Statistics Mid-Term Study Guide

Statistics Mid-Term Study Guide

Statisticsis the science of collecting, organizing, presenting, analyzing, and interpreting numerical data to assist in making more effective decisions

Descriptive Statistics refers to a way or organizing and summarizing in an informative way.

Inferential statistics refers to a decision, estimate or prediction about a population based on a sample.

Types of variables:

Qualitative – refers to attributes or characteristics such as color or shape
Quantitative – refers to numerical classifications
Discrete – Can assume only certain values and there are gaps between the values
1, 2, 5, 8…
Continuous – Can assume any value within a specific range; can be subdivided.
3.1, 5.7, etc…

Levels of measurement (NOIR):

Nominal: Measure of qualitative data that can only be classified and counted
Color, sex, names, etc…
Ordinal: Rank orders specific numbers of qualities.
Rank, etc…
Interval: Includes all characteristics of ordinal but the difference between values is a constant size.
I.e., difference between 10 and 15 degrees is the same as the difference between 50 and 55 degrees, Interval is the same.
Ratio: Highest order, that includes a meaningful zero point and meaningful, ratios between numbers.
0 weight means absence of weight, and 80 is two times 40.

Parameter refers to populations and statistics refer to samples (P-P, S-S)

Frequency Table - A grouping of qualitative data into mutually exclusive classes showing the number of observations in each class.

Relative Class Frequencies – Each of the class frequencies is divided by the total number of observations. Sum of relative class frequencies is equal to 1.

Frequency distributions – A grouping of quantitative data into mutually exclusive classes showing the number of observations in each class.

Class interval – The difference between the lower limit of one class and the lower limit of the next class (15-18, 18-21 = 3)
Class frequency – Refers to the number of observations in each class. Sum is total observations.
Class midpoint: Found by adding the lower limits of two consecutive classes and dividing by two. (15-18, 18-21 = 15 + 18/2 = 16.5)
Relative Frequency – Each class frequency is divided by the total number of observations.

Graphical presentations of relative frequency distributions:

Histograms
Frequency Polygon
Cumulative frequency polygon
Ogive

Determine the number of classes:

2 to the k rule:
Such that 2 to the k power is greater than n.
If n=80, 26 = 64, but 27 = 128. Use 128>80.

Determine class interval or width:

High – low value divided by k
Round to a convenient number.

Frequency table rules:

Classes must be mutually exclusive
Data must be quantitative
There should be at least 100 observations in the sample

Cumulative frequency distribution is the running total of class frequencies.

Cumulative relative frequency distribution is a running total of the relative frequencies.

Measures of Central Tendency

Mean: The sum of all values divided by the total number of values (Average)

Properties of the arithmetic mean:

Every set of interval-level and ratio-level data has a mean.

All the values are included in computing the mean.

The mean is affected by unusually large or small data values.

A set of data has a unique mean.

The sum of the deviations of each value from the mean is zero.

Median: The median is the midpoint (middle) of values that have been ordered from lowest to highest.

For an even set of values the median is the average of the two middle numbers

Mode: The mode is the value that appears most often

Can be determined for all values
Not affected by extremely high or low numbers
Not always one mode and may be multiple modes.

Weighted mean: Special measure of the mean in which there are several observations of the same value.

I.e., GPA

Geometric mean: The geometric mean is useful in determining average change of percentages, ratios, indexes or growth rates over time.

The geometric mean will always be less than or equal to the arithmetic mean.
Defined as the nth root of the product of n values

Finding percent change over time:

Measures of Dispersion

Range: Largest value – smallest value

Mean Deviation: Average of absolute values of the deviations from the mean

Ex.

Population Variance: Average of the squared deviation from the mean

The sum of values of the differences between observation value and the population mean squared, divided by total number of items in the population

Population Standard Deviation is the square root of the population variance.

This allows us to transform the variance into the same unit of measure as the original data.

Sample Variance: The sum of the values of difference between the observation value and the sample mean divided by the sample population minus one squared

Sample Standard Deviation: Te square root of the sample variance

Ex.

