AP Statistics keyterms

Statistics

the science of collecting, analyzing, and drawing conclusions form data.

Descriptive - methods of organizing and summarizing statistics

Inferential - making generalizations from a sample to the population

Population

An entire collection of individuals or objects

Sample

A subset of the population selected for the study

Variable

Any characteristic whose value changes

Data

observations on single or multi-variables

Variables

categorical, numerical, univariate, bivariate, multivariate

Categorical (Quallitative)

-basic characteristics

Numerical (Quantative)

measurements or observations of numerical data.

Discrete- listable sets (counts)

Continuous- any value over an interval of values (measurements)

Univariate

One variable

Bivariate

Two variables

Multivariate

many variables

Types of distributions

symmetrical, uniform, skewed, bimodal

Symmetrical

Data on which both sides are fairly the same shape and size. "Bell curve"

Uniform

Every class has an equal frequency (number) "a rectangle"

Skewed

one side (tail) is longer than the other side. The skewness is in the direction that the tail points (left or right)

Bimodal

data of two or more classes have large frequencies separated by another class between them. "double hump camel"

How to describe numerical graphs - S.O.C.S

Shape, Outliers, Center, Spread

Shape

overall type (symmetrical, skewed right left, uniform or bimodal)

Statistic (x that type of stuff)

a calculated value about a population from a sample(s).

Measures of Center

Median, Mean, Mode

Mean

μ is for a population (parameter) and x is for a sample (statistic)

Variability

allows statisticians to distinguish between usual and unusual occurrences.

Resistant

-not affected by outliers

Median and IQR

Non-resistant

Mean, Range, Variance, Standard Deviation, Correlation Coefficient (r), Least Squares Regression Line (LRSL) and Coefficient of Determination (r^2)

Trimmed Mean

use a % to take observations away from the top and bottom of the ordered data. This possibly eliminates outliers

Z-score

is a standardized score. This tells you how many standard deviations from the mean an observation is. It creates a standard normal curve consisting of z-scores with a μ = 0 & σ = 1.

z= x-μ/σ

5- Number Summary

Minimum, Q1, Median, Q3, Maximum

Probability rules

Sample Space, Event, Complement, Union, Intersection, Mutually Exclusive, Independent, Experimental Probability, Law of Large Numbers

Sample Space

is collection of all outcomes

Event

any sample of outcomes

Complement

all outcomes not in the event

Union

A or B, all the outcomes in both circles. AuB

Intersection

A and B, happening in teh middle of A and B. AnB

Mutually Exclusive (Disjoint)

A and B have no intersection. They cannot happen at the same time.

Independent

if knowing one event does not change the outcome of another

Experimental Probability

is the number of success from an experiment divided by the total amount from the experiment.

Law of Large Numbers

as an experiment is repeated the experimental probability gets close and closer to the true (theoretical) probability. The difference between the two probabilities will approach "0"

Correlation Coefficient - (r)

is a quantitative assessment of the strength and direction of a linear relationship.

Least Squares Regression LIne (LRSL)

is a line of mathematical best fit. Minimizes the deviations (residuals) from teh line. Used with bivariate data.

Residuals (error)

is a vertical difference of a point from the LRSL. All residuals sum up to "0".

Residual Plot

a scatterplot of residual. No matter indicates a linear relationship

Coefficient of Determination (r^2)

gives the proportion of variation in y (response) that is explained by teh relationship of (x,y) Never use the adjusted r^2.

Interpretations

Slope (b)

For unit increase in x, then the y variable will increase/decrease slope amount

Correlation coefficient (r)

There is a strength, direction, linear association between x and y

Coefficient of determination (r^2)

Approximately r^2% of the variation in y can be explained by the LRSL of x any y.

Extrapolation

LRSL cannot be used to find values outside of the range of the original data

Influential Points

are points that if removed significantly change the LSRL.

Outliers (residuals)

are points with large residuals

Sampling Frame

is a list of everyone in the population.

Types of Sampling Designs

SRS, Stratified, Systematic, Cluster Sample

SRS (Simple Random Sample)

one chooses so that each unit has an equal chance and every set of units has an equal chance of being selected.

Advantage's: easy and unbiased

Disadvantages: large σ2 and must know population

Stratified

divide the population into homogeneous groups called strata

Advantages: more precise than an SRS and cost reduced if strata already available.

Disadvantages: difficult to divide into groups, more complex formulas & must know population

Systematic

use a systematic approach (every 50th) after choosing randomly where to begin.

Advantages: unbiased, the sample is evenly distributed across population & don't need to know population

Disadvantages: a large σ2 and can be confounded by trends

Cluster Sample

based on location. Select a random location and sample ALL at that location

Advantages: cost is reduced, is unbiased and don't need to know population

Disadvantages: May not be representative of population and has complex formulas.

