The Standard The Standard Deviation of a Deviation of a

Discrete Random Discrete Random VariableVariableLecture 24Lecture 24

Section 7.5.1Section 7.5.1

Fri, Oct 20, 2006Fri, Oct 20, 2006

The Variance and The Variance and Standard DeviationStandard Deviation

Variance of a Discrete Random VariableVariance of a Discrete Random Variable – The variance of the values that the – The variance of the values that the random variable takes on, in the long random variable takes on, in the long run.run.

This is the average squared deviation of This is the average squared deviation of the values that the random variable the values that the random variable takes on, in the long run.takes on, in the long run.

Standard Deviation of a Discrete Random Standard Deviation of a Discrete Random VariableVariable – The square root of the – The square root of the variance.variance.

Why Do We Need the Why Do We Need the Standard Deviation?Standard Deviation?

Our goal later will be to assign a Our goal later will be to assign a margin of error to our estimates of a margin of error to our estimates of a population mean.population mean.

To do this, we need a measure of the To do this, we need a measure of the variability of our estimatorvariability of our estimatorxx..

This, in turn, requires a measure of This, in turn, requires a measure of the variability of the variability of xx, namely, the , namely, the standard deviation.standard deviation.

The Variance of a The Variance of a Discrete Random Discrete Random

VariableVariable The variance of The variance of XX is denoted by is denoted by

XX22 or Var( or Var(XX))

The standard deviation of The standard deviation of XX is is denoted by denoted by XX..

Usually there are no other variables, Usually there are no other variables, so we may write so we may write 22 and and ..

The VarianceThe Variance

The variance is the expected value of The variance is the expected value of the squared deviations.the squared deviations.

That agrees with the earlier notion That agrees with the earlier notion of the average squared deviation.of the average squared deviation.

Therefore,Therefore,

2Var XEX

The VarianceThe Variance

Since the variance of Since the variance of XX is the is the average value of (average value of (XX – – ))22, we use the , we use the method of weighted averages to method of weighted averages to compute it.compute it.

Example of the VarianceExample of the Variance

x P(x)

00 0.100.10

11 0.300.30

22 0.400.40

33 0.200.20

Again, let Again, let XX be the number of be the number of children in a household.children in a household.

Example of the VarianceExample of the Variance

x P(x) x – µ

00 0.100.10 -1.7-1.7

11 0.300.30 -0.7-0.7

22 0.400.40 +0.3+0.3

33 0.200.20 +1.3+1.3

Subtract the mean (1.70) from each Subtract the mean (1.70) from each value of X to get the deviations.value of X to get the deviations.

Example of the VarianceExample of the Variance

x P(x) x – µ (x – µ)2

00 0.100.10 -1.7-1.7 2.892.89

11 0.300.30 -0.7-0.7 0.490.49

22 0.400.40 +0.3+0.3 0.090.09

33 0.200.20 +1.3+1.3 1.691.69

Square the deviations.Square the deviations.

Example of the VarianceExample of the Variance

x P(x) x – µ (x – µ)2 (x – µ)2P(x)

00 0.100.10 -1.7-1.7 2.892.89 0.2890.289

11 0.300.30 -0.7-0.7 0.490.49 0.1470.147

22 0.400.40 +0.3+0.3 0.090.09 0.0360.036

33 0.200.20 +1.3+1.3 1.691.69 0.3380.338

Multiply each squared deviation by its Multiply each squared deviation by its probability.probability.

Example of the VarianceExample of the Variance

x P(x) x – µ (x – µ)2 (x – µ)2P(x)

00 0.100.10 -1.7-1.7 2.892.89 0.2890.289

11 0.300.30 -0.7-0.7 0.490.49 0.1470.147

22 0.400.40 +0.3+0.3 0.090.09 0.0360.036

33 0.200.20 +1.3+1.3 1.691.69 0.3380.338

0.810 = 2

Add up the products to get the Add up the products to get the variance.variance.

