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context

ideally tells who was measured, what was measured, how the data were collected, where the data were collected, and when and why the study was performed

data

systematically recorded information, whether numbers or labels, together with its context

data table

an arrangement of data in which each row represents a case and each column represents a variable

case

an individual about whom or which we have data

variable

holds information about the same characteristic for many cases

categorical variable

a variable that names categories (whether with words or numerals)

quantitative variable

a variable in which the numbers act as numerical values; always has units

units

a quantity or amount adopted as a standard of measurement, such as dollars, hours, or grams

frequency table

lists the categories in a categorical variable and gives the count or percentage of observations for each category

distribution

gives the possible values of the variable and the relative frequency of each value

area principle

in a statistical display, each data value should be represented by the same amount of area

bar chart

shows a bar representing the count of each category in a categorical variable

pie chart

shows how a "whole" divides into categories by showing a wedge of a circle whose area corresponds to the proportion in each category

contingency table

displays counts and, sometimes, percentages of individuals falling into named categories on two or more variables; categorizes the individuals on all variables at once, to reveal possible patterns in one variable that may be contingent on the category of the other

marginal distribution

the distribution of either variable alone in a contingency table; the counts or percentages are the totals found in the margins (last row or column) of the table

conditional distribution

the distribution of a variable restricting the who to consider only a smaller group of individuals

independence

variables are said to be this if the conditional distribution of one variable is the same for each category of the other

simpson's paradox

when averages are taken across different groups, they can appear to contradict the overall averages

distribution

gives the possible values of the variable and the frequency or relative frequency of each value

histogram

uses adjacent bars to show the distribution of vales in a quantitative variable; each bar represents the frequency (or relative frequency) of values falling in an interval of values

stem-and-leaf display

shows quantitative data values in a way that sketches the distribution of the data

dotplot

graphs a dot for each case against a single axis

shape

to describe this aspect of a distribution, look for single vs. multiple modes, and symmetry vs. skewness

center

a value that attempts the impossible by summarizing the entire distribution with a single number, a "typical" value

spread

a numerical summary of how tightly the values are clustered around the "center"

mode

a hump or local high point in the shape of the distribution of a variable; the apparent locations of these can change as the scale of a histogram is changed

unimodal

having one mode; this is a useful term for describing the shape of a histogram when it's generally mound-shaped

bimodal

distributions with two modes

multimodal

distributions with more than two modes

uniform

a distribution that's roughly flat

symmetric

a distribution is this if the two halves on either side of the center look approximately like mirror images of each other

tails

the parts of a distribution that typically trail off on either side; they can be characterized as long or short

skewed

a distribution is this if it's not symmetric and one tail stretches out farther than the other

outliers

extreme values that don't appear to belong with the rest of the data

timeplot

displays data that change over time

center

summarized with the mean or the median

median

the middle value with half of the data above and half below it

spread

summarized with the standard deviation, interquartile range, and range

range

the difference between the lowest and highest values in a data set

quartile

the lower of this is the value with a quarter of the data below it; the upper of this has a quarter of the data above it

interquartile range

the difference between the first and third quartiles

percentile

the ith ___ is the number that falls above i% of the data

5-number summary

consists of the minimum and maximum, the quartiles Q1 and Q3, and the median

boxplot

displays the 5-number summary as a central box with whiskers that extend to the non-outlying data values

mean

found by summing all the data values and dividing by the count

variance

the sum of squared deviations from the mean, divided by the count minus one

standard deviation

the square root of the variance

comparing distributions

when doing this, consider their shape, center, and spread

shifting

adding a constant to each data value adds the same constant to the mean, the median, and the quartiles, but does not change the standard deviation or IQR

rescaling

multiplying each data value by a constant multiplies both the measures of position and the measures of spread by that constant

standardizing

done to eliminate units; values can be compared and combined even if the original variables had different units and magnitudes

standardized value

value found by subtracting the mean and dividing by the standard deviation

normal model

useful family of models for unimodal, symmetric distributions

parameter

numerically valued attribute of a model

statistic

value calculated from data to summarize aspects of the data

z-score

tells how many standard deviations a value is from the mean; have a mean of zero and a standard deviation of one

standard normal model

a normal model with a mean of 0 and a standard deviation of 1

68-95-99.7 rule

in a normal model, about 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations of the mean, and about 99.7% fall within 3 standard deviations of the mean

normal percentile

this corresponding to a z-score gives the percentage of values in a standard normal distribution found at that z-score or below

normal probability plot

a display to help assess whether a distribution of data is approximately normal; if it is nearly straight, the data satisfy the nearly normal condition

changing center and spread

doing this is equivalent to changing its units

scatterplots

shows the relationship between two quantitative variables measured on the same cases

direction

a positive ____ or association means that, in general, as one variable increases, so does the other; when increases in one variable generally correspond to decreases in the other, the association is negative

form

the ____ we care about most is straight

strength

a scatterplot shows an association that is this if there is little scatter around the underlying relationship

correlation

a numerical measure of the direction and strength of a linear association

outlier

a point that does not fit the overall pattern seen in the scatterplot

lurking variable

a variable other than x and y that simultaneously affects both variables, accounting for the correlation between the two

model

an equation or formula that simplifies and represents reality

linear model

an equation of the form y-hat = b0 + b1x

residuals

the differences between data values and the corresponding values predicted by the regression model; ____ = observed value - predicted value

