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09

01 box

A box of numbered tickets, in which each ticket is numbered either 0 or 1.
See box model.

A

Affine transformation.

See transformation.

Affirming the antecedent.

A valid logical argument that concludes from the premise
A → B
and the premise A that therefore, B
is true.
The name comes from the fact that the argument affirms
(i.e., asserts as true) the
antecedent (A) in the
conditional.

Affirming the consequent.

A logical fallacy that argues from the premise
A → B
and the premise B that therefore, A is true.
The name comes from the fact that the argument affirms
(i.e., asserts as true) the
consequent (B) in the
conditional.

Alternative Hypothesis.

In hypothesis testing, a null
hypothesis (typically that there is no effect) is compared with an alternative
hypothesis (typically that there is an effect, or that there is an effect of a particular
sign).
For example, in evaluating whether a new cancer remedy works, the null hypothesis
typically would be that the remedy does not work, while the alternative
hypothesis would be that the remedy does work.
When the data are sufficiently improbable under the assumption that the null
hypothesis is true, the null hypothesis is rejected in favor of the alternative
hypothesis. (This does not imply that the data are probable under the assumption that the
alternative hypothesis is true, nor that the null hypothesis is false, nor that the
alternative hypothesis is true. Confused? Take a course in Statistics!)

and, &, conjunction, logical conjunction, ∧.

An operation on two logical propositions.
If p and q are two
propositions,
(p & q) is a proposition that
is true if both p and q
are true; otherwise, (p & q) is false.
The operation & is sometimes represented by the symbol ∧.

Ante.

The upfront cost of a bet: the money you must pay to play the game. From Latin for
"before."

Antecedent.

In a conditional p → q,
the antecedent is p.

Appeal to Ignorance.

A logical fallacy: taking the absence of evidence to be evidence of absence.
If something is not known to be false, assume that it is true; or if something is not known to
be true, assume that it is false.
For example, if I have no reason to think that anyone in Tajikistan wish me well,
that is not evidence that nobody in Tajikistan wishes me well.

Applet.

An applet is a small program that is automatically downloaded from a website to your
computer when you visit a particular web page; it allows a page to be interactive—to
respond to your input. The applet runs on your computer, not the computer that hosted the
web page. These materials contain many applets to illustrate
statistical concepts and to help you to analyze data.
Many of them are accessible directly from the
tools page.

Association.

Two variables are associated if some of the variability of one
can be accounted for by the other. In a scatterplot of the two
variables, if the scatter in the values of the variable plotted on the vertical axis is
smaller in narrow ranges of the variable plotted on the horizontal axis (i.e., in
vertical "slices") than it is overall, the two variables are associated.
The correlation coefficient is a measure of linear
association,
which is a special case of association in which large values of one variable tend to occur
with large values of the other, and small values of one tend to occur with small values of
the other (positive association), or in which large values of one tend to occur with small
values of the other, and vice versa (negative association).

Average.

A sometimes vague term. It usually denotes the arithmetic mean, but
it can also denote the median, the mode, the
geometric mean, and weighted means, among other things.

Axioms of Probability.

There are three axioms of probability: (1) Chances are always at least zero. (2) The
chance that something happens is 100%. (3) If two events cannot both occur at the
same time (if they are disjoint or mutually exclusive), the chance
that either one occurs is the sum of the chances that each occurs. For example, consider
an experiment that consists of tossing a coin once. The first axiom says that the chance
that the coin lands heads, for instance, must be at least zero. The second axiom says that
the chance that the coin either lands heads or lands tails or lands on its edge or doesn't
land at all is 100%. The third axiom says that the chance that the coin either lands heads
or lands tails is the sum of the chance that the coin lands heads and the chance that the
coin lands tails, because both cannot occur in the same coin toss. All other mathematical
facts about probability can be derived from these three axioms. For example, it is true
that the chance that an event does not occur is (100% − the chance that the event occurs).
This is a consequence of the second and third axioms.


B

Base rate fallacy.

The base rate fallacy consists of failing to take into account prior probabilities (base rates) when
computing conditional probabilities
from other conditional probabilities. It is related to
the Prosecutor's Fallacy.
For instance, suppose that a test for the presence of some condition has a 1% chance of a
false positive result (the test says the condition is present when it is not) and a 1% chance
of a false negative result (the test says the condition is absent when the condition is present),
so the exam is 99% accurate.
What is the chance that an item that tests positive really has the condition?
The intuitive answer is 99%, but that is not necessarily true: the correct answer depends on the
fraction f of items in the population that have the condition
(and on whether the item tested is selected at random from the population).
The chance that a randomly selected item tests positive is
0.99×f/(0.99×f + 0.01×(1−f)),
which could be much smaller than
99% if f is small.
See Bayes' Rule.

Bayes' Rule.

Bayes' rule expresses the conditional
probability of the event
A given the event
B
in terms of the conditional probability
of the event
B given the event A
and the unconditional probability of A:
P(AB) = P(BA) ×P(A)/(
P(BA)×P(A) + P(BA^{c})
×P(A^{c})
).
In this expression, the unconditional probability of A is also
called the prior probability of A,
because it is the probability assigned to A prior to observing
any data.
Similarly, in this context, P(AB) is called the
posterior probability of A given
B, because it is the probability of A
updated to reflect (i.e., to condition on) the fact that B was observed
to occur.

Bernoulli's Inequality.

The Bernoulli Inequality says that if x ≥ −1 then
(1+x)^{n} ≥ 1 + nx
for every integer n ≥ 0.
If n is even, the inequality holds for all x.

Bias.

A measurement procedure or estimator is said to be biased if,
on the average, it gives an answer that differs from the truth.
The bias is the average (expected) difference between the
measurement and the truth. For
example, if you get on the scale with clothes on, that biases the measurement to be larger
than your true weight (this would be a positive bias). The design of an experiment or of a
survey can also lead to bias. Bias can be deliberate, but it is not necessarily so. See
also nonresponse bias.

Bimodal.

Having two modes.

Bin.

See class interval.

Binomial Coefficient.

See combinations.

Binomial Distribution.

A random variable has a binomial distribution (with parameters
n and p) if
it is the number of "successes" in a fixed number n of
independent random trials, all of which have the same
probability p
of resulting in "success." Under these assumptions, the probability of k
successes (and n−k failures) is
_{n}C_{k} p^{k}(1−p)^{n−k},
where _{n}C_{k} is the number of
combinations
of n objects taken k at a time:
_{n}C_{k} = n!/(k!(n−k)!).
The expected value of a
random
variable with the Binomial distribution is n×p,
and the standard error of a
random variable with the Binomial distribution is
(n×p×(1
− p))^{½}.
This page
shows the probability histogram of the binomial
distribution.

Binomial Theorem.

The Binomial theorem says that (x+y)^{n} = x^{n} + nx^{n−1}y +
… + _{n}C_{k}x^{n−k}y^{k} + … + y^{n}.

Bivariate.

Having or having to do with two variables.
For example, bivariate data are data where we
have two measurements of each "individual." These measurements might be the
heights and weights of a group of people (an "individual" is a person), the
heights of fathers and sons (an "individual" is a fatherson pair), the pressure
and temperature of a fixed volume of gas (an "individual" is the volume of gas
under a certain set of experimental conditions), etc.
Scatterplots,
the correlation coefficient,
and regression
make sense for bivariate data but not univariate data.
C.f.
univariate.

Blind, Blind Experiment.

In a blind experiment, the subjects do not know whether they are
in the treatment group or the
control
group. In order to have a blind experiment with human subjects, it is usually
necessary to administer a placebo to the control group.

Bootstrap estimate of Standard Error.

