alpha the probability of a type I error, represented by the
Greek letter $\alpha$
alternative hypothesis a statistical hypothesis that states a difference between a parameter and a specific value or states that there is a difference between two parameters
analysis of variance (ANOVA) a statistical technique used to test a hypothesis concerning the means of three or more populations
Bayes' theorem a theorem that allows you to compute the revised probability of an event that occurred before another event when the events are dependent
beta the probability of a type II error, represented by the
Greek letter $\beta$
biased sample a sample for which some type of systematic error has been made in the selection of subjects for the sample
bimodal a data set with two modes binomial distribution the outcomes of a binomial experiment and the corresponding probabilities of these outcomes
binomial experiment a probability experiment in which each trial has only two outcomes, there are a fixed number of trials, the outcomes of the trials are independent, and the probability of success remains the same for each trial
boxplot a graph used to represent a data set when the data set contains a small number of values
categorical frequency distribution a frequency distribution used when the data are categorical (qualitative)
central limit theorem a theorem that states that as the sample size increases, the shape of the distribution of the sample means taken from the population with mean $\mu$ and standard deviation $\sigma$ will approach a normal distribution; the distribution will have a mean $\mu$ and and standard deviation $\sigma/\sqrt{n}$
Chebyshev's theorem a theorem that states that theproportion of values from a data set that fall within k standard deviations of the mean will be at least $1-\frac{1}{k^2}$, where k is a number greater than 1
chi-square distribution a probability distribution obtained from the values of $(n-1)s^2/\sigma^2$ when random samples are selected from a normally distributed population whose variance is $\sigma^2$
class boundaries the upper and lower values of a class for a grouped frequency distribution whose values have one additional decimal place more than the data and end in the digit 5
class midpoint a value for a class in a frequency distribution obtained by adding the lower and upper class boundaries (or the lower and upper class limits) and dividing by 2
class width the difference between the upper class boundary and the lower class boundary for a class in a frequency distribution
classical probability the type of probability that uses sample spaces to determine the numerical probability that an event will happen
cluster sample a sample obtained by selecting a preexisting or natural group, called a cluster, and using the members in the cluster for the sample
coefficient of determination a measure of the variation of the dependent variable that is explained by the regression line and the independent variable; the ratio of the explained variation to the total variation
coefficient of variation the standard deviation divided by
the mean; the result is expressed as a percentage
combination a selection of objects without regard to order
complement of an event the set of outcomes in the sample space that are not among the outcomes of the event itself
compound event an event that consists of two or more
outcomes or simple events
conditional probability the probability that an event B
occurs after an event A has already occurred
confidence interval a specific interval estimate of a
parameter determined by using data obtained from a sample and the specific confidence level of the estimate
confidence level the probability that a parameter lies
within the specified interval estimate of the parameter
confounding variable a variable that infiuences the
outcome variable but cannot be separated from the other variables that infiuence the outcome variable
continuity (or finite) population correction factor a correction factor used to correct the standard error of the mean when the sample size is greater than 5% of the population size
consistent estimator an estimator whose value approaches the value of the parameter estimated as the sample size increases
contingency table data arranged in table form for the chi-
square independence test, with R rows and C columns
continuous variable a variable that can assume all values between any two specific values. A variable that can take on any value in a connected interval or set of numbers.
control group a group in an experimental study that is not
given any special treatment
convenience sample sample of subjects used because they
are convenient and available
correction for continuity a correction employed when a continuous distribution is used to approximate a discrete distribution a correction employed when a continuous distribution is used to approximate a discrete distribution
correlation a statistical method used to determine whether a linear relationship exists between variables
correlation coefficient a statistic or parameter that measures the strength and direction of a linear relationship between two variables
critical or rejection region the range of values of the test value that indicates that there is a significant difference and the null hypothesis should be rejected in a hypothesis test
critical value (C.V.) a value that separates the critical
region from the noncritical region in a hypothesis test
cumulative frequency the sum of the frequencies accumulated up to the upper boundary of a class in a frequency distribution
data measurements or observations for a variable
data array a data set that has been ordered
data set a collection of data values
decile a location measure of a data value; it divides the distribution into 10 groups
degrees of freedom the number of values that are free to vary after a sample statistic has been computed; used when a distribution (such as the t distribution) consists of a family of curves
dependent events events for which the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed
dependent samples samples in which the subjects are paired or matched in some way; i.e., the samples are related
dependent variable a variable in correlation and regression analysis that cannot be controlled or manipulated
descriptive statistics a branch of statistics that consists of the collection, organization, summarization, and presentation of data
discrete variable a variable that assumes values that can
be counted
empirical probability the type of probability that uses frequency distributions based on observations to determine numerical probabilities of events.