Chebyshev’sTheorum: For any set of observations (sample or population), the proportion of the values that lie within k standard deviations of the mean is at least 1 – 1k2, where k is a constant greater than 1.

Example. At least what percent of the contributions lie within plus or minus 3.5 stadard deviations of the mean?

1 – 1/k2 = 1 – 1/(3.5)2 = 1 – 1/12.25 = 0.92 or 92%

The Empirical Rule: For a symmetrical bell shaped frequency distribution approximately 68% of the observations will fall within +/- 1 standard deviation of the mean, approximately 95% will fall within +/- 2 standard deviations of the mean and approximately 99.7% within +/- 3 standard deviations of the mean.

BoxPlots:

Five pieces of information are needed (Five Number Summary):

Minimum value
First Quartile
Median
Third Quartile
Maximum value

Skewness:

Finding skewness using Pearsons coefficient:

The Inter quartile range of a set of observations is the difference between the 1st and 3rd quartiles

The data describing the relationship between two variables is referred to as bivariate

A scatter diagram requires that both of the variables be at least interval scale.

A scatter diagram portrays the relationship between two variables
Uses interval or ratio scales

A contingency table is used when one or both variables in nominal or ordinal scale.

Example contingency table:

Probability:

Probability may be thought of in terms of relative frequency, proportion and percentage
Probability helps to determine the amount of error
Measure of the likelihood that an event in the future will happen.
Can only assume a value between 0 and 1

Experiment: The observation of some activity and the process of taking some measurement

Outcome: A particular result of an experiment

Sample space: Consists of all outcomes or results associated with a random experiment

Event: Collection of one or more outcomes of an experiment

Mutual exclusivity: Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time.

I.e., gender, you can either be male or female but not both.

Independence: Events are independent if the occurrence of one event does not affect the probability of the occurrence of another event.

Collectively Exhaustive: Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.

I.e., Die tossing will result in either a even number or an odd number.

Counting Rules:

Multiplication formula: if there are m ways of doing one thing and n ways of doing another, there arem x n ways of doing both.
Permutation formula: Number of ways of arranging r items selected from n objects in order.
Permutation= (nPr)
Combination formula: Number of ways of arranging r items selected from n objects regardless of order. Combination =(nCr)

Multiplication:

Same number of outcomes (k) done n trials (kn)
Example-Flipping coin 4 times
k=2 outcomes; n=4 coin tosses
Total possible outcomes =24=64

Different number of outcomes (k) to nth trial ((k1)(k2)…(kn))
Example - 10 shirts and 8 ties, how many shirt and tie outfits?
k1=10; k2=8
Total possible outcomes =(10)(8)=80

Factorial (n!) - Number of ways n items arranged in order

n! = n(n-1)(n-2)…(1)
0! = 1
How many ways can a set of 5 books be arranged on a shelf
5! = (5)(4)(3)(2)(1)=120

Permutation Formula: The permutation formula is applied to find the possible number of arrangements when there is only one group of objects.

***Order is important****
How many way different ways can I place 2 of 3 books on a shelf?
3P2 = 3!/(3-2)! =6

Combination Formula: The combination formula is applied if the selected order of the objects in NOT important. The formula to count the number of r object combinations from a set of n objects is:

I have three books (A, B, and C) and I want to know how many combinations of 2 books that I can place on the shelf irrespective of order.

Classical Probability (A Priori): Based on the assumption that all outcomes of an experiment are equally likely.

Probability of an event = Number of favorable outcomes/Total number of possible outcomes

Empirical Probability (A Posteriori): Based on the number of times an event occurs as a proportion of a known number of trials.

Probability of an event = Number of times the event has occurred/Total number of observations
I.e., Space shuttle successful flights = number of successful flights/Total number of flights
123/125 = 0.98 or 98%

Law of Large Numbers: The empirical rule is based on the law of large numbers which states that over a large number of trials the empirical probability of an event will approach its true probability.

Subjective Probability: Subjective probability is an estimate based on little known information.

Joint probability: A probability that measures the likelihood that two or more events will happen concurrently.