The name for this idea comes from the idiom "to pull oneself up by one's
bootstraps," which connotes getting out of a hole without anything to stand on.
The idea of the bootstrap is to assume, for the purposes of estimating uncertainties,
that the sample is the population, then use the SE for sampling from the
sample to estimate the SE of sampling from the population.
For sampling from a box of numbers,
the SD of the sample is the bootstrap estimate of the SD of the box from which the
sample is drawn.
For sample percentages, this takes a particularly
simple form:
the SE of the sample percentage
of n
draws from a box, with replacement, is
SD(box)/n^{½},
where for a box that contains only zeros and ones, SD(box) =
((fraction
of ones in box)×(fraction of zeros in box)
)^{½}.
The bootstrap estimate
of the SE of the sample percentage
consists of estimating SD(box) by
((fraction of ones in sample)×(fraction
of zeros in sample))^{½}.
When the sample size is large, this approximation is
likely to be good.

Box model.

An analogy between an experiment and drawing numbered tickets "at random" from
a box with replacement. For example, suppose we are trying to evaluate a cold remedy by
giving it or a placebo to a group of n individuals, randomly choosing half the
individuals to receive the remedy and half to receive the placebo. Consider the median
time to recovery for all the individuals (we assume everyone recovers from the cold
eventually; to simplify things, we also assume that no one recovered in exactly the median
time, and that n is even). By definition, half the individuals got better in less
than the median time, and half in more than the median time. The individuals who received
the treatment are a random sample of
size
n/2 from the set of n subjects, half of whom got better in less than
median time, and half in longer than median time. If the remedy is ineffective, the number
of subjects who received the remedy and who recovered in less than median time is like the
sum of n/2 draws with replacement from a box with two tickets in it: one with a
"1" on it, and one with a "0" on it.
This page illustrates
the sampling distribution of random draws with or without from a box of numbered tickets.

Breakdown Point.

The breakdown point of an estimator is the smallest fraction of
observations one must corrupt to make the estimator take any value one wants.

C

Categorical Variable.

A variable whose value ranges over categories, such as {red,
green, blue}, {male, female}, {Arizona, California, Montana, New York}, {short, tall},
{Asian, AfricanAmerican, Caucasian, Hispanic, Native American, Polynesian}, {straight,
curly}, etc. Some categorical variables are ordinal. The
distinction between categorical variables and
qualitative variables
is a bit blurry. C.f. quantitative variable.

Causation, causal relation.

Two variables are causally related if changes in the value of one cause the other to
change. For example, if one heats a rigid container filled with a gas, that causes the
pressure of the gas in the container to increase.
Two variables can be associated without
having any causal relation, and even if two
variables have a causal relation, their correlation can be
small or zero.

Central Limit Theorem.

The central limit theorem states that the probability
histograms of the sample mean
and sample sum of n draws with replacement
from a box of labeled tickets converge to a
normal curve as the
sample size n grows, in the following sense:
As n grows, the area of the probability histogram for any
range of values approaches the area under the normal curve
for the same range of values, converted to standard units.
See also the normal approximation.

Certain Event.

An event is certain if its
probability is 100%.
Even if an event is certain, it might not occur.
However, by the complement rule,
the chance that it does not occur is 0%.

Chance variation, chance error.

A random variable can be decomposed into
a sum of its expected value and chance variation around
its expected value. The expected value of the chance variation is zero; the
standard error of the chance variation is the same as the
standard error of the random variable—the size of a
"typical" difference between the random variable
and its expected value.
See also sampling error.

Change of Units or Variables.

See also transformation.

Chebychev's Inequality.

For lists: For every number k>0, the fraction of elements in a list that are
k SD's or further from the
arithmetic mean of
the list is at most 1/k^{2}.
For random variables:
For every number k>0, the
probability that a random variable X is k
SEs or further from its expected value is at
most 1/k^{2}.

Chisquare curve.

The chisquare curve is a family of curves that depend on a parameter called
degrees of freedom (d.f.).
The chisquare curve is an approximation to the
probability histogram of the
chisquare statistic
for multinomial model if the
expected number of outcomes in each category is
large.
The chisquare curve is positive, and its total area is 100%, so we can think of
it as the probability histogram of a random variable.
The balance point of the curve is d.f., so the expected value of the
corresponding random variable would equal d.f..
The standard error of the corresponding random variable would be
(2×d.f.)^{½}.
As d.f. grows, the shape of the chisquare curve approaches the shape of
the normal curve.
This page shows
the chisquare curve.

Chisquare Statistic.

The chisquare statistic is used to measure the agreement between
categorical data and a
multinomial model that predicts
the relative frequency of outcomes in each possible category.
Suppose there are n independent trials,
each of which can result in one of k possible outcomes.
Suppose that in each trial, the probability that outcome
i occurs is p_{i},
for i = 1, 2, … , k,
and that these probabilities are the same in every trial.
The expected number of times outcome 1 occurs in the n trials is
n×p_{1}; more generally, the expected number of
times outcome i occurs is
expected_{i} = n×p_{i}.

If the model be correct, we would expect the n trials to result in outcome
i about n×p_{i} times, give or take
a bit.
Let observed_{i} denote the number of times an outcome of type i
occurs in the n trials, for i = 1, 2,
… , k.
The chisquared statistic summarizes the discrepancies between the
expected number of times each outcome occurs (assuming that the model is true)
and the observed number of times each outcome occurs, by summing
the squares of the discrepancies, normalized by the expected numbers, over all
the categories:
chisquared =
(observed_{1} − expected_{1})^{2}/expected_{1}
+
(observed_{2} − expected_{2})^{2}/expected_{2}
+
…
+
(observed_{k} − expected_{k})^{2}/expected_{k}.

As the sample size n increases, if the model is correct,
the sampling distribution of the chisquared statistic
is approximated increasingly well by the chisquared curve with
(#categories − 1) = k − 1

degrees of
freedom (d.f.), in the sense that the chance that the chisquared statistic
is in any given range grows closer and closer to the area under the ChiSquared curve over
the same range.
This page illustrates
the sampling distribution of the chisquare statistic.

Class Boundary.

A point that is the left endpoint of one class interval,
and the right endpoint of another class interval.

Class Interval.

In plotting a histogram, one starts by dividing the range of
values into a set of nonoverlapping intervals, called class intervals, in such a
way that every datum is contained in some class interval.
See the related entries class boundary and
endpoint
convention.

Cluster Sample.

In a cluster sample, the sampling unit is a
collection of population units, not single population units.
For example, techniques for adjusting the U.S. census start with a sample of
geographic blocks, then
(try to) enumerate all inhabitants of the blocks in the sample to obtain a sample
of people.
This is an example of a cluster sample.
(The blocks are chosen separately from different strata, so the overall design is a
stratified cluster sample.)

Combinations.

The number of combinations of n things taken k at a time is the number
of ways of picking a subset of k of the n things, without replacement,
and without regard to the order in which the elements of the subset are picked.
The number
of such combinations is _{n}C_{k} =
n!/(k!(n−k)!),
where k! (pronounced "k factorial")
is k×(k−1)×(k−2)× … × 1.
The numbers _{n}C_{k}
are also called the Binomial coefficients. From a set that has n
elements one can form a total of 2^{n} subsets of all sizes. For example,
from the set {a, b, c}, which has 3 elements, one can form the 2^{3} = 8 subsets
{}, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}.
Because the number of subsets with k
elements one can form from a set with n
elements is _{n}C_{k},
and the total number of subsets of a set is the sum of the numbers of possible subsets of
each size, it follows that
_{n}C_{0}+_{n}C_{1}+_{n}C_{2}+
… +_{n}C_{n} = 2^{n}.
The calculator
has a button (nCm) that lets you compute the number of combinations of
m things chosen from a set of n things.
To use the button, first
type the value of n, then push the nCm button, then type the value of m,
then press the "=" button.

