Eco 72 Midterm "Cheat Sheet"

Formulasfor central tendency: The sample mean can be calculated as

where{X₁,X₂,...X_n} are the observations and n is the size of the sample. If asample has a weight variable w_i then theweighted mean is

Tocalculate the median, simply arrange all of the observations in orderand report the one in the middle; if there is an even number ofobservations, split the difference (the halfway point between twonumbers x and y is (x+y)/2. The mode isthe most common observation.

Formulasfor dispersion:The rangeis the highest value in a sample minus the lowest. The mean absolute deviation is

where, n and {X₁,X₂,...X_n}are defined as above and denotes absolute value. The sample variance can be calculated as

,where , n and{X₁,X₂,...X_n}are defined as above. The standard deviationis the positive square root of the variance. In a population of sizeN,we calculate the population variance using the same method but dividebyN insteadof n-1. If we sort the data, then the interquartilerangeis the median of the top half of the sample minus the median of thebottom half.

Formulasfor probability: For any two events Aand B, . If A and Bare mutually exclusive, then , so . The conditional probability of A given that we know that Bhas happened is . For any events, . If (and only if) A and Bare independent, , so . For complements:

Combinationsof Things: The number of ordered combinations of kobjects from a group of n is n!/(n-k)!.If order is not important, then the number of combinations isn!/[k!(n-k)!].

Distributions:For any discrete random variable taking on Vdifferent values, if we know the probability of observing anyvalue X_i, then the mean is and the variance is .

Theformula for the binomial distribution is P(X=k) ={n!/[k!(n-k)!]}p^k(1-p)^n-k, where n is thenumber of draws/trials, k is the number of ones/successes, and p is the probability of drawing a one/success on any given draw/trial. The mean of a binomial is np and the variance is np(1-p).

The probability of observing the value t in a poisson distributionis , where the parameter is equal to the mean (and, it turns out, the variance as well), and e=2.7183.

IfX is a Normally distributed variable with mean and standard deviation , then for any number t, , where z is the Standard Normal variable whose probabilitiescan be looked up in the Normal chart. A binomial with large nand moderate p can be approximated by a Normal variable with mean and standard deviation ;to figure out ,we compute where the latter reflect outcomes of theNormal variable.