Permutations
A permutation is an ordered arrangement
The number of permutations for n objects is n!
n! = n*(n-1)*(n-2).3*2*1
The number of permutations of n objects taken r at a time is
You hand your friend a list of your 6 favorite movies and ask her to
The Steps Involved in Descriptive Statistics:
1. Decide on the number of classes/categories
a. For this problem use 6 classes.
b. Make sure the Class Width is accurate.
2. Calculate the Minimum Class Width
a. (High value-low value)/# classes and Minimum i
Frequency Tables
Cumulative Frequency Table displays the number of scores, in total, in a class and all
the lower classes
Relative Frequency Table displays the percentage of values that fall in each class
Frequency Distribution Cumulative Frequency
Freque
Normal Distribution and Standard Deviation Example
IQ scores are normally distributed with a mean of 100 and a Standard Deviation of 15. We
decide self-proclaimed Smart people are annoying and should be beaten. If we decide to beat
up the 10.03% smartest
Binomial Distributions
o Could it be .8*.8*.8*.8?
o what about her 5th class?
o Could it be .8*.8*.8*.8*.2?
o If THAT looks good, youre on the way!
o BUT .8*.8*.8*.8*.2 is only one of 5 ways she could get 4As and one non-A.
Binomial Distributions are Disc
Characteristics of a Normal Distribution
Given the symmetry of the Graph it is not difficult to comprehend these:
1.The Mode is the high point, the most often occurring value
2.Since the curve is symmetric, for every value a certain distance above the mod
Important Things to Remember about Probability Distribution
1. The VARIANCE of a Probability Distribution is the SUM of cfw_(x minus the MEAN)
SQUARED TIMES the P(x) for each of the pairs. [Standard Deviation is ALWAYS the
positive square root of the vari
Finding Probabilities
To find the probability that z is less than a given value, read the cumulative area in the
table corresponding to that z-score.
Find P( z < -1.24)
Read down the z-column to -1.2 and across to .04 The cumulative area is 0.1075
P ( z <
Continuous Random Variable
Examples:
x = Your weight in a 24 hour period.
o Continuous: your weight is measured
You cannot list all the possible values: 135, 135.1, 135.01, 135.001
x = The EXACT time you got up this morning.
o Continuous: time is measured
Normal Distribution Graph Facts
68.26% of the values will lie within one standard deviation of the mean.
95.44% of the values will lie within two standard deviations of the mean
99.74% of the values will lie within three standard deviations of the mean
CH
Normal Distribution Tables
You may recall we used the AREA in our Uniform Distribution example as a proxy
for PROBABILITY.
Total area was 1 or 1.00 or 100%; probability of a second hand being between 12 and
3 was 25% of the area, hence a probability of .2
Poisson Distribution
o The Poisson distribution is a discrete probability distribution of a random variable, x,
that satisfies the following conditions
o The experiment consists of counting the number of times an event occurs in a given
interval
o The int
Continuous Random Variables
Examples:
x = Your weight in a 24 hour period.
o Continuous: your weight is measured
o You cannot list all the possible values: 135, 135.1, 135.01, 135.001
x = The EXACT time you got up this morning.
o Continuous: time is measu
A Summary of Probability Rules
o P(A and B) = P(A) * P(B|A)
o P(A and B) = P(B) * P(A|B)
o P(A) = P(A and B)/P(B|A)
o P(B|A) = P(A and B)/P(A)
o P(A or B) = P(A) + P(B) - P(A and B)
o P(A) + P(not A) = 1
o P(A) = 1 - P(not A)
o P(A & B & C) = P(A) * P(B)
Contingency Table
The probability of choosing a lady second depends on whether the first was choice was a
lady. These events are dependent.
Two coins are tossed. A = first coin Heads, B = second coin Heads. P(A) = , P(B|A) =
, P(B) = . These events are in
BAYES THEOREM
o Try it with a contingency table
o 2 by 2 contingency tables (events mutually exclusive and collectively exhaustive)
o I like to start with something like 1000 tires.
o 800 made in Buffalo
o 2% = 16 are blemished
o 200 made in Syracuse
o 30
Complementary Events
o The complement of event E is event E.
o E consists of all the events in the sample space that are not in event E.
GIVEN: the weather in San Diego is excellent 80% of the days. Assuming independence, if
you go to San Diego for two da
Combinations Example
If your friend needed only to give you a list of the top 4, in no particular order, How
many different lists could she give you?
6!
( 6 4 )!4!
6 * 5 * 4 * 3 * 2 *1
15
2 *1* 4 * 3 * 2 *1
6
C4
The 6 of them in order = 720 arrangements
Multiplication Rule Another Example
o In this different case the characteristics Gender and handednessare INDEPENDENT.
o In this case the characteristics (Gender and handedness) are INDEPENDENT.
Independence occurs when the occurrence of one the Probabili
The Multiplication Rule
The probability of BOTH A and B
o We write this as P(A and B) = P(A) * P(B|A) and say The Probability of A
TIMES the Probability of B GIVEN A
Find the probability someone chosen at random is a Man who is Right-handed?
Well, isnt it
Probability Example Questions
Assuming independence, if each team gets a problem to work on (Team A gets a 3-part
problem), with each member responsible for a part
o What is the probability Team A gets theirs INcorrect? 1- P(all correct) or .0297
o What i
Probability Review
We have 2 teams of students:
TEAM A, the Hotshots, consists of 3 students, each of whom gets 99% of the answers
correct.
TEAM B, the Regulars, consists of 4 students, each of whom gets 60% of the answers
correct.
o Team A is given a 3 p
Other Important Terms for Statistics
Sample a subset of a population
Everything we DO count and measure
Why only a sample?
Parameter a number that describes a population characteristic (e.g. Average age of all the
college students in the United States)
St
Probability
Important Terms
Probability experiment an action through which counts, measurements or actions
are obtained
Sample space the set of all possible outcomes
Event a subset of the sample space
Outcome the result of a single trial
CHOOSE A STUDENT
Assigning Event Probabilities
1.
a priori Classical Method
2.
Empirical Classical Method
3.
Subjective Method
Types of Probability
Classical (Equally Probable Outcomes)
P(E)
Number of outcomes in event E
Number of outcomes in sample space
Empirical (Coun
Random Sampling
Each member of the population has the same likelihood of being selected
Examples:
Set up a table at the mall & solicit responses
Set up a table at the Student Union & solicit responses
Types of Sampling
Simple Random Sample - Assign each m
Conditional Probability
The probability an event B will occur, given (on the condition) that another event A
has occurred.
We write this as P(B|A) and say probability of B, given A
Calculate the probability a student chosen at random (from the 586) is lef
Population Variance
Deciles (of which there are nine) and Percentiles (of which there are ninety nine) divide the data
into ten and one hundred parts respectively. Excel will calculate percentiles
Measures of Variation
Range =Max Value Min Value
o If only
Systematic Sampling
If a sample size of n is desired from a population containing N elements, we sample one
element for every N/n elements in the population.
We randomly select one of the first N/n elements from the population list.
We then select every N
Sample Standard Deviation
o To calculate a sample variance divide the sum of squares by n-1.
o The sample standard deviation, s is found by taking the square root of the sample
variance.
o Samples only contain a portion of the population. This portion is