# Standard Deviation

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

## Formula

### Formula

## Example 1

### Example

### Solution

To find the standard deviation,

first find the variance,

then square root the variance.

So this solution will cover

finding the variance of the data.

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.

## Example 2

### Example

### Solution

To find the standard deviation,

first find the variance,

then square root the variance.

So this solution will also cover

finding the variance of the data.

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.