variance of 2, 4, 4

The mean of the 3 values is their sum divided by 3:

\frac{2 + 4 + 4}{3} = \frac{10}{3}

The variance is the mean of the squared deviations from the mean

\frac{10}{3}

: their sum is divided by the number of values, 3 (by 2 for a sample, asked for with "sample variance")

1) Let's evaluate:

\frac{{(2 - \frac{10}{3})}^{2} + {(4 - \frac{10}{3})}^{2} + {(4 - \frac{10}{3})}^{2}}{3}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

2 - \frac{10}{3}

\frac{3 \times 2 - 10}{3}

4 - \frac{10}{3}

\frac{3 \times 4 - 10}{3}

\frac{{(\frac{3 \times 2 - 10}{3})}^{2} + {(\frac{3 \times 4 - 10}{3})}^{2} + {(\frac{3 \times 4 - 10}{3})}^{2}}{3}

ⓘ Multiplications are done before additions and subtractions:

3 \times 2

6

3 \times 4

12

\frac{{(\frac{6 - 10}{3})}^{2} + {(\frac{12 - 10}{3})}^{2} + {(\frac{12 - 10}{3})}^{2}}{3}

ⓘ Calculate the numerator and denominator before dividing:

6 - 10

-4

12 - 10

2

\frac{{(- \frac{4}{3})}^{2} + {(\frac{2}{3})}^{2} + {(\frac{2}{3})}^{2}}{3}

ⓘ Group like terms by adding their coefficients:

{(\frac{2}{3})}^{2} + {(\frac{2}{3})}^{2}

2 \times {(\frac{2}{3})}^{2}

\frac{{(- \frac{4}{3})}^{2} + 2 \times {(\frac{2}{3})}^{2}}{3}

ⓘ The power of a quotient is the quotient of the powers,

{(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}

{(- \frac{4}{3})}^{2}

\frac{{(-4)}^{2}}{3^{2}}

{(\frac{2}{3})}^{2}

\frac{2^{2}}{3^{2}}

\frac{\frac{{(-4)}^{2}}{3^{2}} + 2 \times \frac{2^{2}}{3^{2}}}{3}

ⓘ Powers are computed before multiplications, additions and subtractions. A power with a positive integer exponent is a repeated product:

{(-4)}^{2}

(-4) \times (-4)

16

3^{2}

3 \times 3

9

2^{2}

2 \times 2

4

■To add fractions, put them over the same denominator, then add the numerators:

\frac{16}{9} + 2 \times \frac{4}{9}

\frac{16 + 2 \times \frac{4}{9} \times 9}{9}

\frac{\frac{16 + 2 \times \frac{4}{9} \times 9}{9}}{3}

ⓘ Multiplications are done before additions and subtractions:

2 \times 9

18

■Multiplying a fraction by a number multiplies its numerator:

18 \times \frac{4}{9}

\frac{18 \times 4}{9}

■To add fractions, put them over the same denominator, then add the numerators:

16 + \frac{18 \times 4}{9}

\frac{16 \times 9 + 18 \times 4}{9}

\frac{\frac{\frac{16 \times 9 + 18 \times 4}{9}}{9}}{3}

ⓘ Multiplications are done before additions and subtractions:

16 \times 9

(10 + 6) \times 9

10 \times 9 + 6 \times 9

90 + 54

144

18 \times 4

(10 + 8) \times 4

10 \times 4 + 8 \times 4

40 + 32

72

\frac{\frac{\frac{144 + 72}{9}}{9}}{3}

ⓘ Calculate the numerator and denominator before dividing:

144 + 72

216

\frac{\frac{\frac{216}{9}}{9}}{3}

ⓘ Dividing a fraction by

9

is multiplying its denominator by

9

\frac{\frac{216}{9}}{9}

\frac{216}{9 \times 9}

■Dividing a fraction by

3

is multiplying its denominator by

3

\frac{\frac{216}{9 \times 9}}{3}

\frac{216}{9 \times 9 \times 3}

■A fraction is simplified by dividing the numerator and the denominator by the same nonzero number:

\frac{216}{9 \times 9 \times 3}

\frac{24}{9 \times 3}

\frac{24}{9 \times 3}

ⅰ

9 \times 3

27

\frac{24}{27}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

3

, their greatest common divisor:

\frac{24}{27}

\frac{8 \times 3}{9 \times 3}

\frac{8}{9}

▶Variance:

\frac{8}{9}

2) The numeric value of:

\frac{8}{9}

is:

▷

\approx 0.8889

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