Equations
Statistics and Data Analysis

1.
Sample mean (x̅) => divide by n
SD=>divide by n-1 n-1 = Degree of Freedom (# deviations needed to recover full data set)
V=SD2

2.
Z score tells how many SDs from mean, population = (x - µ)/ σ , for data set= (x – x̅) / s
Empirical rule – 68% in one SD, 95% in 2, 99.7% in 3 … only exact for normal population using known means and variances, but often useful estimate even for non-normal distrib
Z scores
Way to compare the characteristics of two different data sets with normal distributions, with 0=µ

3.
If P(A∩B) is nothing, then mutually exclusive
If P(A|B)=P(A), then independent, otherwise dependent
If independent, P(A∩B)=P(A)P(B)

P(not A=> Ā) = 1-P(A)
Addition rule = P(AUB) = P(A) + P(B) - P(AUB) (which is 0 when mutually exclusive)
P(A|B) = P(A∩B)/P(B)

Since P(A) and P(B) are independent, P(B∩A) = P(B)*P(A)
Therefore, P(B|A)= P(B∩A) / P(A) = P(B)*P(A) / P(A) = P(B)

Expected payout for the policy is E(x), where E(x) = P(x = $0) + P(x = $100,000)
E(x) for a $5 odds bet is P(x=-$5) + P(x=$6)
E(x) = -5*(6/11) + 6*(5/11)

5.
If X discrete RV
Theoretical measures (don’t need data to compute)…
E(X) = µ = weighted average of expected variables = theoretical mean of RV X (LR for pop)
Var(X) = σ2 = E[(x- µ) 2]= ∑all x (x-µ) 2 p(x)
6.
If Binomial distribution, P of x successes in n trials is
P(x) = px qn-x q=1-p = binomial coefficient= n!/(x!(n-x)!), n choose x if order does not matter
P=.5 = symmetric binomial distrib, <.5 skewed right, >.5 skewed left
More ways to get 5 heads than 0 heads
Binomial… E(X) = µ = np
Var(X) = σ2 = n*p*q

P(x≥6) = 1-P(x≤5) = 1-.6230
97% chance machines work, 17 out of 18 work?

7.
Continuous distributions have smooth curves called probability densities=area under curve
Normal distribution the most important type of continuous distrib
> and ≥ are the same
Probability is area of space between the two points
E(X) = µ σ2 = E(X - µ) 2

8.
Normal distribution – no equation, just table
Sample always provides bell curve
Standard normal distribution => standardized so µ = 0, σ2 =