Example of the VarianceExample of the Variance

x P(x) x – µ (x – µ)2 (x – µ)2P(x)

00 0.100.10 -1.7-1.7 2.892.89 0.2890.289

11 0.300.30 -0.7-0.7 0.490.49 0.1470.147

22 0.400.40 +0.3+0.3 0.090.09 0.0360.036

33 0.200.20 +1.3+1.3 1.691.69 0.3380.338

0.810 = 2

0.9 =

Take the square root to get the Take the square root to get the standard deviation.standard deviation.

Alternate Formula for Alternate Formula for the Variancethe Variance

It turns out thatIt turns out that

That is, the variance of That is, the variance of XX is “the is “the expected value of the square of expected value of the square of XX minus the square of the expected minus the square of the expected value of value of XX.”.”

Of course, we could write this asOf course, we could write this as

22Var XEXEX

22Var XEX

Example of the VarianceExample of the Variance

x P(x)

00 0.100.10

11 0.300.30

22 0.400.40

33 0.200.20

One more time, let One more time, let XX be the number be the number of children in a household.of children in a household.

Example of the VarianceExample of the Variance

x P(x) x2

00 0.100.10 00

11 0.300.30 11

22 0.400.40 44

33 0.200.20 99

Square each value of Square each value of XX..

Example of the VarianceExample of the Variance

x P(x) x2 x2P(x)

00 0.100.10 00 0.000.00

11 0.300.30 11 0.300.30

22 0.400.40 44 1.601.60

33 0.200.20 99 1.801.80

Multiply each squared Multiply each squared XX by its by its probability.probability.

Example of the VarianceExample of the Variance

x P(x) x2 x2P(x)

00 0.100.10 00 0.000.00

11 0.300.30 11 0.300.30

22 0.400.40 44 1.601.60

33 0.200.20 99 1.801.80

3.70 = E(X2)

Add up the products to get Add up the products to get EE((XX22).).

Example of the VarianceExample of the Variance

Then use Then use EE((XX22) and µ to compute the ) and µ to compute the variance.variance.

Var(Var(XX) = ) = EE((XX22) – µ) – µ22

= 3.70 – (1.7)= 3.70 – (1.7)22

= 3.70 – 2.89= 3.70 – 2.89

= 0.81.= 0.81. It follows that It follows that = = 0.81 = 0.9.0.81 = 0.9.

TI-83 – Means and TI-83 – Means and Standard DeviationsStandard Deviations

Store the list of values of Store the list of values of XX in L in L11..

Store the list of probabilities of Store the list of probabilities of XX in L in L22.. Select STAT > CALC > 1-Var Stats.Select STAT > CALC > 1-Var Stats. Press ENTER.Press ENTER. Enter LEnter L11, L, L22.. Press ENTER.Press ENTER. The list of statistics includes the mean and The list of statistics includes the mean and

standard deviation of standard deviation of XX.. Use Use x, not Sx, for the standard deviation.x, not Sx, for the standard deviation.

TI-83 – Means and TI-83 – Means and Standard DeviationsStandard Deviations

Let LLet L11 = {0, 1, 2, 3}. = {0, 1, 2, 3}.

Let LLet L22 = {0.1, 0.3, 0.4, 0.2}. = {0.1, 0.3, 0.4, 0.2}. Compute the parameters Compute the parameters and and ..

Example: PowerballExample: Powerball

www.powerball.www.powerball.comcom

Find the Find the standard standard deviation of the deviation of the value of a value of a Powerball ticket.Powerball ticket.

xx PP((xx))

200000020000000000

0.000000000.0000000068446844

200000200000 0.000000280.000000280606

1000010000 0.000001710.0000017111

100100 0.000070150.00007015

100100 0.000083840.00008384

77 0.0034370.003437

77 0.0013410.001341

44 0.0078810.007881

33 0.014500.01450

00 0.97270.9727