Stats Section 1

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a posteriori probability

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a posterior probability is used to describe empirical probability; a revised probability that takes into account new available information; inductive
–relative frequency of past events
– # of times event occurred divided by total # of trials
– e.g., probability of rain on August day in Albuquerque

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a priori probabiity

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The probability estimate prior to receiving new information, a theoretical probability.

# of favorable outcomes divided by total # of outcomes

ex:probability of face card (K, Q, J) in random draw of
card

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Descriptive Statistics

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simply describe characteristics of a set of
numbers (e.g., mean,variance, etc.)

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Inferential Statistics

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infer something about a population from examining a sample. May estimate population parameter from sample statistic. Probability is the basis for this type of inference

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probability

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Asks what is likelihood of some event occurring? Which is a fundamental notion in everyday world.
Asks what is likelihood of some event occurring by chance? which is a fundamental concept in scientific research

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sample space

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set of all possible distinct outcomes for a simple
experiment

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Bayes Theorem

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relates the conditional and marginal probabilities of two random events. Often used to compute posterior probabilities given observations.

P(A|B) = [P(B|A)P(A)]/P(B)

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Generalization of Bayes thm

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P(A_i|B)=

[P(B|A_i)P(A_i)]/
[sum i2r(P(B|A_i)P(A_i))

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independent events

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A and B independent if knowing event B occured does not affect the prob of A occuring. Independent if any one of the 5 rules is met:
P(A|B)=P(A)
P(B|A)=P(B)
P(A&B)=P(A)P(B)
P(A)=0
P(B)=0

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mutually exclusive events

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The occurrence of an event
in one set excludes the occurrence of an event
in another set. The intersection of mutually exclusive sets is the
empty set

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Bernoullis theorem

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law of large numbers but applied to proportions. P(r/n-p)>eps)->0as n->inf. p is proportion of successes in pop. r/n is rate of successes in sample, epsilon is arbitrary small number >0 and n is sample size. Diff between sample stat and pop parameter gets smaller as n approaches N.

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nominal

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simplest scale (classification only), classifies objects or events into categories.. Each object is placed in one and only one class making the classes mutually exclusive and exhaustive. Example: numbers on athletic jerseys; blondes =1; red heads =2; sex: M-0 and F-1

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biased sample

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is one in which some members of the population are more likely to be included than others.

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binomial probability law

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computes probabilities for binomial random variables:
The probability of r successes from n events is given by

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Central limit theorem

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Distribution of sample means from samples of N (N has variance and mean ) independent observations  normal distribution with variance /N and mean as N increases. When N very large, distribution of the means is ~ normal

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central moments

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Special class of moments where C=µ. Then the rth central moment of X is given by: . When r=1, the first central moment says that the expected deviation of a units value from the population mean is 0. r=2 tells us the average squared deviation of a units value from the mean of its population, or the variance of the population.

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central tendency

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Measures of central tendency are measures of the location of the middle or the center of a distribution. Ways of measuring central tendency include the mean, median and mode.

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chance

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May be defined as the following:
- (in philosophy) a contingency; a concept denoting the status of facts that are not logically necessarily true or false
- a probability or likelihood
randomness, lack of order, purpose, cause, or predictability.
luck (vernacular) or outside of someone’s control.
unintentional (vernacular) or without intention or planning.

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combination

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counting the number of ways that r objects might be selected from among some N objects (r<=N). The number of ways of selecting and arranging r objects from among N distinct objects is

N!/(N-r)!

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conditional probability

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a probability based on only part of the total sample space. Formally, let A and B be events in a sample space, then the conditional probability of B given A is p(B/A)=p(A∩B)/p(A) , provided that p(A) is not 0.

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covariance

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given the random variable X with expectation E(X) and the random variable Y with expectation E(Y), then the covariance of X and Y is cov(X,Y)=E(XY)-E(X)E(Y). It’s a reflection of the departure from independence of X and Y. When X and Y are independent cov(X,Y)=0. When random variables are independent, their covariance is 0, but just because their covariance =0, doesn’t mean they are independent. However, when cov(X,Y) ≠0 X and Y are NOT independent.

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descriptive statistics

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values used to describe characteristics of a
set of numbers (e.g., mean, variance, etc.)

