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)
(Saturday+Sunday)

-$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)

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