sample standard deviation of 2, 4, 4, 4, 5, 5, 7, 9

The mean of the 8 values is their sum divided by 8:

\frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5

For a sample, the variance is the sum of the squared deviations from the mean

5

divided by the number of values minus one, 7, and the standard deviation is its square root

1) Let's calculate:

\sqrt{\frac{{(2 - 5)}^{2} + {(4 - 5)}^{2} + {(4 - 5)}^{2} + {(4 - 5)}^{2} + {(5 - 5)}^{2} + {(5 - 5)}^{2} + {(7 - 5)}^{2} + {(9 - 5)}^{2}}{7}}

ⓘ Parentheses are computed first:

2 - 5

=

-3

,

4 - 5

=

-1

,

5 - 5

=

0

,

7 - 5

=

2

,

9 - 5

=

4

\sqrt{\frac{{(-3)}^{2} + {(-1)}^{2} + {(-1)}^{2} + {(-1)}^{2} + 0^{2} + 0^{2} + 2^{2} + 4^{2}}{7}}

ⓘ Powers are computed before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product:

{(-3)}^{2}

=

(-3) \times (-3)

=

9

,

{(-1)}^{2}

=

(-1) \times (-1)

=

1

,

2^{2}

=

2 \times 2

=

4

,

4^{2}

=

4 \times 4

=

16

■Adding 0 changes nothing:

0 + 0 + 2^{2}

=

2^{2}

\sqrt{\frac{9 + 1 + 1 + 1 + 4 + 16}{7}}

ⓘ Calculate the numerator and denominator before dividing:

9 + 1

=

10

\sqrt{\frac{10 + 1 + 1 + 4 + 16}{7}}

ⓘ Calculate the numerator and denominator before dividing:

10 + 1

=

11

\sqrt{\frac{11 + 1 + 4 + 16}{7}}

ⓘ Calculate the numerator and denominator before dividing:

11 + 1

=

12

\sqrt{\frac{12 + 4 + 16}{7}}

ⓘ Calculate the numerator and denominator before dividing:

12 + 4

=

16

\sqrt{\frac{16 + 16}{7}}

ⓘ Calculate the numerator and denominator before dividing:

16 + 16

=

32

\sqrt{\frac{32}{7}}

ⓘ The root of a quotient is the quotient of the roots,

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

:

\sqrt{\frac{32}{7}}

=

\frac{\sqrt{32}}{\sqrt{7}}

\frac{\sqrt{32}}{\sqrt{7}}

ⓘ Let's look for the largest square dividing

32

,

16

=

4^{2}

, and take it out of the root:

\sqrt{32}

=

\sqrt{16 \times 2}

=

\sqrt{16} \times \sqrt{2}

=

4 \times \sqrt{2}

\frac{4 \times \sqrt{2}}{\sqrt{7}}

By convention, we prefer numerical square roots in the numerator:

ⅰ

\frac{4 \times \sqrt{2}}{\sqrt{7}}

=

\frac{4 \times \sqrt{2} \times \sqrt{7}}{7}

\frac{4 \times \sqrt{2} \times \sqrt{7}}{7}

ⓘ Multiply two square roots by taking the square root of their product:

\sqrt{2} \times \sqrt{7}

=

\sqrt{2 \times 7}

\frac{4 \times \sqrt{2 \times 7}}{7}

ⅰ

2 \times 7

=

14

▶Sample standard deviation:

\frac{4 \times \sqrt{14}}{7}

2) The numeric value of:

\frac{4 \times \sqrt{14}}{7}

is:

\frac{4 \times \approx 3.7417}{7}

\frac{\approx 14.9666}{7}

▷

\approx 2.1381

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