# Standard Deviation

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

## Formula

### Formula

σ(X) = √V(X)

σ(X): Standard deviation
V(X): Variance

Just like the variance,
the standard deviation σ(X) also means
how far the values are from the mean x.

Square root is used
to make the unit of σ(X)
the same as the mean.
(= to undo the square part of the variance)

## Example 1

### 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
xi, xi - x, and (xi - x)2.

Write the values in the xi column:
70, 75, 80, 85, 90.

Make an empty row
at the bottom of the table.

Write the xi - x column.

x = 80
So write xi - 80.

70 - 80 = -10
75 - 80 = -5
80 - 80 = 0
85 - 80 = 5
90 - 80 = 10

Write the (xi - x)2 column.

Square the xi - x column.

(-10)2 = 100
(-5)2 = 25
02 = 0
52 = 25
102 = 100

Add up the (xi - x)2 column.

100 + 25 + 0 + 25 + 100
= 200 + 50
= 250

The sum of (xi - 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 = 52⋅2

Prime Factorization

52⋅2 = 5√2

## Example 2

### 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
xi, fi, xifi, xi - x, (xi - x)2, and , (xi - x)2fi.

Write the scores in the xi column:
0, 1, 2, 3, 4, 5.

Write the frequencies in the fi column:
1, 8, 5, 3, 2, 1.

Make an empty row
at the bottom of the table.

1 + 8 + 5 + 3 + 2 + 1
= 9 + 8 + 3
= 17 + 3
= 20

The sum of the frequencies is n.

So n = 20.

Write the xifi column.

Multiply xi and fi.

0⋅1 = 0
1⋅8 = 8
2⋅5 = 10
3⋅3 = 9
4⋅2 = 8
5⋅1 = 5

0 + 8 + 10 + 9 + 8 + 5
= 18 + 17 + 5
= 18 + 22
= 40

The sum of xifi 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 xi - x column.

x = 2

So write xi - 2.

0 - 2 = -2
1 - 2 = -1
2 - 2 = 0
3 - 2 = 1
4 - 2 = 2
5 - 2 = 3

Write the (xi - x)2 column.

Square the xi - x column.

(-2)2 = 4
(-1)2 = 1
02 = 0
12 = 1
22 = 4
32 = 9

Write the (xi - x)2fi column.

Multiply fi and (xi - x)2.

1⋅4 = 4
8⋅1 = 8
5⋅0 = 0
3⋅1 = 3
2⋅4 = 8
1⋅9 = 9

Add up the (xi - x)fi column.

4 + 8 + 0 + 3 + 8 + 9
= 12 + 11 + 9
= 12 + 20
= 32

The sum of (xi - x)fi 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

8
= √23
= √22⋅2
= 2√2

Rationalize the denominator5
by multiplying √5/√5.

Then 2√10/5.