Complement.

The complement of a subset of a given set is
the collection of all elements of the set that
are not elements of the subset.

Complement rule.

The probability of the complement of an event
is 100% minus the probability of the event: P(A^{c}) = 100% − P(A).

Compound proposition.

A logical proposition formed from other
propositions using logical operations such as
!, , XOR,
&,
→ and ↔.

Conditional Probability.

Suppose we are interested in the probability that some event A
occurs, and we learn that the event B occurred. How should we update
the probability of A to reflect this new knowledge? This is what the conditional
probability does: it says how the additional knowledge that B occurred should affect the
probability that A occurred quantitatively. For example, suppose that A and B are
mutually exclusive. Then if B occurred, A did not, so the
conditional
probability that A occurred given that B occurred is zero. At the other extreme,
suppose that B is a subset of A, so that A must occur whenever B
does. Then if we learn that B occurred, A must have occurred too, so the conditional
probability that A occurred given that B occurred is 100%. For inbetween cases,
where A and B intersect, but B is not a subset of A, the conditional
probability of A given B is a number between zero and 100%. Basically, one
"restricts" the outcome space S to
consider only the part of S that is in B, because we know that B
occurred. For A to have happened given that B happened requires that
AB happened, so we are interested in the event
AB. To have a legitimate probability requires that
P(S)
= 100%, so if we are restricting the outcome space to B, we need to divide by the
probability of B to make the probability of this new S be 100%. On this
scale, the probability that AB happened is P(AB)/P(B). This is the definition of the
conditional probability of A given B, provided P(B) is not zero (division by zero is
undefined). Note that the special cases AB = {} (A and B are mutually
exclusive) and AB = B (B is a subset of A) agree with our
intuition as described at the top of this paragraph. Conditional probabilities satisfy the
axioms of probability, just as ordinary probabilities
do.

Confidence Interval.

A confidence interval for a parameter is a random interval
constructed from data in such a way that the probability that the interval contains the
true value of the parameter can be specified before the data are collected.
Confidence intervals are demonstrated in this
page.

Confidence Level.

The confidence level of a confidence interval is the
chance that the interval that will result once data are collected will contain the
corresponding parameter. If one computes confidence intervals
again and again from independent data, the longterm limit of the fraction of intervals
that contain the parameter is the confidence level.

Confounding.

When the differences between the treatment and
control groups other than the treatment produce differences in
response that are not distinguishable from the effect of the
treatment,
those differences between the groups are said to be confounded with the effect of
the treatment (if any). For example, prominent statisticians questioned whether
differences between individuals that led some to smoke and others not to (rather than the
act of smoking itself) were responsible for the observed difference in the frequencies
with which smokers and nonsmokers contract various illnesses. If that were the case,
those factors would be confounded with the effect of smoking. Confounding is quite likely
to affect observational studies and
experiments
that are not randomized.
Confounding tends to be decreased by randomization.
See also Simpson's Paradox.

Continuity Correction.

In using the normal approximation to the
binomial probability histogram,
one can get more accurate answers by finding the area under the normal curve corresponding
to halfintegers, transformed to standard units.
This is clearest if we are seeking the chance of a particular number of successes.
For example, suppose we seek to approximate the chance of 10 successes in 25
independent
trials, each with probability p = 40% of success.
The number of successes in this
scenario has a binomial distribution with parameters n =
25 and p = 40%. The expected
number of successes is np
= 10, and the standard error is
(np(1−p))^{½}
= 6^{½} = 2.45. If we consider the area under the
normal
curve at the point 10 successes, transformed to standard
units, we get zero: the area under a point is always zero. We get a better
approximation by considering 10 successes to be the range from 9 1/2 to 10 1/2 successes.
The only possible number of successes between 9 1/2 and 10 1/2 is 10, so this is exactly
right for the binomial distribution.
Because the
normal curve is
continuous
and a binomial random variable
is discrete, we need to "smear out"
the binomial
probability over an appropriate range. The lower endpoint of the range, 9 1/2 successes,
is (9.5 − 10)/2.45 = −0.20 standard units.
The upper endpoint of the range, 10 1/2 successes, is (10.5 − 10)/2.45 = +0.20
standard units.
The area under the normal
curve between −0.20 and +0.20 is about 15.8%.
The true binomial
probability is
_{25}C_{10}×(0.4)^{10}×(0.6)^{15}
= 16%. In a similar way, if we seek the normal
approximation to the probability that a binomial
random variable is in the range from
i successes to k
successes, inclusive, we should find the area under the normal
curve from i−1/2 to k+1/2 successes, transformed to
standard units.
If we seek the probability of more than i
successes and fewer than k successes, we should find the area under
the normal curve corresponding to the range
i+1/2 to k−1/2
successes, transformed to standard units. If we seek the
probability of more than i but no more than k successes, we should find
the area under the normal curve corresponding to
the range i+1/2
to k+1/2 successes, transformed to
standard units.
If we seek the probability of at least i but fewer than k successes, we
should find the area under the normal curve corresponding to
the range i−1/2 to k−1/2 successes, transformed to
standard units.
Including or excluding the halfinteger ranges
at the ends of the interval in this manner is called the continuity correction.

Consequent.

In a conditional p → q,
the consequent is q.

Continuous Variable.

A quantitative variable is continuous if its set of
possible values is uncountable. Examples include temperature, exact height, exact age
(including parts of a second). In practice, one can never measure a continuous variable to
infinite precision, so continuous variables are sometimes approximated by
discrete variables.
A random variable
X is also called continuous if its set of possible values is uncountable, and the
chance that it takes any particular value is zero (in symbols, if P(X = x) = 0
for every real number x). A random variable is continuous if and
only if its cumulative probability distribution function
is a continuous function (a function with no jumps).

Contrapositive.

If p and q are two logical propositions,
then the contrapositive of the proposition
(p → q)
is the proposition
((! q) →
(!p) ).
The contrapositive is logically equivalent
to the original proposition.

Control.

There are at least three senses of "control" in statistics: a
member of the control
group, to whom no treatment is given;
a controlled experiment, and to
control
for a possible confounding variable.

Controlled experiment.

An experiment that uses the method of
comparison to evaluate the effect of a
treatment
by comparing treated subjects with a control group, who do not
receive the treatment.

Controlled, randomized experiment.

A controlled experiment in which the
assignment of subjects to the treatment
group or control group is done at random, for example,
by tossing a coin.

Control for a variable.

To control for a variable is to try to separate its effect from the treatment
effect, so it will not confound with the treatment.
There are many methods that try to control for variables.
Some are based on matching individuals between treatment and control; others
use assumptions about the nature of the effects of the variables to try
to model the effect mathematically, for example, using regression.

Control group.

The subjects in a controlled
experiment who do not receive the treatment.

Convenience Sample.

A sample drawn because of its convenience; it is not a
probability
sample.
For example, I might take a sample of opinions in Berkeley (where I live) by
just asking my 10 nearest neighbors. That would be a sample of convenience, and would be
unlikely to be representative of all of Berkeley.
Samples of convenience are not typically representative, and it is not possible to quantify
how unrepresentative results based on samples of convenience are likely to be.
Convenience samples are to be avoided, and results based on convenience samples are to be
viewed with suspicion.
See also quota sample.

Converge, convergence.

A sequence of numbers x_{1}, x_{2},
x_{3}
… converges if there is a number
x such that for any number
E>0,
there is a number k (which can depend on E) such that
x_{j} − x < E whenever j >
k. If such a number x exists, it is called the
limit of the sequence x_{1},
x_{2}, x_{3} … .

