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. Consider, we cannot simply add square roots.

STDs To solve this in Stats, we Square, Add and Square-Root. Dinner The STD for making dinner is 4 minutes The STD for eating dinner is 11 minutes The STD for making and eating is

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.