# Standard Deviation

How to find the standard deviation of the given data: formula, 2 examples, and their solutions.

## Formula

## Example

To find the standard deviation,

first find the variance,

then square root the variance.

Find the mean x of the data.

(sum) = 70 + 75 + 80 + 85 + 90

70 + 90 = 160

75 + 85 = 160

160 + 160 = 320

320 + 80 = 400

There are 5 values.

So n = 5.

(sum) = 400

n = 5

Then x = 400/5.

400/5 = 80

So x = 80.

Make a 3 column table like this.

Name the titles

x_{i}, x_{i} - x, and (x_{i} - x)^{2}.

Write the values in the x_{i} column:

70, 75, 80, 85, 90.

Make an empty row

at the bottom of the table.

Write the x_{i} - x column.

x = 80

So write x_{i} - 80.

70 - 80 = -10

75 - 80 = -5

80 - 80 = 0

85 - 80 = 5

90 - 80 = 10

Write the (x_{i} - x)^{2} column.

Square the x_{i} - x column.

(-10)^{2} = 100

(-5)^{2} = 25

0^{2} = 0

5^{2} = 25

10^{2} = 100

Add up the (x_{i} - x)^{2} column.

100 + 25 + 0 + 25 + 100

= 200 + 50

= 250

The sum of (x_{i} - x)^{2} is 250.

n = 5

So V(X) = 250/5.

250/5 = 50

V(X) = 50

Then the standard deviation σ(X) is,

square root V(X),

√50.

50 = 5^{2}⋅2

Prime Factorization

√5^{2}⋅2 = 5√2

Simplify a Radical

So 5√2 is the answer.

## Examplefrom a Frequency Table

To find the standard deviation,

first find the variance,

then square root the variance.

To find the variance from a frequency table,

make a 6 column table like this.

Name the titles

x_{i}, f_{i}, x_{i}f_{i}, x_{i} - x, (x_{i} - x)^{2}, and , (x_{i} - x)^{2}f_{i}.

Write the scores in the x_{i} column:

0, 1, 2, 3, 4, 5.

Write the frequencies in the f_{i} column:

1, 8, 5, 3, 2, 1.

Make an empty row

at the bottom of the table.

Add up the f_{i} column.

1 + 8 + 5 + 3 + 2 + 1

= 9 + 8 + 3

= 17 + 3

= 20

The sum of the frequencies is n.

So n = 20.

Write the x_{i}f_{i} column.

Multiply x_{i} and f_{i}.

0⋅1 = 0

1⋅8 = 8

2⋅5 = 10

3⋅3 = 9

4⋅2 = 8

5⋅1 = 5

Add up the x_{i}f_{i} column.

0 + 8 + 10 + 9 + 8 + 5

= 18 + 17 + 5

= 18 + 22

= 40

The sum of x_{i}f_{i} is the sum of the values.

So (sum) = 40.

(sum) = 40

n = 20

Then the mean is

x = 40/20.

40/20 = 2

So x = 2.

Write the x_{i} - x column.

x = 2

So write x_{i} - 2.

0 - 2 = -2

1 - 2 = -1

2 - 2 = 0

3 - 2 = 1

4 - 2 = 2

5 - 2 = 3

Write the (x_{i} - x)^{2} column.

Square the x_{i} - x column.

(-2)^{2} = 4

(-1)^{2} = 1

0^{2} = 0

1^{2} = 1

2^{2} = 4

3^{2} = 9

Write the (x_{i} - x)^{2}f_{i} column.

Multiply f_{i} and (x_{i} - x)^{2}.

1⋅4 = 4

8⋅1 = 8

5⋅0 = 0

3⋅1 = 3

2⋅4 = 8

1⋅9 = 9

Add up the (x_{i} - x)f_{i} column.

4 + 8 + 0 + 3 + 8 + 9

= 12 + 11 + 9

= 12 + 20

= 32

The sum of (x_{i} - x)f_{i} is 32.

n = 20

So V(X) = 32/20.

Reduce 32 to, 32/4, 8

and reduce 20 to, 20/4, 5.

V(X) = 8/5

Then the standard deviation σ(X) is,

square root V(X),

√8/5.

√8/5 = √8/√5

Divide Radicals

√8

= √2^{3}

= √2^{2}⋅2

= 2√2

Rationalize the denominator √5

by multiplying √5/√5.

Then 2√10/5.

Multiply Radicals

So 2√10/5 is the answer.