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exhaustive events

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a set of events is collectively exhaustive if at least one of the events must occur. For example, when rolling a six-sided die, the outcomes 1, 2, 3, 4, 5, and 6 are collectively exhaustive, because they encompass the entire range of possible outcomes. Another description is that their union must cover all the events within the entire sample space. For example, events A and B are said to be collectively exhaustive if A U B=S where S is the sample space.

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expected value

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the expected value (or mathematical expectation, or mean) represents the average amount one "expects" as the outcome of the random trial when identical odds are repeated many times. Mathematically, if X is a discrete random variable with probability mass funtion p, then . Expected values follow linear laws.

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Expected value, independence and correlation relationship:

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1. E(XY)=E(X)E(Y)  X and Y independent
2. if E(XY)=E(X)E(Y) then X and Y are uncorrelated (zero correlation). (If independent, then uncorrelated)
3. Uncorrelation does not imply independence

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hypergeometric probability law

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Used for population of N depletable units. Not independence from trial to trial.

R= successes
N-R = # failures
n= # items selected from N items
r= #successes in n items.

Tells us the prob that a random sample of n units will contain exactly r successes



P(r)= ways of getting r successes out of r times ways of getting n-r failures out of N-R total failures divided by number of ways of choosing n things from N.

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independent events

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If knowing event B occurred does not affect the probability that event A occurs then A and B are independent
If any one of the properties are met then A and B are independent:

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indices of dispersion

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tells how spread out the distribution is
according to some criterion. Range, variance and standard dev all all measures.

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indicies of location

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allows us to represent an entire distribution or a special characteristic of it by indicating where some namable part of a distribution is on the horizontal axis. The mode is the crudest index of location, also have median and mean, and percentile.

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interval scale

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- classification + ordering + meaningful differences. rule: equal differences between sets of numbers on the scale must represent equal differences in whatever attribute is being measured. Has true unit of measurement, its meaningful to talk about one unit of whatever is being measured.

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kurtosis

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for standardized central moment, when r=4. Tells us the shape of the populations distribution. Sensitive to peakedness or concentration of X’s about the mean. Most meaningful in cases of symmetric distribution.

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Law of Large Numbers

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Given a random variable with a finite expected value, if its values are repeatedly sampled, as the number of these observations increases, their mean will tend to approach and stay close to the expected value.

P[(X-u)>eps]->0 as sample size n --> infinity

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validity

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to the degree to which a study accurately reflects or assesses the specific concept that the researcher is attempting to measure.

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mean

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best known index of location, its simply the arithmetic average of the N unit values. Mean is sensitive to the exact value of all the scores in the distribution. Mean is least subject to sampling variation. Mean is mathematically tractable.

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measurement

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process of assigning numbers or other symbols to the things in such a way that relationships of the numbers or symbols reflect relationships of the attribute being measured.

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measurement error

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occurs when an observation of a unit’s value differs from the unit’s true value

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median

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consider the N units arranged in a line in order of increasing value, then median is the value of the middle unit, if N is odd. It is the midpoint between the values of
the two midmost units when N is even. median is the 50th percentile

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mode

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crudest index of location. Its the value that occurs with the greatest
frequency, in other words, the value that is possessed by the greatest number of units
in the set. Two or more values may be tied for the greatest frequency, in which case the
distribution is called bimodal or multimodal, respectively

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moment

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are special cases of expected values that describes the characteristics of a population. For a population with variance , mean , and units with value X, and for C is a constant, the rth moment of X about the arbitrary point C is

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multinomial prob law

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generalization of binomial probability law to case where sampled population contains>2 kinds of units. Random sample of n observations is drawn either from infinite pop. or with replacements from finite. Population not depleted. Independent outcomes:

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multivariate hypergeometric prob law

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An extension of the hypergeometric probability law where now there are more than two categories. Tells us the probability for a certain number of successes from each category:

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mutually exclusive events

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set of outcomes which are mutually exclusive, which means that at most one of the events may occur. The set of all possible die rolls is both collectively exhaustive and mutually exclusive. The outcomes 1 and 6 are mutually exclusive but not collectively exhaustive. The outcomes "even" (2,4 or 6) and "not-6" (1,2,3,4, or 5) are collectively exhaustive but not mutually exclusive.

