QTM 100 Unit 1

Contingency Table

Table that summarizes data for two categorical variables

Stacked bar plot

Graphical display of contingency table information, compare sto variables on top of each other *not added

Side by side bar plot

Compares two variables side by side

When is stacked, side by side, or standardized stacked bar plot the most useful?

Stacked bar plot: assign one variable as explanatory and other as response

Side by side: does not show which variable causes the other, easy to compare cases but requires more horizontal space, also not favorable if groups are different sizes

Standardized bar plot: used if primary variable is relatively imbalanced, shows proportions

Mosaic plot

Resembles standardized stacked bar lot

(Widths = proportions)

Use areas to represent number of cases in each category

Side by side plot

Traditional tool for comparing across groups

Hollow histograms

Compare numerical data across groups (shown with outline)

Hollow histograms vs side by side plots

Hollow histograms: useful for seeing distribution and skew

Side by side boxplots: useful for comparing centers and spreads

Scatterplot

Case by case data for two numerical variables

Dot plot

One variable scatterplot

Formula for end of whiskers

Q1-1.5 times IQR or Q3+1.5 times IQR

Should it surprise you if the actual outcome of a potential random variable problem is slightly below or under the probability you calculated?

No— there is natural variability

Random variable

Random process with numerical outcome

Expected value

Sum of X*P(X)

Expected value can also be known as

Mean

How to find proportion of certain bin in continuous distribution (histogram)

See count divide by total sample size

Probability density function

Smooth curve over bars on histogram

total area under a curve in density distribution

1

Two types of random variables

Discrete and continuous

Probability distribution in the real world shows you

Your chances of winning

Columns for probability distribution

x, p(x), x*p(x), x²*p(x)

When you have money you should always round

Up

Variability

Difference in value of random variable (how much variation is expected)

Why do you need to find variability?

To find standard deviation

Formula for variability (to get ______)

SD*2= sum of x²*p(x)-(x*p(x))²

Fair game

costs as much as payout so expected profit =0

How do you prove a game is fair?

Sum of X*p(x) =0

Normal distribution

Most variable are normally distributed aka uni-modal and bell shaped

Two ways to prove symmetry

1. Mean= median=mode

2. Peason’s index

Pearson’s Index

I=[3(mean-median)]/SD

*must be between -1 and 1

What does z score tell you

How far away you are from the mean

Why do you need z score?

Standardize data to compare

Percentile

Percentage of data below specific data point

How do you round percentile?

Always round up

Gold standard for z-score

Mean =0

SD = 1

Z score formula

Z= (given-average)/SD

How to use z-score

Find z score then look at chart for probability and percentile

unusual vs outlier stats wth z score

|z|>2 = unusual

|z|>3= outlier

Empirical Rule

68-95-99.7

1 SD-2 SD- 3 SD

Area under curve

Probability

How do you get right side of a Norma distribution

Subtract from 1

How to look at normal distribution graph

Left to right (DO NOT START FROM MEAN)

How do you find cutoff?

1. Give you probability => look for closest z score (NUMBER IN PROBLEM IS NOT Z SCORE)

2. Plug into equation (x or given should be blank)

3. Solve algebra

Upper vs lower limit questions normal distribution

Will be 2 SD away BECAUSE otherwise they are outliers and we do not plot outlier (only use 3 SD if it says to include outliers)

How to write up normal distribution with parameters

N (mean =0, SD =1)

How to compare z scores

Higher |z| score = unusual

probability

proportion of times the outcome would occur if we observed the random process an infinite number of times

Law of Large Numbers

the tendency of proportion of outcomes to stabilize around calculated probability (empirical => classical)

think experiments

how to write probability of xx happening

p(xx)

disjoint

two outcomes cannot occur at the same time

another term for disjoint is

mutually exclusive

how do you calculate disjoint probability

add up probabilities of each thing occuring

what word is associated with disjoint outcome

or

addition rule

P(A1 or A2)= P(A1)+P(A2)-P(A1andA2)

events

sets/collections of outcomes

ex: group a: if you roll 1 or 2, group b: if you roll 2 or 3

nondisjoint outcomes

when outcomes can overlap

ex: roll 2 and even

face card

jack, queen, king

how many cards in a deck of cards

52

Or is

inclusive (so it is and/or)

Why would (P and B) be 0 if outcome is disjoint

because they would never overlap

probability distribution

table of all disjoint outcomes and their probabiliteis

rules for probability distributions*

1. outcomes listed must be disjoint

2. each probability must be between 0 and 1

3. probabilities must total 1

sample space

all possible outcomes

complement of x

all outcomes that are not x

P(x or x1)= 1

independent

when outcome of one provides no useful info about another outcome (like flipping a coin and rolling a dice)

when do you use multiplication rule vs addition rule

multiplication: independent

addition: disjoint

multipplication rule

P (A and B) = P(A) x P (B)

who is the father of probability

Jerome Cardan

classical probability

#ways e can occur / total possible outcomes

empirical probability

#ways E occured/total # attempts

limitation of probability

cannot PROVE anything with probability

can disjoint outcome also be independent?

no because disjoint occurs at differnet times and independent is at same time

@ least 1 means

1 OR MORE

complement of at least 1 is

0 cuz otherwise you have no options

Contingency table includes

cases are horizontal and results are vertical

marginal probabilities

probability based on single variable

joint probability

probability of outcomes for two ore more variables

table proportions

data presented shows a proportional relationship between two variables

conditional proabability

compute probability under a condition

two parts of conditional probability

outcome of interest and condition

how to read out P( X| Y)

probability of x given y

conditional probability formula

P(A|B)= P(A and B)/P(B)

general multiplication rule

P(A and B)= P(A|B) x P (B)

gamblers fallacy

casinos post last several outcomes of betting games to trick gamblers into believing odds are in their favor

ex: all black last time, you believe it is unlikely you will get black

tree diagram

organize outcome and proability around structure of data

primary v secondary branch

primary: first branch (split)

secondary: other splits

false negative and false positive

shows up as positive/negative (true/untrue) even when it isn’t

without replacement

you cannot sample the same case twice

what do you have to do with a “without replacement” situation

remove from possibilities

when should you use at least rule

when you need to calculate the likelihood of an event happening at least once in a given set of trials

when sample size is nearly less than 10% of pop, observations are

independent

If P(A and B) = P(A) P(B), A and B are

independent

If P(B) = P(B |A), then A and B are

independent

Binomial distribution

Describes number of success in fixed number trials (usually one or the other)

Difference between binomial and geometric distribution

Geometric: describe number of trails you must wait before success

Binomial: number of success in fixed number trials

Binomial distribution formula

(N!/K!(n-k)!)*p^k(1-p)^n-k

What do variables in formula mean?

N= number of trials

K= number successes

Mean, variance, and SD formula of observed successes

Mean: np

SD²=np(1-p)

Four conditions to check if binomial

1. Independent trials

2. Number of trials n is fixed

3. Each trial can be classified as success or failure

4. Probability of success p is the same for each trial

How to solve binomial distribution with normal distribution steps

ONLY TO BE USED WITH LARGE SAMPLES!!!

Treat all steps normal except add 0.5