Convergence in probability.

A sequence of random variables
X_{1}, X_{2}, X_{3}
… converges in probability if there is a random
variable X such that for any number E>0, the sequence of numbers
P(X_{1} − X < e), P(X_{2} − X < e),
P(X_{3} − X < e),
…

converges to 100%.

Converse.

If p and q are two logical propositions,
then the converse of the proposition
(p → q)
is the proposition (q → p).

Correlation.

A measure of linear association
between two (ordered) lists.
Two variables can be strongly correlated without having any causal
relationship, and two variables can have a causal
relationship and yet be uncorrelated.

Correlation coefficient.

The correlation coefficient r is a measure of how nearly a
scatterplot
falls on a straight line. The correlation coefficient is always between −1 and +1. To
compute the correlation coefficient of a list of pairs of measurements (X,Y),
first transform X and Y individually into
standard
units.
Multiply corresponding elements of the transformed pairs to get a single list
of numbers.
The correlation coefficient is the mean of that list of
products.
This page
contains a tool that lets you generate bivariate
data with any correlation coefficient you want.

Counting.

To count a set of things is to put it in one to one correspondence with a consecutive subset of the
positive integers, starting with 1.

Countable Set.

A set is countable if its elements can be put in onetoone correspondence with a subset
of the integers. For example, the sets {0, 1, 7, −3}, {red, green, blue},
{…,−2, −1, 0,
1, 2, …}, {straight, curly}, and the set of all fractions,
are countable.
If a set is not countable, it is uncountable.
The set of all real numbers is uncountable.

Cover.

A confidence interval is said to cover if
the interval contains the true value of the parameter. Before the
data are collected, the chance that the confidence interval will contain the parameter
value is the coverage probability,
which equals the confidence level
after the data are collected and the
confidence interval is actually computed.

Coverage probability.

The coverage probability of a procedure for making
confidence intervals is the chance that the
procedure produces an interval that covers the truth.

Critical value

The critical value in an hypothesis test
is the value of the test statistic beyond which we
would reject the null hypothesis.
The critical value is set so that the probability that the
test statistic is beyond the critical value is
at most equal to the significance level if the
null hypothesis be true.

Crosssectional study.

A crosssectional study compares different individuals to each
other at the same time—it looks at a crosssection of a population. The differences
between those individuals can confound with the effect being
explored. For example, in trying to determine the effect of age on sexual promiscuity, a
crosssectional study would be likely to confound
the effect of
age with the effect of the mores the subjects were taught as children: the older
individuals were probably raised with a very different attitude towards promiscuity than
the younger subjects.
Thus it would be imprudent to attribute differences in promiscuity
to the aging process. C.f. longitudinal study.

Cumulative Probability Distribution Function (cdf).

The cumulative distribution function of a random variable
is the chance that the random variable is less than or equal to x, as a function
of x. In symbols, if F is the cdf of the
random
variable X, then F(x) = P( X ≤ x). The cumulative
distribution function must tend to zero as x approaches minus infinity, and must
tend to unity as x approaches infinity.
It is a positive function, and increases monotonically:
if y > x, then
F(y) ≥ F(x).
The cumulative distribution function completely characterizes the
probability distribution of a
random variable.

D

de Morgan's Laws

de Morgan's Laws are identities involving logical operations:
the negation of a conjunction
is logically equivalent to
the disjunction of the negations, and the negation of
a disjunction is logically equivalent to the conjunction of the negations.
In symbols, !(p & q) = !p  !q and
!(p  q) = !p & !q.

Deck of Cards.

A standard deck of playing cards contains 52 cards, 13 each of four suits: spades,
hearts, diamonds, and clubs. The thirteen cards of each suit are {ace, 2, 3, 4, 5, 6, 7,
8, 9, 10, jack, queen, king}. The face cards are {jack, queen, king}. It is
typically assumed that if a deck of cards is shuffled well, it is equally likely to be in
each possible ordering. There are 52!
(52 factorial)
possible orderings.

Dependent Events, Dependent
Random Variables.

Two events or random variables are
dependent if they are not independent.

Dependent Variable.

In regression, the variable whose values are supposed to be
explained by changes in the other variable
(the the independent
or explanatory variable). Usually one regresses the
dependent variable on the independent variable.

Density, Density Scale.

The vertical axis of a histogram has units of percent per unit of the horizontal axis.
This is called a density scale; it measures how "dense" the observations are in
each bin. See also probability density.

Denying the antecedent.

A logical fallacy that argues from the premise
A → B
and the premise !A that therefore, !B.
The name comes from the fact that the operation denies
(i.e., asserts the negation of) the
antecedent (A) in the
conditional.

Denying the consequent.

A valid logical argument that concludes from the premise
A → B
and the premise !B that therefore, !A.
The name comes from the fact that the operation denies
(i.e., asserts the logical negation) the
consequent (B) in the
conditional.

Deviation.

A deviation is the difference between a datum and some reference value, typically the
mean
of the data. In computing the SD, one finds the rms
of the deviations from the mean, the differences between the
individual data and the mean of the data.

Discrete Variable.

A quantitative variable whose set of possible
values is countable. Typical examples of discrete
variables are variables
whose possible values are a subset of the integers, such as Social Security numbers, the
number of people in a family, ages rounded to the nearest year, etc. Discrete
variables are "chunky." C.f. continuous
variable.
A discrete random variable is one whose set of possible
values is countable. A random variable is discrete if and only if
its cumulative probability distribution function is a stairstep
function; i.e., if it is piecewise constant and only increases by jumps.

Disjoint or Mutually Exclusive Events.

Two events are disjoint or mutually exclusive if the occurrence of
one is incompatible with the occurrence of the other; that is, if they can't both happen
at once (if they have no outcome in common).
Equivalently, two events
are disjoint if their intersection is the
empty set.

Disjoint or Mutually Exclusive Sets.

Two sets are disjoint or mutually exclusive if they have no element
in common. Equivalently, two sets are disjoint if their
intersection is the empty set.

Distribution.

The distribution of a set of numerical data is how their values are distributed over the
real numbers. It is completely characterized by the empirical distribution
function. Similarly, the probability distribution of
a random variable is completely characterized by its probability
distribution function. Sometimes the word "distribution" is used as a
synonym for the empirical distribution function or the
probability
distribution function.
If two or more random variables are defined for the same experiment, they have
a joint probability distribution.

Distribution Function, Empirical.

The empirical (cumulative) distribution function of a set of numerical data is, for each
real value of x, the fraction of observations that are less than or equal to
x.
A plot of the empirical distribution function is an uneven set of stairs. The width of the
stairs is the spacing between adjacent data; the height of the stairs depends on how many
data have exactly the same value. The distribution function is zero for small enough
(negative) values of x, and is unity for large enough values of x. It
increases monotonically:
if y > x, the empirical distribution function
evaluated at y is at least as large as the empirical distribution function
evaluated at x.

DoubleBlind, DoubleBlind Experiment.

In a doubleblind experiment, neither the subjects nor the people
evaluating the subjects knows who is in the treatment group
and who is in the control group.
This mitigates the placebo effect and guards
against conscious and unconscious
prejudice for or against the treatment on the part of the evaluators.

E

Ecological Correlation.

The correlation between
averages of groups of individuals, instead of individuals.
Ecological correlation can be misleading about the association of individuals.

Element of a Set.

See member.

Empirical Law of Averages.

The Empirical Law of Averages lies at the base of the
frequency
theory of probability. This law, which is, in fact, an assumption about how the world
works, rather than a mathematical or physical law, states that if one repeats a
random experiment
over and over, independently and under
"identical" conditions, the fraction of trials that result in a given outcome
converges to a limit as the number of trials grows without bound.

