Sanderson M. Smith

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VARIANCES AND MEANS CAN ADD

(BUT STANDARD DEVIATIONS CAN'T)

Here is an illustration of these important statistical formulas:

If X and Y are any two random variables, then

m

_{X+Y}= m_{X}+ m_{Y}If X and Y are

independentrandom variables, thens

^{2}_{X+Y}= s^{2}_{X}+ s^{2}_{Y}

**Note:** These "nice"
rules do not hold for standard deviations.

Here's a story about my friend, Herkimer, to illustrate what is stated above:

Herkimer works only on weekends. On Saturday there are three possible financial outcomes for him, each with probability = 1/3 = 33 1/3% He can end up the day $100 in debt, he can end the day up with a gain of $50, or he can end up the day with a gain of $200. On Sunday, he has a similar situation, although on that day there are five possible financial outomes, each with probability = 1/5 = 20%. The following table shows Herkimer's possible gains and losses for Saturday and Sunday, along with the set (Sa + Su), containing the 5x3 = 15 gains and losses he can have at the end of a weekend.

Sa (Saturday)
Su (Sunday)
(Sa + Su)
-$100
-$80
-$180
$50
-$40
-$140
$200
$25
-$75
$90
-$10
$160
$60
------------
------------
-$30
Means
$50.00
$31.00
$10
Standard Deviations
$122.47
$86.63
$75
Variances (sq. dollars)
$15,000.00
$7,504.00
$140
$210
$120
$160
$225
$290
$360
------------
Mean(Sa) + Mean (Su)
<------ <--Mean(Sa+Su)
St.Dev.(Sa)+ St.Dev.(Su)
$197.47
<---STANDARD DEVIATIONS DO NOT
ADD-->
$150.01
<--St. Dev. (Sa+Su)
Var(Sa) + Var(Su)
<--- <--Variance(Sa+Su)

(Saturday+Sunday)
**$81.00**
**MEANS
ADD**-------->
**$81.00**
**$22,504.00**
**VARIANCES ADD**--->
**$22,504.00**

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