Statistics Examples

$1$ , $2$ , $4$ , $7$ , $9$ , $10$

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.

$\sqrt{\frac{{(1)}^{2} + {(2)}^{2} + {(4)}^{2} + {(7)}^{2} + {(9)}^{2} + {(10)}^{2}}{6}}$

Step 2

Simplify the result.

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Step 2.1

Simplify the expression.

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Step 2.1.1

One to any power is one.

$\sqrt{\frac{1 + {(2)}^{2} + {(4)}^{2} + {(7)}^{2} + {(9)}^{2} + {(10)}^{2}}{6}}$

Step 2.1.2

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

$\sqrt{\frac{1 + 4 + {(4)}^{2} + {(7)}^{2} + {(9)}^{2} + {(10)}^{2}}{6}}$

Step 2.1.3

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

$\sqrt{\frac{1 + 4 + 16 + {(7)}^{2} + {(9)}^{2} + {(10)}^{2}}{6}}$

Step 2.1.4

Raise $7$ to the power of $2$ .

$\sqrt{\frac{1 + 4 + 16 + 49 + {(9)}^{2} + {(10)}^{2}}{6}}$

Step 2.1.5

Raise $9$ to the power of $2$ .

$\sqrt{\frac{1 + 4 + 16 + 49 + 81 + {(10)}^{2}}{6}}$

Step 2.1.6

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

$\sqrt{\frac{1 + 4 + 16 + 49 + 81 + 100}{6}}$

Step 2.1.7

Add $1$ and $4$ .

$\sqrt{\frac{5 + 16 + 49 + 81 + 100}{6}}$

Step 2.1.8

Add $5$ and $16$ .

$\sqrt{\frac{21 + 49 + 81 + 100}{6}}$

Step 2.1.9

Add $21$ and $49$ .

$\sqrt{\frac{70 + 81 + 100}{6}}$

Step 2.1.10

Add $70$ and $81$ .

$\sqrt{\frac{151 + 100}{6}}$

Step 2.1.11

Add $151$ and $100$ .

$\sqrt{\frac{251}{6}}$

Step 2.2

Cancel the common factor of $251$ and $6$ .

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Step 2.2.1

Rewrite $251$ as $1 (251)$ .

$\sqrt{\frac{1 (251)}{6}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Rewrite $6$ as $1 (6)$ .

$\sqrt{\frac{1 \cdot 251}{1 \cdot 6}}$

Step 2.2.2.2

Cancel the common factor.

$\sqrt{\frac{1 \cdot 251}{1 \cdot 6}}$

Step 2.2.2.3

Rewrite the expression.

$\sqrt{\frac{251}{6}}$

Step 2.3

Rewrite $\sqrt{\frac{251}{6}}$ as $\frac{\sqrt{251}}{\sqrt{6}}$ .

$\frac{\sqrt{251}}{\sqrt{6}}$

Step 2.4

Multiply $\frac{\sqrt{251}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{251}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{\sqrt{251}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{251} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 2.5.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{251} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 2.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{251} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 2.5.4

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

$\frac{\sqrt{251} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 2.5.5

Add $1$ and $1$ .

$\frac{\sqrt{251} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 2.5.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{251} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 2.5.6.2

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

$\frac{\sqrt{251} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 2.5.6.3

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

$\frac{\sqrt{251} \sqrt{6}}{6^{\frac{2}{2}}}$

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.

$\frac{\sqrt{251} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 2.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{251} \sqrt{6}}{6^{1}}$

Step 2.5.6.5

Evaluate the exponent.

$\frac{\sqrt{251} \sqrt{6}}{6}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Combine using the product rule for radicals.

$\frac{\sqrt{251 \cdot 6}}{6}$

Step 2.6.2

Multiply $251$ by $6$ .

$\frac{\sqrt{1506}}{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{1506}}{6}$

Decimal Form:

$6.46786930 \dots$