empirical rule a rule that states that when a distribution is bell-shaped (normal), approximately 68% of the data values will fall within 1 standard deviation of the mean; approximately 95% of the data values will fall within 2 standard deviations of the mean; and approximately 99.7% of the data values will fall within 3 standard deviations of the mean<
equally likely events the events in the sample space that
have the same probability of occurring
estimation the process of estimating the value of a
parameter from information obtained from a sample
estimator a statistic used to estimate a parameter
event outcome of a probability experiment
expected value the theoretical average of a variable that has a probability distribution
expected frequency the frequency obtained by calculation (as if there were no preference) and used in the chi-square test
experimental study a study in which the researcher manipulates one of the variables and tries to determine how the manipulation influences other variables
explanatory variable a variable that is being manipulated
by the researcher to see if it affects the outcome variable
exploratory data analysis the act of analyzing data to determine what information can be obtained by using stem and leaf plots, medians, interquartile ranges, and boxplots
extrapolation use of the equation for the regression line to predict y' for a value of x which is beyond the range of the data values of x
F distribution the sampling distribution of the variances when two independent samples are selected from two normally distributed populations in which the variances are equal and the variances $s_1^2$ and $s_2^2$ are compared as $s_1^2 \div s_2^2$
F test a statistical test used to compare two variances or
three or more means factors the independent variables in ANOVA tests
five-number summary five specific values for a data set that consist of the lowest and highest values, Q1 and Q3, and the median
frequency the number of values in a specific class of a
frequency distribution
frequency distribution an organization of raw data in
table form, using classes and frequencies
frequency polygon a graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes
goodness-of-fit test a chi-square test used to see whether a
frequency distribution fits a specific pattern
grouped frequency distribution a distribution used when the range is large and classes of several units in width are needed
histogram a graph that displays the data by using vertical bars of various heights to represent the frequencies of a distribution
homogeneity of proportions test a test used to determine the equality of three or more proportions
hypergeometric distribution the distribution of a variable that has two outcomes when sampling is done without replacement
hypothesis testing a decision-making process for evaluating claims about a population
independence test a chi-square test used to test the independence of two variables when data are tabulated in table form in terms of frequencies
independent events events for which the probability of the first occurring does not affect the probability of the second occurring
independent samples samples that are not related
independent variable a variable in correlation and
regression analysis that can be controlled or manipulated
inferential statistics a branch of statistics that consists of generalizing from samples to populations, performing hypothesis testing, determining relationships among variables, and making predictions
influential observation an observation which when removed from the data values would markedly change the position of the regression line
interaction effect the effect of two or more variables on each other in a two-way ANOVA study
interquartile range $Q_3 - Q_1$ (see "quartiles" definition). The range of the middle 50% of the data.
interval estimate a range of values used to estimate a
parameter
interval level of measurement a measurement level that ranks data and in which precise differences between units of measure exist. See also nominal, ordinal, and ratio levels of measurement
Kruskal-Wallis test a nonparametric test used to compare three or more means
law of large numbers when a probability experiment is repeated a large number of times, the relative frequency probability of an outcome will approach its theoretical probability
least-squares line another name for the regression line
left-tailed test a test used on a hypothesis when the critical
region is on the left side of the distribution
level a treatment in ANOVA for a variable
level of significance the maximum probability of committing a type I error in hypothesis testing
lower class limit the lower value of a class in a frequency distribution that has the same decimal place value as the data
lurking variable a variable that influences the relationship between x and y, but was not considered in the study
marginal change the magnitude of the change in the dependent variable when the independent variable changes 1 unit
maximum error of estimate the maximum likely difference between the point estimate of a parameter and the actual value of the parameter
mean the sum of the values, divided by the total number
of values. The formula is $\bar{x}=\frac{\sum x}{n}$ if the data is from a sample and $\mu=\frac{\sum x}{N}$ if the data includes all measurements in the population.
