Transcript Law of Large Numbers Powerpoint

Law of Large Numbers
Means and STDs
Law of Large Numbers
• The more the merrier.
• There is no Law of Small Numbers. We know
what to expect in the long run only, not the
short run.
• How many is enough to fit the Law of Large
Numbers? Enough (depends on the situation).
Means
• We can add means with the same units.
• Dinner
– It takes 30 minutes on average to make dinner.
– It takes 10 minutes on average to eat dinner.
– It therefore takes, on average, 40 minutes on
average to make and eat dinner.
STDs
• We CANNOT add STDs.
• Consider how we calculate a STD.


2
1
xi  x

n
– Consider, we cannot simply add square roots.
6  10  6  10
STDs
• To solve this in Stats, we Square, Add and
Square-Root.
2
2
2
 6    10   
6  10

• Dinner
– The STD for making dinner is 4 minutes
– The STD for eating dinner is 11 minutes
– The STD for making and eating is
42  112  137  11.70
Consider…
• Let’s consider an example.
– Mr. Shahin is learning a new way to tie his shoes.
The left shoe takes him a mean of 6 seconds to tie
with a standard deviation of 0.5 seconds. The right
shoe takes him 8 seconds to tie with a standard
deviation of 0.6 seconds.
– Mr. Shahin is Very particular, and must repeat the
tying of his right shoe once(to try and improve his
speed).
– Determine the time, on average, it takes Mr.
Shahin to tie both shoes.
Twice?
• So, since he ties his right shoe twice, we can
easily consider the mean.
– It takes 6 seconds for him to tie his left shoe, and
16 seconds for him to tie his right shoe (2*8 for
the twice he ties it). Thus, it will take him, on
average, 24 seconds to tie his shoes.
– The standard deviations for tying are 0.5 and 1.2
seconds between left and right shoe, respectively
(again, 2*0.6 to account for the tying of the right
shoe twice).
Twice!
– Thusly, we find that the standard deviations
(which are themselves square roots) must be
squared before being summed.
– For left, that would be 0.5^2 = 0.25, and for right
that would be 1.2 ^ 2 = 1.44.
– Together, they are 1.69 (as a variance, since we
squared them).
– Square rooting, we get the standard deviation of
the whole process: SQRT(1.69) = 1.3 seconds
So…
• The process of Mr. Shahin tying his shoes in
the morning take a mean of 24 seconds with a
standard deviation of 1.3 seconds.
• We went from knowing how to tie shoes
separately to the total time to tie shoes (since
obviously that’s what matters as you should
not leave the house with just one shoe on).
Subtracting IS Adding
• Weird Concept…subtracting Standard
Deviations is the same as adding them.
– New example, we wish to find the difference
between the heights of men and women.
– Men have an average height of 69 inches with a
standard deviation of 2.5 inches.
– Women have an average height of 64 inches with
a standard deviation of 2.5 inches.
Differences
• So, the average difference in the heights between
men and women is clearly 5 inches (69-64).
• The average difference in the standard deviation
of the heights would appear to be 0 (2.5^2 –
2.5^2)…but, that makes no sense! The average
difference between every man and woman is 5?
There is NO room for error?
• This by itself helps to show us that we should
NEVER subtract standard deviations (or
variances).
Hrm…
• So, consider why we still Square, Add and
Square Root.
– Let X – Y represent the difference between
distributions.
– This would be the same as X + (-Y).
– Thus, we would take the negative value for every
element in Y, determine it’s mean, and it’s
standard deviation. The mean would in fact be
negative…but remember that a standard deviation
always comes out positive. Thus, X + (-Y) still adds
the STDs.
In Our Case
• 2.5^2 + 2.5^2 = 6.25 + 6.25 = 12.5
• SQRT(12.5) = 3.5355
• Thus, the difference between the distribution
of heights between men and women is 5
inches with a standard deviation of 3.5355
inches.