Variance, Standard Deviation

Variance, Standard Deviation

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

Formulas

V(X) = [the sum of (xi - x bar)^2 as i goes from 0 to n]/n, sigma(X) = sqrt(V(X)). V(X): Variance, sigma(X): Standard deviation, xi: Value of the data, x bar: Mean (average) n: Number of the data

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

The following data show 5 test scores of a student. 60, 70, 80, 90, 100. 1. Find the variance of the data. 2. Find the standard deviation of the data.

First, find the mean of the data: 80

Average (mean)

Make a 3 column table.

[xi]: Scores

[xi - x]: (column 1) - x
60 - 80 = -20, 70 - 80 = -10, 80 - 80 = 0 ...

[(xi - x)2]: (column 2)2
(-20)2 = 400, (-10)2 = 100, 02 = 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

The following data show 5 test scores of a student. 70, 75, 80, 85, 90. 1. Find the variance of the data. 2. Find the standard deviation of the data.

Find the mean of the data: 80

Average (mean)

Make a 3 column table.

[xi]: Scores

[xi - x]: (column 1) - x
70 - 80 = -10, 75 - 80 = -5, 80 - 80 = 0 ...

[(xi - x)2]: (column 2)2
(-10)2 = 100, (-5)2 = 25, 02 = 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

The following table shows quiz scores of 20 students: Scores(Frequency): 0(1), 1(1), 2(4), 3(7), 4(5), 5(2). Find the standard deviation of the data.

Make a 6 column table.

Write the first three columns.

[xi]: Scores

[fi]: Frequencies

[fixi]: (column 2)⋅(column 1)
1⋅0 = 0, 1⋅1 = 1, 4⋅2 = 8, ...

Write the sums of fi (20) and fixi (60)
below each column.

Find the mean.

x = ∑ fixi / ∑ fi
= 60 / 20
= 3

Write the next three columns.

[xi - x]: (column 1) - x
0 - 3 = -3, 1 - 3 = -2, 2 - 3 = -1 ...

[(xi - x)2]: (column 4)2
(-3)2 = 9, (-2)2 = 4, (-1)2 = 1 ...

[fi⋅(xi - 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

Rationalizing a Denominator