Empty Set.

The empty set, denoted {} or Ø, is the set that
has no members.

Endpoint Convention.

In plotting a histogram, one must decide whether to include a
datum that lies at a class boundary with the class interval
to the left or the right of the boundary. The rule for making this assignment is called an
endpoint convention. The two standard endpoint conventions are (1) to include the
left endpoint of all class intervals and exclude the right, except for the rightmost class
interval, which includes both of its endpoints, and (2) to include the right endpoint of
all class intervals and exclude the left, except for the leftmost interval, which includes
both of its endpoints.

Estimator.

An estimator is a rule for "guessing" the value of a population
parameter based on a random sample
from the population. An estimator is a random variable,
because its value depends on which particular sample is obtained, which is random.
A canonical example of an estimator is the sample mean,
which is an estimator of the population mean.

Event.

An event is a subset of
outcome space.
An event determined by a random variable
is an event of the form A=(X is in A). When the random variable X is observed, that
determines
whether or not A occurs: if the value of X happens to be in A, A occurs; if
not, A does not occur.

Exhaustive.

A collection of events {A_{1}, A_{2}, A_{3},
… }
exhausts the set A
if, for the event A to occur, at least one of those sets must also
occur; that is, if
S ⊂ A_{1} ∪ A_{2}
∪ A_{3} ∪ …
If the event A is not specified, it is assumed to be the entire
outcome space S.

Expectation, Expected Value.

The expected value of a random variable is the longterm
limiting average of its values in independent repeated experiments. The expected value of
the random variable X is denoted EX or E(X). For a discrete random variable (one that has
a countable number of possible values) the expected value is the
weighted average of its possible values, where the weight assigned to each possible value
is the chance that the random variable takes that value. One can think of the expected
value of a random variable as the point at which its
probability
histogram would balance, if it were cut out of a uniform material. Taking the expected
value is a linear operation: if X and Y are two random variables,
the expected value of their sum is the sum of their expected values (E(X+Y) = E(X) +
E(Y)), and the expected value of a constant a times a random variable X is the
constant times the expected value of X (E(a×X ) =
a× E(X)).

Experiment.

What distinguishes an experiment from an observational study is
that in an experiment, the experimenter decides who receives the
treatment.

Explanatory Variable.

In regression, the explanatory or independent variable
is the one that is supposed to "explain" the other. For example, in examining
crop yield versus quantity of fertilizer applied, the quantity of fertilizer would be the
explanatory or independent variable, and the crop
yield would be the dependent variable. In
experiments, the explanatory variable is the one that is
manipulated; the one that is observed is the dependent
variable.

Extrapolation.

See interpolation.

F

Factorial.

For an integer k that is greater than or equal to 1, k! (pronounced
"k factorial") is
k×(k−1)×(k−2)×
…×1. By convention, 0! = 1. There are k!
ways of ordering k
distinct objects. For example, 9! is the number of batting orders of 9 baseball players,
and 52! is the number of different ways a standard deck of playing cards
can be ordered. The calculator above has a button to compute
the factorial of a number. To compute k!, first type the value of k,
then press the button labeled "!".

Fair Bet.

A fair bet is one for which the expected value of the payoff
is zero, after accounting for the cost of the bet. For example, suppose I offer to pay you
$2 if a fair coin lands heads, but you must ante up $1 to play. Your
expected payoff is
−$1+ $0×P(tails) + $2×P(heads)
= −$1 + $2×50%
= $0. This is a fair bet—in the long run, if you made this bet over and over again, you
would expect to break even.

False Discovery Rate.

In testing a collection of hypotheses, the false discovery rate is the fraction of
rejected null hypotheses that are rejected erroneously (the number of Type I errors
divided by the number of rejected null hypotheses), with the convention that if no
hypothesis is rejected, the false discovery rate is zero.

Finite Population Correction.

When sampling without replacement, as in a simple random
sample, the SE of sample sums and sample means depends on the
fraction of the population that is in the sample: the greater the fraction, the smaller
the SE. Sampling with replacement is like sampling from an infinitely
large population. The adjustment to the SE for sampling without replacement is called the
finite population correction. The SE for sampling without replacement is
smaller than the SE for sampling with replacement by the finite
population correction factor ((N −n)/(N −
1))^{½}. Note that for sample size n=1,
there is
no difference between sampling with and without replacement; the finite population
correction is then unity. If the sample size is the entire population of N units,
there is no variability in the result of sampling without replacement (every member of the
population is in the sample exactly once), and the SE should be zero.
This is indeed what the finite population correction gives (the numerator vanishes).

Fisher's exact test (for the equality of two
percentages)

Consider two populations of zeros and ones.
Let p_{1} be the proportion of ones in the first population,
and let p_{2} be the proportion of ones in the second population.
We would like to test the null hypothesis that
p_{1} = p_{2}
on the basis of a simple random sample
from each population.
Let n_{1} be the size of the sample from population 1, and
let n_{2} be the size of the sample from population 2.
Let G be the total number of ones in both samples.
If the null hypothesis be true, the two samples are like one larger sample from
a single population of zeros and ones.
The allocation of ones between the two samples would be expected
to be proportional to the relative sizes of the samples, but would have
some chance variability.
Conditional on G and the two
sample sizes, under the null hypothesis, the tickets in the first sample are like
a random sample of size n_{1} without replacement from a collection of
N = n_{1} + n_{2} units of
which G are labeled with ones.
Thus, under the null hypothesis, the number of tickets labeled with ones
in the first sample has (conditional on G)
an hypergeometric distribution
with parameters N, G, and n_{1}.
Fisher's exact test uses this distribution to set the ranges of observed values of
the number of ones in the first sample for which we would reject the null hypothesis.

FootballShaped Scatterplot.

In a footballshaped scatterplot, most of the points lie within a tilted oval, shaped
moreorless like a football. A footballshaped scatterplot is one in which the
data are homoscedastically
scattered about a straight
line.

Frame, sampling frame.

A sampling frame is a collection of units from which
a sample will be drawn. Ideally, the frame is identical to the
population we want to learn about; more typically, the frame
is only a subset of the
population of interest. The difference between the
frame and the population can be a source of
bias in sampling design, if the parameter
of interest has a different value for the frame than it does for the
population. For example, one might desire to estimate
the current annual average income of 1998 graduates of the University of California
at Berkeley. I propose to use the sample mean income
of a sample of graduates drawn at random. To facilitate taking the sample and contacting
the graduates to obtain income information from them,
I might draw names at random from the list of 1998 graduates for whom the alumni
association has an accurate current address.
The population is the collection of 1998 graduates; the frame is those graduates
who have current addresses on file with the alumni association.
If there is a tendency for graduates with higher incomes to have uptodate
addresses on file with the alumni association,
that would introduce a positive bias into the annual average
income estimated from the sample by the sample mean.

FPP.

Statistics, third edition, by Freedman, Pisani, and Purves,
published by W.W. Norton, 1997.

Frequency theory of probability.

See Probability, Theories of.

Frequency table.

A table listing the frequency (number) or relative frequency (fraction or percentage) of
observations in different ranges, called
class intervals.

Fundamental Rule of Counting.

If a sequence of experiments or trials T_{1}, T_{2}, T_{3},
…, T_{k} could result, respectively, in n_{1},
n_{2}
n_{3}, …, n_{k }possible outcomes, and the
numbers n_{1},
n_{2} n_{3}, …, n_{k }do not depend on
which outcomes actually occurred, the entire sequence of k experiments has
n_{1}× n_{2} ×
n_{3}×
…× n_{k} possible outcomes.

