# Standard Deviation

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

## Formula

The standard deviation σ(*X*)

is the square root of the variance V(*X*).

So, just like the variance,

the standard deviation also means

how far the values are from the mean *x*.

The reason to write a square root

is to undo the effect of the square in V(*X*):

(*x*_{i} - *x*)^{2}.

By adding a square root,

the unit of σ(*X*)

and the unit of *x*

are the same.

So it's convineient to compare

the standard deviation and the mean.

## Example 1: Standard Deviation of 30, 40, 50, 60, 70

First, find the mean of the data.

The values are 30, 40, 50, 60, 100.

There are 5 values.

So the mean *x* is

(30 + 40 + 50 + 60 + 70)/5.

Mean (Average)

30 + 70 = 100

40 + 60 = 50

100 + 100 + 50 = 250

250/5 = 50

So *x* = 50.

Draw a three column table like this.

Write the titles*x*_{i}, *x*_{i} - *x*, and (*x*_{i} - *x*)^{2}.

Write the given values in the *x*_{i} column.

*x* = 50

So write the value of *x*_{i} - *x*, *x*_{i} - 50,

in the next column.

30 - 50 = -20

40 - 50 = -10

50 - 50 = 0

60 - 50 = 10

70 - 50 = 20

Write the squares of the second column

in the (*x*_{i} - *x*)^{2} column.

(-20)^{2} = 400

(-10)^{2} = 100

0^{2} = 0

10^{2} = 100

20^{2} = 400

Then write the sum of the third column

in the next row:

400 + 100 + 0 + 100 + 400 = 1000.

This 1000 is the sum of (*x*_{i} - *x*)^{2}.

The sum of (*x*_{i} - *x*)^{2} is 1000.*n* = 5

So V(*X*) = 1000/5.

1000/5 = 200

So V(*X*) = 200.

The standard deviation σ(*X*)

is the square root of the variance V(*X*).

V(*X*) = 200

So σ(*X*) = √200.

200 = 2⋅100

100 = 10^{2}

Take 10 out from the square root sign.

Then σ(*X*) = 10√2.

Simplify a radical

*x* = 50 [points]

V(*X*) = 200 [points^{2}]

σ(*X*) = 10√2 [points]

Unlike the unit of V(*X*) [points^{2}],

the unit of σ(*X*) [points]

is the same as the unit of *x* [points].

This is why

the standard deviation is widely used

in analyzing data.

## Example 2: Standard Deviation from a Frequency Table

If a frequency table is given,

the solution is quite different.

First, make a six column table.

Write the titles*x*_{i} (score), *f*_{i} (frequency),*x*_{i}*f*_{i}, *x*_{i} - *x*, (*x*_{i} - *x*)^{2}, and *f*_{i}(*x*_{i} - *x*)^{2}.

Write the scores in the *x*_{i} column:

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

And write the frequencies in the *f*_{i} column:

1, 1, 4, 7, 5, 2.

Then write the sum of the *f*_{i} column

in the ∑ row:

1 + 1 + 4 + 7 + 5 + 2 = 20.

So *n* = 20.

Write *x*_{i}*f*_{i} column

by multiplying *x*_{i} and *f*_{i}.

0⋅1 = 0

1⋅1 = 1

2⋅4 = 8

3⋅7 = 21

4⋅5 = 20

5⋅2 = 10

Write the sum of the *x*_{i}*f*_{i} column

in the ∑ row:

0 + 1 + 8 + 21 + 20 + 10 = 60.

This 60 is the sum of the values.

(sum) = 60*n* = 20

So *x* = 60/20.

So *x* = 3.

Mean from the Frequency Table

*x* = 3

So write the value of *x*_{i} - *x*, *x*_{i} - 3,

in the next column.

0 - 3 = -3

1 - 3 = -2

2 - 3 = -1

3 - 3 = 0

4 - 3 = 1

5 - 3 = 2

Write (*x*_{i} - *x*)^{2} column

by squaring the (*x*_{i} - *x*) column.

(-3)^{2} = 9

(-2)^{2} = 4

(-1)^{2} = 1

0^{2} = 0

1^{2} = 1

2^{2} = 4

Write *f*_{i}(*x*_{i} - *x*)^{2} column

by multiplying *f*_{i} and (*x*_{i} - *x*)^{2}.

1⋅9 = 9

1⋅4 = 4

4⋅1 = 4

7⋅0 = 0

5⋅1 = 5

2⋅4 = 8

Write the sum of the *f*_{i}(*x*_{i} - *x*)^{2} column

in the ∑ row:

9 + 4 + 4 + 0 + 5 + 8 = 30.

This 30 is the total sum of (*x*_{i} - *x*)^{2}.

The total sum of (*x*_{i} - *x*)^{2} is 30.*n* = 20

So V(*X*) = 30/20.

30/20 = 3/2

So V(*X*) = 3/2.

σ(*X*) = √V(*X*)

V(*X*) = 3/2

So σ(*X*) = √3/2.

√3/2 = √3/√2

Divide radicals

To rationalize the denominator √2,

multiply √2/√2.

Then σ(*X*) = √6/2.

Rationalizing a Denominator