P(A ∩ B) = (P(A and B)

Rules for computing Probabilities

Rules of Addition:

Special rule of addition: If events are mutually exclusive, the special rule of addition states that the probability of one or the other events occurring is equal to the sum of their probabilities.
P(A or B) = P(A) + P(B)
P(A or B or C) = P(A) + P(B) + P(C)
Therefore (compliment rule) P(A) = 1 – P(not A)

General rule of addition: If events are NOT mutually exclusive (may include both), the general rule of addition states that the probability of one or the other event happening is equal to the sum of their probabilities minus their joint probability (or the events both occurring).
P(A or B) = P(A) + P(B) – P(A and B)
P(A U B) = P(A) + P(B) - P(A ∩ B)
P(A or B) = P(A) + P(B) - P(A and B)
= 4/52 + 13/52 - 1/52
= 16/52, or .3077

Multiplication Rules:

Special Rule of Multiplication: The special rule of multiplication requires that events be independent; that is, the occurrence of one event does not alter the probability that the other will occur.
It states that for two independent events A and B, the probability that A and B will both occur is found by multiplying the two probabilities.
P(A and B) = P(A)P(B)
P(A ∩ B) = P(A)P(B)

General Rule of Multiplication:The general rule of multiplication is used when events are not independent.
The joint probability that both events (A and B) will happen is found by multiplying the probability that event A will happen by the conditional probability of event B occurring given that A has occurred
P(A ∩ B) = P(A) x P(B|A)
Example:
A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry.
What is the likelihood both shirts selected are white?
P(W1 and W2) = P(W1)*P(W2|W1)
= 9/12*8/11 = 0.55

Contingency Table: A table used to classify sample observations according to two or more identifiable characteristics.

Example: King of Hearts. Two variables, Kings and Hearts

Marginal Probability: The probability of one variable taking a specific value irrespective of the values of the other variables. I.e., P(A)

Probability of getting a King (P(A)) = 4/52

Probability of getting a Heart (P(B)) = 13/52

Joint Probability: The probability of two or more events occurring concurrently. I.e., P (A∩ B)

= 4/52 * 13/52 = 1/52

Conditional Probability: The probability of a particular event occurring, given that another event has already occurred.

Written as P(A|B) (Probability of A given B)
What is P(Kings|Hearts)

P(King|Hearts) = 1/13

Some probability examples:

Say you roll 2 dice and want to know what the probability of getting 12 is.

1/6*1/6 = 1/36
Hint (How many possibilities are there? It can only happen one way

Say you roll two dice and want to know the probability of getting 8?

How many possibilities are there?
2/6, 6/2, 3/5, 5/3, 4/4
5 possibilities of 36 total outcomes
= 5/36

You decide to try your luck at a Craps table in Vegas. You make a pass line bet when a new shooter rolls the dice (2 dice). You win if the shooter rolls a 7 or an 11 on the first roll. What is the probability that you will win?

How many possibilities are there?
2/5, 5/2, 3/4, 4/3, 1/6, 6/1, 5/6, 6/5
8 possibilities/36 total outcomes
= 8/36

Reviewing probability computations:

General Rule of Addition - If events A and B are
NOT MUTUALLY EXCLUSIVE:
P(A or B) = P(A) + P(B) - P(A and B)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Special Rule of Addition – If events A and B are
MUTUALLY EXCLUSIVE:
P(A or B) = P(A) + P(B)
P(A ∪ B) = P(A) + P(B)
General Rule of Multiplication – If events A and B are
not independent:
P(A and B) = P(A) * P(B given A)
P(A ∩ B) = P(A) * P(B|A)
Special Rule of Multiplication – If events A and B are independent:
P(A and B) = P(A) * P(B)
P(A ∩ B) = P(A) * P(B)

Events are independent if the occurrence of one event does not affect the probability of occurrence of another event -- P(A|B) = P(A)

In a standard deck of cards, are events A & B independent?
Event A (King) = P(4/52)
Event B (Card is red) = P(26/52)
P (A|B) = 2/26
P (A) = 2/26
Does P (A|B) = P(A) ? Yes

Random Variables:

Discrete Random Variable can assume only certain clearly separated values. It is usually the result of counting something

Continuous Random Variable can assume an infinite number of values within a given range. It is usually the result of some type of measurement

Characteristics of a probability function distribution:

Probability of each particular outcome is between 0 and 1, inclusive

Outcomes are mutually exclusive

List is collectively exhaustive - sum of probabilities of all events equal 1

Measures of Central Tendencies and Dispersion for probability distribution

Mean of a probability distribution (expected value), is the sum of the values (x) times their probabilities.