mean square the variance found by dividing the sum of the squares of a variable by the corresponding degrees of freedom; used in ANOVA
measurement scales a type of classification that tells how variables are categorized, counted, or measured; the four types of scales are nominal, ordinal, interval, and ratio
median the midpoint of a data array
midrange the sum of the lowest and highest data values,
divided by 2
modal class the class with the largest frequency
mode the value that occurs most often in a data set Monte Carlo method a simulation technique using
random numbers
multimodal a data set with three or more modes
multinomial distribution a probability distribution for an experiment in which each trial has more than two outcomes
multiple correlation coefficient a measure of the strength of the relationship between the independent variables and the dependent variable in a multiple regression study
multiple regression a study that seeks to determine if several independent variables are related to a dependent variable
multiple relationship a relationship in which many
variables are under study
multistage sampling a sampling technique that uses a
combination of sampling methods
mutually exclusive events probability events that cannot occur at the same time
negative relationship a relationship between variables such that as one variable increases, the other variable decreases, and vice versa
negatively skewed or left-skewed distribution a distribution in which the majority of the data values fall to the right of the mean
nominal level of measurement a measurement level that classifies data into mutually exclusive (nonoverlapping) exhaustive categories in which no order or ranking can be imposed on them. See also interval, ordinal, and ratio levels of measurement
noncritical or nonrejection region the range of values of the test value that indicates that the difference was probably due to chance and the null hypothesis should not be rejected
nonparametric statistics a branch of statistics for use when the population from which the samples are selected is not normally distributed and for use in testing hypotheses that do not involve specific population parameters
nonrejection region see noncritical region
normal distribution a continuous, symmetric, bell-shaped
distribution of a variable
normal quantile plot graphical plot used to determine
whether a variable is approximately normally distributed
null hypothesis a statistical hypothesis that states that there is no difference between a parameter and a specific value or that there is no difference between two parameters
observational study a study in which the researcher merely observes what is happening or what has happened in the past and draws conclusions based on these observations
observed frequency the actual frequency value obtained
from a sample and used in the chi-square test
ogive a graph that represents the cumulative frequencies
for the classes in a frequency distribution
one-tailed test a test that indicates that the null hypothesis should be rejected when the test statistic value is in the critical region on one side of the mean
one-way ANOVA a study used to test for differences among means for a single independent variable when there are three or more groups
open-ended distribution a frequency distribution that has
no specific beginning value or no specific ending value
ordinal interaction an interaction between variables in
ANOVA, indicated when the graphs of the lines connecting the means do not intersect
ordinal level of measurement a measurement level that classifies data into categories that can be ranked; however, precise differences between the ranks do not exist. See also interval, nominal, and ratio levels of measurement
outcome the result of a single trial of a probability
experiment
outcome variable a variable that is studied to see if it has changed significantly due to the manipulation of the explanatory variable
outlier an extreme value in a data set; it is omitted from a boxplot
parameter a characteristic or measure obtained by using
all the data values for a specific population
parametric tests statistical tests for population parameters such as means, variances, and proportions that involve assumptions about the populations from which the samples were selected
Pareto chart chart that uses vertical bars to represent
frequencies for a categorical variable
Pearson product moment correlation coefficient (PPMCC) a statistic used to determine the strength of a relationship when the variables are normally distributed
Pearson's index of skewness value used to determine the
degree of skewness of a variable
percentile a location measure of a data value; it divides the
distribution into 100 groups
permutation an arrangement of n objects in a specific order
pie graph a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution
point estimate a specific numerical value estimate of a
parameter
Poisson distribution a probability distribution used when n is large and p is small and when the independent variables occur over a period of time
positive relationship a relationship between two variables such that as one variable increases, the other variable increases or as one variable decreases, the other decreases
population the totality of all subjects possessing certain
common characteristics that are being studied
population correlation coefficient the value of the correlation coefficient computed by using all possible pairs of data values (x, y) taken from a population
positively skewed or right-skewed distribution a distribution in which the majority of the data values fall to the left of the mean
power of a test the probability of rejecting the null
hypothesis when it is false
prediction interval a confidnece interval for a predicted
value y
probability the chance of an event occurring
probability distribution the values a random variable can
assume and the corresponding probabilities of the values
probability experiment a chance process that leads to
well-defined results called outcomes
proportion a part of a whole, represented by a fraction, a
decimal, or a percentage
P-value the actual probability of getting the sample mean value if the null hypothesis is true
qualitative variable a variable that can be placed into distinct categories, according to some characteristic or attribute
quantiles values that separate the data set into
approximately equal groups
quantitative variable a variable that is numerical in nature
and that can be ordered or ranked
quartile a location measure of a data value; it divides the distribution into four groups
quasi-experimental study a study that uses intact groups rather than random assignment of subjects to groups
random sample a sample obtained by using random or chance methods; a sample for which every member of the population has an equal chance of being selected
random variable a variable whose values are determined
by chance
range the highest data value minus the lowest data value
range rule of thumb dividing the range by 4, given an
approximation of the standard deviation
ranking the positioning of a data value in a data array
according to some rating scale
ratio level of measurement a measurement level that possesses all the characteristics of interval measurement and a true zero; it also has true ratios between different units of measure. See also interval, nominal, and ordinal levels of measurement
raw data data collected in original form
regression a statistical method used to describe the nature of the relationship between variables, that is, a positive or negative, linear or nonlinear relationship
regression line the line of best fit of the data rejection region see critical region
relative frequency graph a graph using proportions
instead of raw data as frequencies
relatively efficient estimator an estimator that has the smallest variance from among all the statistics that can be used to estimate a parameter
residual the difference between the actual value of y and
the predicted value y' for a specific value of x
resistant statistic a statistic that is not affected by an
extremely skewed distribution
right-tailed test a test used on a hypothesis when the
critical region is on the right side of the distribution
run a succession of identical letters preceded by or
followed by a different letter or no letter at all, such as the beginning or end of the succession
runs test a nonparametric test used to determine whether data are random
sample a group of subjects selected from the population
sample space the set of all possible outcomes of a
probability experiment
sampling distribution of sample means a distribution
obtained by using the means computed from random samples taken from a population
sampling error (sampling variability) the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population. The variability that exists from one sample to another.
