Download Document

Mean and Standard
Deviation
Lecture 24
Section 7.5.1
Wed, Oct 27, 2004
The Mean and Standard
Deviation


Mean of a Discrete Random Variable – The
average of the values that the random variable
takes on, in the long run.
Standard Deviation of a Discrete Random
Variable – The standard deviation of the values
that the random variable takes on, in the long
run.
The Mean of a Discrete Random
Variable



The mean is also called the expected value.
However, that does not mean that it is literally
the value that we expect to see.
“Expected value” is simply a synonym for the
mean or average.
The Mean of a Discrete Random
Variable



The mean, or expected value, of X may be
denoted by either of two symbols.
µ or E(X)
If another random variable is called Y, then we
would write E(Y).
Or we could write them as µX and µY.
Computing the Mean

Given the pdf of X, the mean is computed as
  x1  P X  x1     xn  P X  xn 
 xi  P X  xi 

This is a weighted average of X.

Each value is weighted by its likelihood.
Example of the Mean

Recall the example where X was the number of
children in a household.
x
0
1
2
3
P(X = x)
0.10
0.30
0.40
0.20
Example of the Mean

Multiply each x by the corresponding probability.
x
0
1
2
3
P(X = x)
0.10
0.30
0.40
0.20
xP(X = x)
0.00
0.30
0.80
0.60
Example of the Mean

Add up the column of products to get the mean.
x
0
1
2
3
P(X = x)
0.10
0.30
0.40
0.20
xP(X = x)
0.00
0.30
0.80
0.60
1.70 = µ
Let’s Do It!

Let’s do it! 7.23, p. 430 – Profits and Weather.
The Variance of a Discrete
Random Variable



Variance of a Discrete Random Variable – The
square of the standard deviation of that random
variable.
The variance of X is denoted by
2 or Var(X)
The standard deviation of X is denoted by .
The Variance and Expected
Values



The variance is the expected value of the
squared deviations.
That agrees with the earlier notion of the
average squared deviation.
Therefore,

Var  X   E  X   
2

Example of the Variance

Again, let X be the number of children in a
household.
x
P(X = x)
0
1
2
0.10
0.30
0.40
3
0.20
Example of the Variance

Subtract the mean (1.70) from each value of X to
get the deviations.
x
P(X = x)
x–µ
0
1
2
0.10
0.30
0.40
-1.7
-0.7
+0.3
3
0.20
+1.3
Example of the Variance

Square the deviations.
x
P(X = x)
x–µ
(x – µ)2
0
1
2
0.10
0.30
0.40
-1.7
-0.7
+0.3
2.89
0.49
0.09
3
0.20
+1.3
1.69
Example of the Variance

Multiply each squared deviation by its probability.
x
P(X = x)
x–µ
(x – µ)2
0
1
2
3
0.10
0.30
0.40
0.20
-1.7
-0.7
+0.3
+1.3
2.89
0.49
0.09
1.69
(x – µ)2P(X = x)
0.289
0.147
0.036
0.338
Example of the Variance

Add up the products to get the variance.
x
P(X = x)
x–µ
(x – µ)2
0
1
2
3
0.10
0.30
0.40
0.20
-1.7
-0.7
+0.3
+1.3
2.89
0.49
0.09
1.69
(x – µ)2P(X = x)
0.289
0.147
0.036
0.338
0.810 = 2
Example of the Variance

Add up the products to get the variance.
x
P(X = x)
x–µ
(x – µ)2
0
1
2
3
0.10
0.30
0.40
0.20
-1.7
-0.7
+0.3
+1.3
2.89
0.49
0.09
1.69
(x – µ)2P(X = x)
0.289
0.147
0.036
0.338
0.810 = 2
0.9 = 
Alternate Formula for the
Variance

It turns out that
   E  X 
Var  X   E X


2
2
That is, the variance of X is “the expected value
of the square of X minus the square of the
expected value of X.”
Of course, we could write this as
 
Var X   E X 2   2
Example of the Variance

One more time, let X be the number of children
in a household.
x
0
P(X = x)
0.10
1
2
3
0.30
0.40
0.20
Example of the Variance

Square each value of X.
x
0
P(X = x)
0.10
x2
0
1
2
3
0.30
0.40
0.20
1
4
9
Example of the Variance

Multiply each squared X by its probability.
x
0
P(X = x)
0.10
x2
0
x2P(X = x)
0.00
1
2
3
0.30
0.40
0.20
1
4
9
0.30
1.60
1.80
Example of the Variance

Add up the products to get E(X2).
x
0
P(X = x)
0.10
x2
0
x2P(X = x)
0.00
1
2
3
0.30
0.40
0.20
1
4
9
0.30
1.60
1.80
3.70 = E(X2)
Example of the Variance



Then use E(X2) to compute the variance.
Var(X) = E(X2) – µ2
= 3.70 – (1.7)2
= 3.70 – 2.89
= 0.81.
It follows that  = 0.81 = 0.9.
TI-83 – Means and Standard
Deviations








Store the list of values of X in L1.
Store the list of probabilities of X in L2.
Select STAT > CALC > 1-Var Stats.
Press ENTER.
Enter L1, L2.
Press ENTER.
The list of statistics includes the mean and standard
deviation of X.
Use x, not Sx, for the standard deviation.
TI-83 – Means and Standard
Deviations




Let L1 = {0, 1, 2, 3}.
Let L2 = {0.1, 0.3, 0.4, 0.2}.
Compute the statistics.
Compute µ and  for the Indoor and Outdoor
distributions in Let’s Do It! 7.23, p. 430.
Let’s Do It!

Return once more to Let’s Do It! 7.23, p. 430.
The standard deviation of Profit Outdoors is 23.9.
 Use the original formula to compute the standard
deviation of Profit Indoors.
 Use the alternate formula to compute the standard
deviation of Profit Indoors.
 Use the TI-83 to find the standard deviation of
Profit Indoors.