G

Game Theory.

A field of study that bridges mathematics, statistics, economics, and psychology. It is
used to study economic behavior, and to model conflict between nations, for example,
"nuclear stalemate" during the Cold War.

Geometric Distribution.

The geometric distribution describes the number of trials up to and including the first
success, in independent trials with the same probability of success. The geometric
distribution depends only on the single parameter p, the probability of success in
each trial. For example, the number of times one must toss a fair coin until the first
time the coin lands heads has a geometric distribution with parameter p = 50%.
The geometric distribution assigns probability
p×(1 − p)^{k−1}to
the event that it takes k trials to the first success.
The expected
value of the geometric distribution is 1/p, and its SE is
(1−p)^{½}/p.

Geometric Mean.

The geometric mean of n numbers {x_{1},
x_{2},
x_{3}, …, x_{n}}
is the nth root of their product:

(x_{1}×x_{2}×x_{3}×
…
×x_{n})^{1/n}.

Graph of Averages.

For bivariate data, a graph of averages is a plot of the
average values of one variable (say y) for small ranges of values of the other
variable (say x), against the value of the second variable (x) at the
midpoints of the ranges.

H

Heteroscedasticity.

"Mixed scatter." A scatterplot or
residual plot shows heteroscedasticity if the scatter in
vertical slices through the plot depends on where you take the slice.
Linear regression is not usually a good idea if the data are
heteroscedastic.

Histogram.

A histogram is a kind of plot that summarizes how data are distributed. Starting with a
set of class intervals, the histogram is a set of rectangles
("bins") sitting on the horizontal axis. The bases of the
rectangles are the class intervals, and their heights are
such that their areas are proportional to the fraction of observations in the
corresponding class intervals. That is, the height of a
given rectangle is the fraction of observations in the corresponding
class interval, divided by the length of the corresponding
class interval. A histogram does not need a vertical scale,
because the total area of the histogram must equal 100%. The units of the vertical axis
are percent per unit of the horizontal axis. This is called the density scale.
The horizontal axis of a histogram needs a scale. If any observations coincide with the
endpoints of class intervals, the
endpoint convention is important.
This page
contains a histogram tool, with controls to highlight ranges of values and read their
areas.

Historical Controls.

Sometimes, the a treatment group is compared with
individuals from another epoch who did not receive the treatment; for example, in studying
the possible effect of fluoridated water on childhood cancer, we might compare cancer
rates in a community before and after fluorine was added to the water supply. Those
individuals who were children before fluoridation started would comprise an historical
control group. Experiments and studies with historical controls tend to be more
susceptible to confounding than those with contemporary controls, because many factors
that might affect the outcome other than the treatment tend to
change over time as well. (In this example, the level of other potential carcinogens in
the environment also could have changed.)

Homoscedasticity.

"Same scatter." A scatterplot or
residual plot shows homoscedasticity if the scatter
in vertical slices through the plot does not depend much on where you take the slice.
C.f. heteroscedasticity.

House Edge.

In casino games, the expected payoff to the bettor
is negative: the house (casino) tends to win money in the
long run. The amount of money the house would expect to win for each $1 wagered on
a particular bet (such as a bet on "red" in roulette) is
called the house edge for that bet.

HTLWS.

The book How to lie with
Statistics by D. Huff.

Hypergeometric Distribution.

The hypergeometric distribution with parameters N, G and
n is the distribution of the number of "good"
objects in a simple random sample of size n
(i.e., a
random sample without replacement in which every subset of size n has the same
chance of occurring) from a population of N objects of which
G are "good."
The chance of getting exactly g good objects in such a sample is
_{G}C_{g} ×
_{N−G}C_{n−g}/_{N}C_{n},

provided g ≤ n, g ≤ G, and
n − g ≤ N − G.
(The probability is zero otherwise.)
The expected value of the hypergeometric distribution is
n×G/N,
and its standard error is
((N−n)/(N−1))^{½}
× (n ×
G/N × (1−G/N)
)^{½}.

Hypothesis testing.

Statistical hypothesis testing is formalized as making a decision between rejecting or
not rejecting a null hypothesis, on the basis of a set of
observations.
Two types of errors can result from any decision rule (test): rejecting the
null hypothesis when it is true (a Type I error), and failing to
reject the null hypothesis when it is false (a Type II error).
For any hypothesis, it is possible to develop many different decision rules (tests).
Typically, one specifies ahead of time the chance of a Type I error one is willing to
allow.
That chance is called the significance level of the
test or decision rule.
For a given significance level, one way of deciding which decision
rule is best is to pick the one that has the smallest chance of a Type II error when a
given alternative hypothesis is true.
The chance of correctly
rejecting the null hypothesis when a given alternative hypothesis is true is
called the power of the test against that alternative.
I

iff, if and only if, ↔

If p and q are two logical propositions,
then(p ↔ q) is a proposition that is true when
both p and q are true, and when both p and q are
false.
It is logically equivalent to the proposition

( (p → q)
&
(q → p) )

and to the proposition

( (p & q)
 ((!
p) & (!q)) ).

Implies,
logical implication, → , conditional, ifthen

Logical implication is an operation on two logical propositions.
If p and q are two logical propositions,
(p → q), pronounced "p implies q" or "if p then q"
is a logical proposition that is
true if p is false, or if both p and q are true.
The proposition (p → q) is
logically equivalent to the proposition
((!p)  q).
In the conditional p → q, the
antecedent is p
and the consequent is q.

Independent, independence.

Two events A and B are (statistically) independent if the chance
that they both happen simultaneously is the product of the chances that each occurs
individually; i.e., if P(AB) = P(A)P(B). This is essentially equivalent to saying
that learning that one event occurs does not give any information about whether the other
event occurred too: the conditional probability of A given B is the same as the
unconditional probability of A, i.e., P(AB) = P(A). Two
random variables X and Y are independent if all events
they
determine are independent, for example, if the event
{a < X ≤ b}
is independent of the event {c < Y ≤ d} for
all
choices of a, b, c, and d.
A collection of more than two random variables is independent if for every proper subset
of the variables, every event determined
by that subset of the variables is independent of every event determined by the variables
in the complement of the subset. For example, the three random variables X, Y, and Z are
independent if every event determined by X is independent of every event
determined by Y and
every event determined by X is independent of every event determined by Y and Z
and every event determined by Y is
independent of every event determined by X and Z and every event determined by Z
is independent of every event determined by X and Y.

Independent and identically distributed (iid).

A collection of two or more random variables {X_{1}, X_{2},
… , }
is independent and identically distributed if the variables have the same
probability distribution,
and are independent.

Independent Variable.

In regression, the independent variable is the one that is
supposed to explain the other; the term is a synonym for "explanatory variable."
Usually, one regresses the "dependent variable" on the "independent
variable." There is not always a clear choice of the independent variable. The
independent variable is usually plotted on the horizontal axis. Independent in this
context does not mean the same thing as
statistically independent.

Indicator Random Variable.

The indicator [random variable] of the
event A, often written 1_{A}, is the
random variable that
equals unity if A occurs, and zero if A does not occur.
The expected
value of the indicator of A is the probability of A, P(A), and the
standard error of the indicator of A is
(P(A)×(1−P(A))^{½}.
The sum

1_{A} + _{}1_{B} + 1_{C} +
…

of the indicators of a
collection of events {A, B, C, …}
counts how many of the
events {A, B, C, …} occur in a given
trial.
The product of the indicators of a collection of events is the indicator of the
intersection of the events (the product equals one if and only if all of
indicators equal one).
The maximum of the indicators of a collection of events is the indicator
of the union of the events (the maximum equals one if any of the indicators equals one).

