{182} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{182} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.6.4.1

Cancel the common factor.

$\frac{\sqrt{182} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{182} \sqrt{3}}{3^{1}}$

Step 2.5.6.5

Evaluate the exponent.

$\frac{\sqrt{182} \sqrt{3}}{3}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Combine using the product rule for radicals.

$\frac{\sqrt{182 \cdot 3}}{3}$

Step 2.6.2

Multiply $182$ by $3$ .

$\frac{\sqrt{546}}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{546}}{3}$

Decimal Form:

$7.78888096 \dots$