Variance and Standard Deviation
Measures the amount of spread in a distribution
Variance – Sum of the squared deviation from mean times the probability
Standard deviation - square root of the variance

Discrete Probability Distributions

Binomial Probability Distribution:

There are only two possible outcomes on a particular trial of an experiment (success/failure)
The random variable (x) is the result of counts of successes in a fixed number of trials (n)
Probability (p) of event remains constant between trials
Each trial is independent of any other trials

Binomial Probability Formula:

C denotes Combination
n is the number of trials
x is the random variable defined as number of successes
p is the probability of a success on each trial (π sometimes used for p)
Mean and Variance of a binomial distribution:
μ=np
σ2 =np(1-p)

Hypergeometric probability Distribution

•An outcome on each trial of an experiment is classified into one of two mutually exclusive categories—a success or a failure.

•The probability of success and failure changes from trial to trial.

•The trials are not independent, meaning that the outcome of one trial affects the outcome of any other trial.

•Note: Use hypergeometric distribution if experiment is binomial, but sampling is without replacement from a finite population where n/N > 5%

•Must know N, S, n and x.

Poisson Probability Distribution

Poisson probability distribution describes the number of times some event occurs during a specified interval - usually used for rare events with low probabilities
The interval may be time, distance, area, or volume

Assumptions of the Poisson Distribution

The probability is proportional to the length of the intervals

The intervals are independent and do not overlap

Mean
μ = np, where n is the number of trials in the interval and p= probability of occurrence
Variance
Equals mean (σ2 = μ = np)
Mean = Variance = np

Continuous Probability Distributions

Random variable is continuous
Uniform, normal, Student’s t, chi-square, F distributions
Infinite number of values between any two data points; therefore, probability that a continuous random variable will assume a particular value equals zero
Probability distribution is the graph of an equation that links each possible value that a random variable can assume with its probability of occurrence

The Normal Distribution:

It is bell-shaped and has a single peak at the center of the distribution.
It is symmetricalabout the mean
It is asymptotic: The curve gets closer and closer to the X-axis but never actually touches it.
The location of a normal distribution is determined by the mean,, the dispersion or spread of the distribution is determined by the standard deviation,σ .
The arithmetic mean, median, and mode are equal
The total area under the curve is 1.00; half the area under the normal curve is to the right of this center point and the other half to the left of it
A z-value (also called z-score) is the distance between a selected value, designated X, and the population mean , divided by the population standard deviation, σ

Remember the Empirical rule!

MegaStat vs. Appendix B:

Characteristics of the t-distribution:

Like z distribution, bell-shaped and symmetrical

All t distributions have a mean of 0, but their standard deviations differ according to the sample size, n

There is not one t distribution, but rather a family of t distributions defined by degrees of freedom (df=n-1)

The t distribution is more spread out and flatter at the center than the standard normal distribution

Most useful for small sample sizes (<30)

As the sample size increases, t distribution approaches the standard normal distribution

Characteristics of the f-distribution:

Family of F Distributions -Specific Fdistrodetermined by two parameters: the numerator df and the denominator df

df num = nnum-1;
df den = nden-1

F cannot be negative- its values range from 0 to

The F distribution is positively skewed

As F  the curve approaches the X-axis

Usually used to compare two variances

Characteristics of the Chi Square:

The major characteristics of the chi-square distribution are:

Positively skewed

Non-negative

Based on degrees of freedom (df = n-1)

Usually used to compare expected versus observed values

Why sample the population

Cost

Time consuming

Destructive nature of certain tests

Impossible to check whole population

Sample results usually adequate

Probability sampling:

Probability sample is one in which elements are selected based on known probabilities - known likelihood of being included in the sample from the population

Non-probability sample - inclusion in the sample is based on judgment of the person selecting sample.