predicted value

found by substituting the x-value in the regression equation; they're the values on the fitted line

slope

gives a value in "y-units per x-unit"; changes of one unit in x are associated with changes of b1 units in predicted values of y

regression to the mean

each predicted y-hat tends to be fewer standard deviations from its mean than its corresponding x was from its mean

regression line

the linear equation y-hat = b0 + b1x that satisfies the least squares criterion

intercept

this, b0, gives a starting value in y-units; it's the y-hat-value when x is 0

least squares

this criterion specifies the unique line that minimizes the variance of the residuals or, equivalently, the sum of the squared residuals

r2

the square of the correlation between y and x; gives the fraction of the variability of y accounted for by the least squares linear regression on x; an overall measure of how successful the regression is in linearly relating y to x

subset

if data consist of two or more groups that have been thrown together, it is usually best to fit different linear models to each group than to try to fit a single model to all of the data

extrapolation

although linear models provide an easy way to predict values of y for a given value of x, it is unsafe to predict for values of x far from the ones used to find the linear model equation; predictions should not be trusted

outlier

any data point that stands away from the others; can be extraordinary by having a large residual or by having high leverage

leverage

data points whose x-values are far from the mean of x are said to exert ____ on a linear model; with high enough ____, residuals can appear to be deceptively small

influential point

when omitting a point from the data results in a very different regression model, the point is an ____

lurking variable

a variable that is not explicitly part of a model but affects the way the variables in the model appear to be related

re-express data

we do this by taking the logarithm, the square root, the reciprocal, or some other mathematical operation on all values in the data set

ladder of powers

places in order the effects that many re-expressions have on the data

random

an event is this if we know what outcomes could happen, but not which particular values will happen

random numbers

these are hard to generate, but several websites offer an unlimited supply of equally likely random values

simulation

models random events by using random numbers to specify event outcomes with relative frequencies that correspond to the true real-world relative frequencies we are trying to model

simulation component

the most basic situation in a simulation in which something happens at random

outcome

an individual result of a component of a simulation

trial

the sequence of several components representing events that we are pretending will take place

response variable

values of this record the results of each trial with respect to what we were interested in

population

the entire group of individuals or instances about whom we hope to learn

sample

a representative subset of a population, examined in hope of learning about the population

sample survey

a study that asks questions of a sample drawn from some population in the hope of learning something about the entire population

bias

any systematic failure of a sampling method to represent its population; common errors are voluntary response, undercoverage, nonresponse ____, and response ____

randomization

the best defense against bias, in which each individual is given a fair, random chance of selection

matching

any attempt to force a sample to resemble specified attributes of the population

sample size

the number of individuals in a sample

census

a sample that consists of the entire population

population parameter

a numerically valued attribute of a model for a population

representative

a sample is this if the statistics computed from it accurately reflect the corresponding population parameters

simple random sample

this of sample size n is one in which each set of n elements in the population has an equal chance of selection

sampling frame

a list of individuals from whom the sample is drawn

sampling variability

the natural tendency of randomly drawn samples to differ

stratified random sample

a sampling design in which the population is divided into several subpopulations, and random samples are then drawn from each stratum

cluster sample

a sampling design in which entire groups are chosen at random

multistage sample

sampling schemes that combine several sampling methods

systematic sample

a sample drawn by selecting individuals systematically from a sampling frame

voluntary response bias

bias introduced to a sample when individuals can choose on their own whether to participate in the sample

convenience sample

consists of the individuals who are conveniently available

undercoverage

a sampling scheme that biases the sample in a way that gives a part of the population less representation than it has in the population

nonresponse bias

bias introduced to a sample when a large fraction of those sampled fails to respond

response bias

anything in a survey design that influences response

observational study

a study based on data in which no manipulation of factors has been employed

retrospective study

an observational study in which subjects are selected and then their previous conditions or behaviors are determined

prospective study

an observational study in which subjects are followed to observe future outcomes

experiment

manipulates factor levels to create treatments, randomly assigns subjects to these treatment levels, and then compares the responses of the subject groups across treatment levels

random assignment

to be valid, an experiment must assign experimental units to treatment groups at random

factor

a variable whose levels are controlled by the experimenter

response

a variable whose values are compared across different treatments

experimental units

individuals on whom an experiment is performed

level

the specific values that the experimenter chooses for a factor

treatment

the process, intervention, or other controlled circumstance applied to randomly assigned experimental units

principles of experimental design

control, randomize, replicate, block

statistically significant

when an observed difference is too large for us to believe that is is likely to have occurred naturally

control group

the experimental units assigned to a baseline treatment level, typically either the default treatment, which is well understood, or a null, placebo treatment

blinding

any individual associated with an experiment who is not aware of how subjects have been allocated to treatment groups

single-blind

when either those who could influence or evaluate the results is blinded

double-blind

when both those who could influence and evaluate the results are blinded

placebo

a treatment known to have no effect, administered so that all groups experience the same conditions

placebo effect

the tendency of many human subjects (often 20% or more of experiment subjects) to show a response even when administered a placebo

block

when groups of experimental units are similar, it is a good idea to gather them together into these

matched

in a retrospective or prospective study, subjects who are similar in ways not under study may be ____ and then compared with each other on the variables of interest

randomized block design

randomization occurring within blocks

completely randomized design

all experimental units have an equal chance of receiving any treatment

confounded

when the levels of one factor are associated with the levels of another factor so their effects cannot be separated