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nominal scale

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objects are arranged into equivalence classes. Objects in same class are thought of as qualitatively the same and objects in different classes as qualitatively different. Each object is placed in one and only one class making the classes mutually exclusive and exhaustive. Cannot be arranged in ordering scheme. Arithmetic operations are non-sensical for nominal data.

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ordinal scale

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classification + rank ordering. rule: the rank order (ordinal position)
represents the rank order of the attribute measured. Gives order of preference only, not difference in preference. both nominal and ordinal scales lack a unit of measurement, nothing that corresponds to one unit of something.

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parameter

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defines certain characteristics about a population.

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partition

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sets that are mutully exclusive and exhaustive. Example:1,2,3,4,5,6 are mutually exclusive but exhaustive sets for the possible rolls of a die

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percentile

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- the value of a variable below which a certain percent of observations fall. So the 20th percentile is the value (or score) below which 20 percent of the observations may be found. The 25th percentile is also known as the first quartile; the 50th percentile as the median

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permissible transformations

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the set of operations that can be performed on members of a set that result in a sensicle output.

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permutation

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the notion of mapping the elements of a set to other elements of the same set, i.e., exchanging (or "permuting") elements of a set.

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population

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the group being studied. We often use samples of a population determine or infer caharacetersitics of the populaiton.

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probability

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(# of times A can occur)/(# of events in reference set)

Likelihood of something happening.

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range

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for a data set is equal to the max value in the data set minus the minimum value in the data set.

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ratio scale

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classification + order + meaningful differences + a meaningful zero point
An interval scale with a rational (true) zero rather than an arbitrary zero
Ex: weight, height, time, money. Rule: ratios between numbers on the scale at different
points reflect meaning differences of attributes. Ratio scale contains most information
– most powerful of all scales
– knowing data on ratio scale can derive all other scales
- no psychological variables at ratio scale

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reliability

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the consistency of a set of measurements or measuring instrument, often used to describe a test. This can either be whether the measurements of the same instrument give or are likely to give the same measurement (test-retest), or in the case of more subjective instruments, such as personality or trait inventories, whether two independent assessors give similar scores (inter-rater reliability). Reliability is inversely related to random error.

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sample

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elements taken from a population in order to find out something about the population. so like a for a political poll, the population is all registered voters and the sample is the group of people the pollers call and ask for whom they will vote

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sample space

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(reference set) – set of all possible distinct outcomes for a simple experiment

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sampling error

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occurs when a statistic derived from a sample differs from the corresponding population parameter

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sample variance

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the computed variance of a sample size n which can then be used to estimate the population variance.

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scales of measurement

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classification that is used to describe the nature of information contained within numbers assigned to objects and, therefore, within the variable. The levels were proposed by Stanley Smith Stevens in his 1946 article On the theory of scales of measurement. According to Stevens' theory of scales, different mathematical operations on variables are possible, depending on the level at which a variable is measured.

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skewness

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index of asymmetry of the X or Z population. Is given by the third standardized central moment:

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standard deviation

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Measurement of spread of the data. Specifically, tells us the spread of the data about the mean. Square root of the variance.

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standardized variable

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shows relative status of a score in a distribution
• z scores tell us how many standard deviation units a score is away from the mean
• mean of distribution of standardized scores is 0
• standard deviation of standardized scores is 1.

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statistic

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is a single element of a data set or a value of a function of a data set.

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Tchebycheff’s inequality

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states that in any data sample or probability distribution, nearly all the values are close to the mean value, and provides a quantitative description of "nearly all" and "close to".
In particular,
No more than 1/4 of the values are more than 2 standard deviations away from the mean;
No more than 1/9 are more than 3 standard deviations away;
No more than 1/25 are more than 5 standard deviations away;
and so on. In general:
No more than 1/k2 of the values are more than k standard deviations away from the mean.

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validity

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the degree to which a study accurately reflects or assesses the specific concept that the researcher is attempting to measure. While reliability is concerned with the accuracy of the actual measuring instrument or procedure, validity is concerned with the study's success at measuring what the researchers set out to measure.

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variability

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spread in a variable or a probability distribution. Common examples of measures of statistical variability are the variance, standard deviation and interquartile range.

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variance

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measure of statistical dispersion, averaging the squared distance of its possible values from the mean. Whereas the mean is a way to describe the location of a distribution, the variance is a way to capture its scale or degree of being spread out.

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variance of the mean

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standard error of the mean.