# Variance, Standard Deviation

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

## Formulas

The variance V(*X*) means

'how far the values of data are from the mean (*x*).

(*x* - *x*)^{2} part has that meaning.

The square is added

to undo the effect of signs (+, -).

Sigma Notation

The standard deviation σ(*X*)

has the same meaning:

how far the values of the data are from the mean (*x*).

σ is the small greek letter 'sigma'.

σ(*X*) is the square root of V(*X*).

The square root is added

to undo the 'square' effect of (*x* - *x*)^{2}.

## Example 1

First, find the mean of the data: 80

Average (mean)

Make a 3 column table.

[*x*_{i}]: Scores

[*x*_{i} - *x*]: (column 1) - *x*

60 - 80 = -20, 70 - 80 = -10, 80 - 80 = 0 ...

[(*x*_{i} - *x*)^{2}]: (column 2)^{2}

(-20)^{2} = 400, (-10)^{2} = 100, 0^{2} = 0 ...

Write the sum of (column 3): 1000.

Then V(*X*) = 1000/5

= 200.

σ(*X*) = √V(*X*)

= √200

= 10√2

Simplifying a Radical (Part 1)

## Example 2

Find the mean of the data: 80

Average (mean)

Make a 3 column table.

[*x*_{i}]: Scores

[*x*_{i} - *x*]: (column 1) - *x*

70 - 80 = -10, 75 - 80 = -5, 80 - 80 = 0 ...

[(*x*_{i} - *x*)^{2}]: (column 2)^{2}

(-10)^{2} = 100, (-5)^{2} = 25, 0^{2} = 0 ...

Write the sum of (column 3): 250.

Then V(*X*) = 250/5

= 50.

σ(*X*) = √V(*X*)

= √50

= 5√2

The answer (5√2) is lesser than

the previous problem's standard deviation (10√2).

This is true because

the given scores (70, 75, 80, 85, 90)

are more closer to the mean (80)

then the previous problem's scores (60, 70, 80, 90, 100).

Simplifying a Radical (Part 1)

## Example 3

Make a 6 column table.

Write the first three columns.

[*x*_{i}]: Scores

[*f*_{i}]: Frequencies

[*f*_{i}*x*_{i}]: (column 2)⋅(column 1)

1⋅0 = 0, 1⋅1 = 1, 4⋅2 = 8, ...

Write the sums of *f*_{i} (20) and *f*_{i}*x*_{i} (60)

below each column.

Find the mean.*x* = ∑ *f*_{i}*x*_{i} / ∑ *f*_{i}

= 60 / 20

= 3

Write the next three columns.

[*x*_{i} - *x*]: (column 1) - *x*

0 - 3 = -3, 1 - 3 = -2, 2 - 3 = -1 ...

[(*x*_{i} - *x*)^{2}]: (column 4)^{2}

(-3)^{2} = 9, (-2)^{2} = 4, (-1)^{2} = 1 ...

[*f*_{i}⋅(*x*_{i} - *x*)^{2}]: (column 2)⋅(column 5)

1⋅9 = 9, 1⋅4 = 4, 4⋅1 = 4, ...

Write the sum of (column 6): 30.

Then V(*X*) = 30/20

= 3/2.

σ(*X*) = √V(*X*)

= √3/2

= √6/2