Random Digit Table

each entry is equally likely and each digit is independent of the rest

Random # Generator

Calculator or computer program

Bias-

Error, favors a certain outcome, has to do with center of sampling distributions - if centered over true parameter then considered unbiased

Sources of Bias

Voluntary Response, Convenience Sampling, Undercoverage, Non-response, Response, Wording of the Questions

Voluntary Response

People choose themselves to participate

Convenience Sampling

ask people who are easy, friendly, or comfortable asking

Undercoverage

some group(s) are left out of the selection process.

Non-response

someone cannot or does not want to be contacted or participate.

Response

false answers- can be caused by a variety of things

Wording of Questions

leading questions

Types of Experimental Designs

Observational study, experiment, experimental unit, factor, level, response variable, treatment, control group, placebo, blinding, double blinding.

Observational study

observe outcomes with out giving a treatment

Experiment

actively imposes a treatment on the subjects

Experimental unit

single individual or object that receives a treatment

Factor

Is the explanatory variable, what is being tested.

Level

a specific value for the factor

Response Variable

What you are measuring with the experiment

Treatment

experimental condition applied to each unit

Control Group

a group used to compare the factor to for effectiveness - does NOT have to be placebo

Placebo

a treatment with no active ingredients (provides control)

Blinding

a method used so that the subjects are unaware of the treatment (who gets a placebo or the real treatment).

Double Blinding

neither the subjects nor the evaluators know which treatment is being given.

Principles

Control, Replication, Randomization

Control

Keep all extraneous variables (not being stated) constant

Replication

uses many subjects to quantify the natural variation in the response

Randomization

uses chance to assign the subjects to the treatments.

How to create proper cause and effect

it is with a well designed, well controlled experiment

Experimental Designs

Completely Randomized, Randomized Block, Matched Pairs, Confounding Variables, Randomization, Blocking

Completely Randomized

all units are allocated to all the treatments randomly

Randomized Block

units are blocked and then randomly assigned in each block - reduces variation

Matched Pairs-

are matched up units by characteristics and then randomly assigned. Once a pair receives a certain treatment, then the other pair automatically receives the second treatment. OR individuals do both treatments in random order (before/ after or pretest/post-test). Assignment in dependent

Confounding Variables

are where the effect of the variable on the response cannot be separated from teh effects of the factor being tested - happens in observational studies - when you use random assignment to treatments you do NOT have confounding variables.

Randomization (Designs)

reduces bias by spreading extraneous variables to all groups in the experiment

Blocking

helps reduce variability. Another was to reduce variability is to increase sample size

Random variable

a numerical value that depends on teh outcome of an experiment

Discrete

a count of a random variable

Continuous

a measure of a random variable

Discrete Probability Distributions

gives values and probabilities associated with each possible x.

calculator shortcut - 1 VARSTAT L1, L2

Fair game

a fair game is one in which all pay-ins equal all pay-outs

Special discrete distributions

binomial distributions and geometric distributions

Binomial distribution

Properties- two mutually exclusive outcomes, fixed number of trails (n), each trial is independent, the probability (p) of success is the same for all trials.

Random variable- is the number of successes out of the fixed # of trials. Starts at X = 0 and is finite.

μx = np σ = sqrt(npq)

Calculator: binomialpdf (n, p, x) - single outcome P(X=x)

binomialcdf (n, p, x) = cumulative outcome P(X < x)

1 - binomialcdf (n, p, (x-1)) = cumulative outcome P(X>x)

Geometric Distributions

Properties - two mutually exclusive outcomes, each trial is independent, probability (p) of success is the same for all trials. (NOT a fixed number of trials)

Random Variable - when the FIRST succcess occurs. Starts at 1 and is infinite

Calculator: geometricpdf (p, a) = single outcome P(X = a)

geometriccdf (p, a) = cumulative outcomes P(X < a)

1 - geometriccdf (n, p, (a-1)) = cumulative outcome P(X > a)

Continuous Random Variable

numerical values that fall within a range of interval (measurements), use density curves where the area under the curve always = 1. The find probabilities, find area under the curve

Unusual Density Curves - any shape (triangles, etc.)

Uniform Distributions - uniformly (evenly) distributed, shape of a rectangle.

Normal Distributions - symmetircal, unimodal, bell shaped curves defined by the parameters μ and σ.

Calculator: Normalpdf - used for graphing only

Normalcdf (lower bound, upper bound, μ, σ) - finds probability

InvNorm(p) - z-score OR InvNorm (p, μ, σ) - gives x-value

To assess Normality

Use Graphs - dotplots, boxplots, histograms, or normal probability plot.

Distribution

is all of the values of a random variable

Sampling Distribution

of a statistic is the distribution of all possible values of all possible samples. Use normalcdf to calculate probabilities - be sure to use correct SD

Standard error

estimate of the standard deviation of the statistic

Central Limit Theorem

when n is sufficiently large (n>30) the sampling distribution is approximately normal even if the population distribution is not normal