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
mX+Y = mX + mY
If X and Y are independent random variables, then
s2X+Y = s2X + s2Y
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)
$81.00
<------MEANS
ADD-------->
$81.00
<--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)
$22,504.00
<---VARIANCES ADD--->
$22,504.00
<--Variance(Sa+Su)
(Saturday+Sunday)
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