Interquartile Range (IQR).

The interquartile range of a list of numbers is the upper
quartile
minus the lower quartile.

Interpolation.

Given a set of bivariate data (x, y), to
impute a value of y corresponding to some value of x at which there is
no measurement of y is called interpolation, if the value of x is within
the range of the measured values of x. If the value of x is outside the
range of measured values, imputing a corresponding value of y is called
extrapolation.

Intersection.

The intersection of two or more sets is the set of elements that all the sets have in
common; the elements contained in every one of the sets.
The intersection of the events A and B is written "A∩B,"
"A and B," and "AB." C.f.
union. See also Venn diagrams.

Invalid (logical) argument.

An invalid logical argument is one in which
the truth of the premises does not guarantee the truth
of the conclusion.
For example, the following logical argument is invalid:
If the forecast calls for rain, I will not wear sandals.
The forecast does not call for rain.
Therefore, I will wear sandals.
See also valid argument.

J

Joint Probability Distribution.

If X_{1}, X_{2}, … ,
X_{k} are
random variables defined for the same experiment,
their joint probability distribution gives the probability
of events determined by the collection of random variables:
for any collection of sets of numbers
{A_{1}, … , A_{k}},
the joint probability distribution determines
P( (X_{1} is in A_{1}) and
(X_{2} is in A_{2}) and … and
(X_{k} is in A_{k})
).
For example, suppose we roll two fair dice independently.
Let X_{1} be the number of spots that show on the first die,
and let X_{2} be the total number of spots that show on both dice.
Then the joint distribution of X_{1} and
X_{2}
is as follows:
P(X_{1} = 1, X_{2} = 2) =
P(X_{1} = 1, X_{2} = 3) =
P(X_{1} = 1, X_{2} = 4) =
P(X_{1} = 1, X_{2} = 5) =
P(X_{1} = 1, X_{2} = 6) =
P(X_{1} = 1, X_{2} = 7) =
P(X_{1} = 2, X_{2} = 3) =
P(X_{1} = 2, X_{2} = 4) =
P(X_{1} = 2, X_{2} = 5) =
P(X_{1} = 2, X_{2} = 6) =
P(X_{1} = 2, X_{2} = 7) =
P(X_{1} = 2, X_{2} = 8) = …
… P(X_{1} = 6, X_{2} = 7) =
P(X_{1} = 6, X_{2} = 8) =
P(X_{1} = 6, X_{2} = 9) =
P(X_{1} = 6, X_{2} = 10) =
P(X_{1} = 6, X_{2} = 11) =
P(X_{1} = 6, X_{2} = 12) = 1/36.
If a collection of random variables is independent,
their joint probability distribution is the product of their
marginal probability distributions, their
individual probability distributions without regard for the value of the other variables.
In this example, the marginal probability distribution of X_{1}
is
P(X_{1} = 1) = P(X_{1} = 2) = P(X_{1} = 3) =
P(X_{1} = 4) = P(X_{1} = 5) = P(X_{1} = 6) = 1/6,
and the marginal probability distribution of X_{2} is
P(X_{2} = 2) = P(X_{2} = 12) = 1/36
P(X_{2} = 3) = P(X_{2} = 11) = 1/18
P(X_{2} = 4) = P(X_{2} = 10) = 3/36
P(X_{2} = 5) = P(X_{2} = 9) = 1/9
P(X_{2} = 6) = P(X_{2} = 8) = 5/36
P(X_{2} = 7) = 1/6.
Note that P(X_{1} = 1, X_{2} = 10) = 0,
while P(X_{1} = 1)×P(X_{2} = 10) = (1/6)(3/36) = 1/72.
The joint probability is not equal to the product of the marginal probabilities:
X_{1} and X_{2}
are dependent random variables.
K
L

Law of Averages.

The Law of Averages says that the average of
independent
observations of random variables
that have the same probability distribution is
increasingly likely to be close
to the expected value of the
random
variables as the number of observations grows.
More precisely, if X_{1}, X_{2},
X_{3}, …, are independent
random variables with
the same probability distribution, and
E(X) is their
common expected value, then for
every number ε > 0,

P{(X_{1} + X_{2} + … +
X_{n})/n
− E(X)  < ε}

converges to 100% as n grows.
This is equivalent to saying that the sequence of sample means

X_{1}, (X_{1}+X_{2})/2,
(X_{1}+X_{2}+X_{3})/3, …

converges in probability to E(X).

Law of Large Numbers.

The Law of Large Numbers says that in repeated, independent
trials with the same probability p of success in each trial, the percentage of
successes is increasingly likely to be close to the chance of success as the number of
trials increases. More precisely, the chance that the percentage of successes differs from
the probability p by more than a fixed positive amount, e > 0,
converges to zero as the number of trials n goes to infinity, for every number
e > 0. Note that in contrast to the difference between the percentage of
successes and the probability of success, the difference between the number of
successes and the expected number of successes,
n×p,
tends to grow as n grows.
The following tool illustrates the law of large numbers; the button toggles between
displaying the difference between the number of successes and the expected number of
successes, and the difference between the percentage of successes and the expected
percentage of successes.
The tool on this page illustrates
the law of large numbers.

Limit.

See converge.

Linear Operation.

Suppose f is a function or operation that acts on things we shall denote
generically by the lowercase Roman letters x and y. Suppose it makes
sense to multiply x and y by numbers (which we denote by a),
and that it makes sense to add things like x and y together. We say that
f is linear if for every number a and every value of x
and y for which f(x) and f(y) are defined,
(i) f( a×x ) is defined and equals
a×f(x),
and (ii) f( x + y ) is defined and equals
f(x)
+ f(y). C.f. affine.

Linear association.

Two variables are linearly associated if a change in one is associated with a
proportional change in the other, with the same constant of proportionality throughout the
range of measurement. The correlation coefficient measures
the degree of linear association on a scale of −1 to 1.

Location, Measure of.

A measure of location is a way of summarizing what a "typical" element of a
list is—it is a onenumber summary of a distribution. See
also arithmetic mean, median, and
mode.

Logical argument.

A logical argument consists of one or more premises,
propositions
that are assumed to be true, and a conclusion, a proposition that is
supposed to be guaranteed to be true (as a matter of pure logic) if the premises
are true.
For example, the following is a logical argument:
This argument has two premises: p → q,
and p.
The conclusion of the argument is q.
If a logical argument is valid if the truth of the premises
guarantees the truth of the conclusion; otherwise, the argument is
invalid.
That is, an argument with premises p_{1}, p_{1},
… p_{n} and conclusion q is valid if the
compound proposition
(p_{1} & p_{2} & … & p_{n})
→ q
is logically equivalent to TRUE.
The argument given above is valid because if it is true that p → q
and that p is true (the two premises), then q
(the conclusion of the argument) must also be true.

Logically equivalent, logical equivalence.

Two propositions are logically equivalent if they always
have the same truth value.
That is, the propositions p and q are logically equivalent
if p is true
whenever q is true and p is false whenever q is false.
The proposition (p ↔ q) is always true if and only if p and
q are logically equivalent.
For example, p is logically equivalent to p, to
(p & p), and to (p  p);
(p  (!p))
is logically equivalent to TRUE;
(p & !p) is logically equivalent to
FALSE;
(p ↔ p) is logically equivalent to TRUE;
and (p → q) is
logically equivalent to (!p  q).

Longitudinal study.

A study in which individuals are followed over time, and compared
with themselves at different times, to determine, for example, the effect of aging on some
measured variable. Longitudinal studies provide much more persuasive
evidence about the effect of aging than do crosssectional
studies.

