Ekstrom Math 115b Mathematics for Business Decisions, part II Variance Math 115b

Ekstrom Math 115b

Mathematics for Business Decisions, part II

Variance

Math 115b

Variance gives a measure of the dispersion of data

Larger variance means the data is more spread out

Several formulas for different types of variables

Variance

Finite R.V.

Easiest if a table is set up

Variance: Finite R.V.

x

XX xfxXV all

2

x

Given

Given Compute Compute Compute

xf X Xx 2Xx xfx XX 2


Ex. Find the variance for the following finite R.V., X

x fX (x)

1 0.30

3 0.15

5 0.10

7 0.15

9 0.30

First find the mean


x fX (x)

1 0.30

3 0.15

5 0.10

7 0.15

9 0.30

5

30.0915.0710.0515.0330.01

X

Calculate x-value minus the mean


-4

-2

0

2

4

x

1 0.30

3 0.15

5 0.10

7 0.15

9 0.30

xf X Xx

Calculate the square of x-value minus the mean


16

4

0

4

16

x

1 0.30 -4

3 0.15 -2

5 0.10 0

7 0.15 2

9 0.30 4

xf X 2

Xx Xx

Multiply the squared values by their respective p.d.f. values


4.80

0.60

0.00

0.60

4.80

x

1 0.30 -4 16

3 0.15 -2 4

5 0.10 0 0

7 0.15 2 4

9 0.30 4 16

xfx XX 2 2Xx Xx xf X

Add the final column to find the variance

Variance: Finite R.V.x

1 0.30 -4 16 4.80

3 0.15 -2 4 0.60

5 0.10 0 0 0.00

7 0.15 2 4 0.60

9 0.30 4 16 4.80

2Xx Xx xfx XX 2 xf X

Variance = 10.8

Ex. Find the variance for the following finite R.V., X


x

1 0.10

3 0.25

5 0.30

7 0.25

9 0.10

xf X

First find the mean


x fX (x)

1 0.10

3 0.25

5 0.30

7 0.25

9 0.10

5

10.0925.0730.0525.0310.01

X

Calculate x-value minus the mean


-4

-2

0

2

4

x

1 0.10

3 0.25

5 0.30

7 0.25

9 0.10

xf X Xx

Calculate the square of x-value minus the mean


16

4

0

4

16

x

1 0.10 -4

3 0.25 -2

5 0.30 0

7 0.25 2

9 0.10 4

xf X 2

Xx Xx

Multiply the squared values by their respective p.d.f. values


1.60

1.00

0.00

1.00

1.60

x

1 0.10 -4 16

3 0.25 -2 4

5 0.30 0 0

7 0.25 2 4

9 0.10 4 16

xfx XX 2 2Xx Xx xf X

Add the final column to find the variance

Variance: Finite R.V.x

1 0.10 -4 16 1.60

3 0.25 -2 4 1.00

5 0.30 0 0 0.00

7 0.25 2 4 1.00

9 0.10 4 16 1.60

2Xx Xx xfx XX 2 xf X

Variance = 5.2

Variance is used to find standard deviation

Standard deviation is ALWAYS the square root of variance

Standard deviation is represented by sigma

or

Standard deviation is the “typical amount” of variation from the mean (approx. 2/3 of all data lies within 1 standard deviation of mean)

Variance

XVX 2XXV

Shortcut for binomial R.V.

Variance: Binomial R.V.

ppn

XVX

1

ppnXV 1

Ex. Suppose X represents to total number of students that pass a particular class in a given semester. If there are 34 students in the class and historically 83% of the students pass, find the standard deviation of X.

Soln:


1903.2

7974.4

83.0183.034

1

ppnX

A solution could also be attempted using the BINOMDIST function

Recall binomial R.V.’s are finite R.V.’s

Complete table as done in previous examples


Similar formula for continuous R.V.

Value is found using Integrating.xlsm

Recall,

Variance: Continuous R.V.

x

XX dxxfxXV all

2

XVX

Ex. Suppose is a p.d.f. on the interval [0, 1]. Find the mean of X, the variance of X, and the standard deviation of X.

Soln:

Variance: Continuous R.V. 5.033 2 xxxf

5.0

5.0331

0

2

1

0

dxxxx

dxxfx XX

Soln:


2582.0

0667.0

XVX

0667.0

5.0335.01

0

22

1

0

2

dxxxx

dxxfxXV XX

Ex. Let T be an exponential random variable with parameter . Find and .

Soln:


8 TV T

64

80

8/812

0

2

dtet

dttftTV

t

TT

8

64

TVT

Note that for an exponential random variable, the variance is equal to and the standard deviation is equal to .

Ex. Let W be a uniform random variable on the interval [0, 30]. Find and .


WV W

2

Estrom Math 115b

Soln.:


75

1530

0 3012

30

0

2

dww

dwwfwWV WW

6603.8

75

WVW

Note that for a uniform random variable, the variance is equal to .


12

2ab

Different types of variables can have similar parameters (mean & std. dev.)

We can transform the variables

New variable S defined as

Variance: Standardization

X

XXS

We say S is the standardization of X.

The mean of S will be 0

The standard deviation of S will be 1.


Determining probabilities using standardized values

Ex. Suppose X is an exponential random variable with parameter . Determine where S is the standardization of X.

Soln: Recall


5 2SP

X

XXS

So,

Then,

15

105

25

5

22

XP

XP

XP

XPSP

X

X

9502.0

1515

0

5/51

dxeXP x


Formula:

A sample is a collection of data from some random variable (finite or continuous)

Variance: Sample

n

ii xx

ns

1

22

1

1

Standard deviation of a sample is found by taking the square root of variance

Formula:

Ex. Find the mean, variance, and standard deviation of the sample 14, 16, 17, 21, 22.

Variance: Sample

n

ii xx

nss

1

22

1

1

Soln:

Mean:

AVERAGE function in Excel

Variance: Sample

185

2221171614

x

Variance:

VAR function in Excel

Variance: Sample

5.11

16914164

1

1822182118171816181415

1

1

1

22222

1

22

n

ii xx

ns

Standard Deviation:

STDEV function in Excel

Variance: Sample

3912.3

5.11

2

ss

Why are sample values important?

Sometimes unreasonable/impossible to achieve all values for a random variable

We can assume and

Samples help predict values for the random variable

Variance: Sample

Xx XVs 2

Formula:

Compares a group of sample means

Variance: Sample Mean

n

XVxV

nX

x

Ex. Find the variance and standard deviation of the sample mean for the following data set:

14, 16, 17, 21, 22

Soln:

Variance: Sample Mean

3.25

5.11

n

XVxV

5166.15

5.11

nX

x

How do you interpret ?

If there were samples of size 5, the sample mean would be about 18 (from previous calculation) and, on average, the sample mean would be within 1.5166 units about 2/3 of the time.

Variance: Sample Mean5166.1x

What to do:

1. Calculate standard deviation of errors (use STDEV function)

2. My standard deviation is about 13.53

3. We assume mean is 0 even though we calculated a value that was different

Variance: Project

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Ekstrom Math 115b Mathematics for Business Decisions, part II Variance Math 115b