scatter plot a graph of the independent and dependent
variables in regression and correlation analysis
sequence sampling a sampling technique used in quality control in which successive units are taken from production lines and tested to see whether they meet the standards set by the manufacturing company
sign test a nonparametric test used to test the value of the median for a specific sample or to test sample means in a comparison of two dependent samples
simple event an outcome that results from a single trial of
a probability experiment
simple relationship a relationship in which only two
variables are under study
simulation techniques techniques that use probability
experiments to mimic real-life situations
Spearman rank correlation coefficient the nonparametric equivalent to the correlation coefficient, used when the data are ranked
standard deviation the square root of the variance standard error of the estimate the standard deviation of the observed y values about the predicted y' values in regression and correlation analysis
standard error of the mean the standard deviation of the sample means for samples taken from the same population
standard normal distribution a normal distribution for which the mean is equal to 0 and the standard deviation is equal to 1
standard score the difference between a data value and the
mean, divided by the standard deviation
statistic a characteristic or measure obtained by using the
data values from a sample
statistical hypothesis a conjecture about a population
parameter, which may or may not be true
statistical test a test that uses data obtained from a sample to make a decision about whether the null hypothesis should be rejected
statistics the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data
stem and leaf plot a data plot that uses part of a data value as the stem and part of the data value as the leaf to form groups or classes
stratified sample a sample obtained by dividing the population into subgroups, called strata, according to various homogeneous characteristics and then selecting members from each stratum
subjective probability the type of probability that uses a probability value based on an educated guess or estimate, employing opinions and inexact information
sum of squares between groups a statistic computed in the numerator of the fraction used to find the between-group variance in ANOVA
sum of squares within groups a statistic computed in the numerator of the fraction used to find the within-group variance in ANOVA
symmetric distribution a distribution in which the data
values are uniformly distributed about the mean
systematic sample a sample obtained by numbering each element in the population and then selecting every kth number from the population to be included in the sample
t distribution a family of bell-shaped curves based on degrees of freedom, similar to the standard normal distribution with the exception that the variance is greater than 1; used when you are testing small samples and when the population standard deviation is unknown
t test a statistical test for the mean of a population, used when the population is normally distributed and the population standard deviation is unknown
test value the numerical value obtained from a statistical test, computed from (observed value - expected value) / standard error
time series graph a graph that represents data that occur
over a specific time
treatment group a group in an experimental study that has
received some type of treatment
treatment groups the groups used in an ANOVA study
tree diagram a device used to list all possibilities of a
sequence of events in a systematic way
Tukey test a test used to make pairwise comparisons of means in an ANOVA study when samples are the same size
two-tailed test a test that indicates that the null hypothesis should be rejected when the test value is in either of the two critical regions
two-way ANOVA a study used to test the effects of two or more independent variables and the possible interaction between them
type I error the error that occurs if you reject the null
hypothesis when it is true
type II error the error that occurs if you do not reject the null hypothesis when it is false
unbiased estimator an estimator whose value approximates the expected value of a population parameter, used for the variance or standard deviation when the sample size is less than 30; an estimator whose expected value or mean must be equal to the mean of the parameter being estimated
unbiased sample a sample chosen at random from the population that is, for the most part, representative of the population
ungrouped frequency distribution a distribution that uses
individual data and has a small range of data
uniform distribution a distribution whose values are
evenly distributed over its range
upper class limit the upper value of a class in a frequency distribution that has the same decimal place value as the data
variable a characteristic or attribute that can assume different values
variance the average of the squares of the distance that
each value is from the mean
Venn diagram a diagram used as a pictorial representative for a probability concept or rule
weighted mean the mean found by multiplying each value by its corresponding weight and dividing by the sum of the weights
Wilcoxon rank sum test a nonparametric test used to test
independent samples and compare distributions
Wilcoxon signed-rank test a nonparametric test used to
test dependent samples and compare distributions
within-group variance a variance estimate using all the sample data for an F test; it is not affected by differences in the means
z distribution see standard normal distribution
z score see standard score population, used when the population is normally distributed and the population standard deviation is known
z test a statistical test for means and proportions of a population, used when the population is normally
distributed and the population standard deviation is
known