Lower Quartile (LQ).

See quartiles.

M

Margin of error.

A measure of the uncertainty in an estimate of a
parameter; unfortunately, not everyone
agrees what it should mean.
The margin of error of an estimate is typically
one or two times the estimated standard error of the estimate.

Marginal probability distribution.

The marginal probability distribution of a random variable that has a
joint probability distribution
with some other random variables is the probability distribution of that
random variable without regard for the values that the other random variables take.
The marginal distribution of a discrete random variable X_{1}
that has a joint distribution with other discrete random variables can be found from the
joint distribution by summing over all possible values of the other variables.
For example, suppose we roll two fair dice independently.
Let X_{1} be the number of spots that show on the first die,
and let X_{2} be the total number of spots that show on both dice.
Then the joint distribution of X_{1} and
X_{2}
is as follows:
P(X_{1} = 1, X_{2} = 2) =
P(X_{1} = 1, X_{2} = 3) =
P(X_{1} = 1, X_{2} = 4) =
P(X_{1} = 1, X_{2} = 5) =
P(X_{1} = 1, X_{2} = 6) =
P(X_{1} = 1, X_{2} = 7) =
P(X_{1} = 2, X_{2} = 3) =
P(X_{1} = 2, X_{2} = 4) =
P(X_{1} = 2, X_{2} = 5) =
P(X_{1} = 2, X_{2} = 6) =
P(X_{1} = 2, X_{2} = 7) =
P(X_{1} = 2, X_{2} = 8) = …
… P(X_{1} = 6, X_{2} = 7) =
P(X_{1} = 6, X_{2} = 8) =
P(X_{1} = 6, X_{2} = 9) =
P(X_{1} = 6, X_{2} = 10) =
P(X_{1} = 6, X_{2} = 11) =
P(X_{1} = 6, X_{2} = 12) = 1/36.
The marginal probability distribution of X_{1}
is
P(X_{1} = 1) = P(X_{1} = 2) = P(X_{1} = 3) =
P(X_{1} = 4) = P(X_{1} = 5) = P(X_{1} = 6) = 1/6.
We can verify that the marginal probability that X_{1} = 1
is indeed the sum of the joint probability distribution
over all possible values of X_{2} for which
X_{1} = 1:
P(X_{1} = 1) = P(X_{1} = 1, X_{2} = 2) +
P(X_{1} = 1, X_{2} = 3) +
P(X_{1} = 1, X_{2} = 4) +
P(X_{1} = 1, X_{2} = 5) +
P(X_{1} = 1, X_{2} = 6) +
P(X_{1} = 1, X_{2} = 7) = 6/36 = 1/6.
Similarly, the marginal probability distribution of X_{2} is
P(X_{2} = 2) = P(X_{2} = 12) = 1/36
P(X_{2} = 3) = P(X_{2} = 11) = 1/18
P(X_{2} = 4) = P(X_{2} = 10) = 3/36
P(X_{2} = 5) = P(X_{2} = 9) = 1/9
P(X_{2} = 6) = P(X_{2} = 8) = 5/36
P(X_{2} = 7) = 1/6.
Again, we can verify that the marginal probability that X_{2} = 4
is 3/36 by adding the joint probabilities for all possible values of
X_{1} for which X_{2} = 4:
P(X_{2} = 4) = P(X_{1} = 1, X_{2} = 4) +
P(X_{1} = 2, X_{2} = 4) +
P(X_{1} = 3, X_{2} = 4) = 3/36.

Markov's Inequality.

For lists: If a list contains no negative numbers, the fraction of numbers in the list
at least as large as any given constant a>0 is no larger than the
arithmetic mean of the list, divided by a.
For random variables: if a random variable X must be
nonnegative, the chance that X exceeds any given constant a>0 is no larger than
the expected value of X, divided by a.

Maximum Likelihood Estimate (MLE).

The maximum likelihood estimate of a parameter from data is the
possible value of the parameter for which the chance of observing
the data largest. That is, suppose that the parameter is p,
and that we observe data x. Then the maximum likelihood estimate of
p is

estimate p by the value q that makes P(observing x when the
value of p is q) as large as possible.

For example, suppose we are trying to estimate the chance that a (possibly biased) coin
lands heads when it is tossed. Our data will be the number of times x the coin
lands heads in n independent tosses of the coin. The distribution of the number
of times the coin lands heads is binomial with
parameters n (known) and p (unknown). The chance
of observing x heads in n trials if the chance of heads in a given trial
is q is
_{n}C_{x} q^{x}(1−q)^{n−x}.
The maximum likelihood estimate of p would be the value of q that
makes that chance largest. We can find that value of q explicitly using calculus;
it turns out to be q = x/n, the fraction of times the coin is
observed to land heads in the n tosses. Thus the maximum likelihood estimate of
the chance of heads from the number of heads in n independent tosses of the coin
is the observed fraction of tosses in which the coin lands heads.

Mean, Arithmetic mean.

The sum of a list of numbers, divided by the number of numbers.
See also average.

Mean Squared Error (MSE).

The mean squared error of an estimator of a
parameter is the expected value of the
square of the difference between the estimator and the parameter. In symbols, if X is an
estimator of the parameter t, then
MSE(X) = E( (X−t)^{2} ).

The MSE measures how far the estimator is off from what it is trying to estimate, on the
average in repeated experiments. It is a summary measure of the accuracy of the estimator.
It combines any tendency of the estimator to overshoot or undershoot the truth
(bias), and the variability of the estimator (SE).
The MSE can be written in terms of the bias and
SE of the estimator:

MSE(X) = (bias(X))^{2} +
(SE(X))^{2}.

Median.

"Middle value" of a list. The smallest number such that at least half the
numbers in the list are no greater than it. If the list has an odd number of entries, the
median is the middle entry in the list after sorting the list into increasing order. If
the list has an even number of entries, the median is the smaller of the two middle
numbers after sorting. The median can be estimated from a histogram by finding the
smallest number such that the area under the histogram to the left of that
number is 50%.

Member of a set.

Something is a member (or element) of a set if it is one of the
things in the set.

Method of Comparison.

The most basic and important method of determining whether a
treatment
has an effect: compare what happens to individuals who are treated
(the treatment group) with what happens to
individuals who are not
treated (the control group).

Minimax Strategy.

In game theory, a minimax strategy is one that minimizes one's maximum loss, whatever
the opponent might do (whatever strategy the opponent might choose).

Mode.

For lists, the mode is a most common (frequent) value. A list can have more than one
mode. For histograms, a mode is a relative maximum
("bump").

Moment.

The kth moment of a list is the average value of the elements raised to
the kth power; that is, if the list consists of the N elements
x_{1}, x_{2}, … ,
x_{N},
the kth moment of the list is

(
x_{1}^{k} +
x_{2}^{k} +
x_{N}^{k}
)/N.

The kth moment of a random variable X is
the expected value of X^{k},
E(X^{k}).

Monotone, monotonic function.

A function is monotone if it only increases or only decreases:
f increases monotonically (is monotonic increasing)
if x > y, implies thatf(x)
≥ f(y).
A function f decreases monotonically (is monotonic decreasing)
if x > y, implies thatf(x)
≤ f(y).
A function f is strictly monotonically increasing
if x > y, implies thatf(x)
> f(y), and strictly monotonically decreasing if
if x > y, implies thatf(x)
< f(y).

Multimodal Distribution.

A distribution with more than one mode.
The histogram
of a multimodal distribution has more than one "bump."

Multinomial Distribution

Consider a sequence of n independent trials,
each of which can result in an outcome in any of k categories.
Let p_{j} be the probability that each trial results
in an outcome in category j, j = 1, 2, … ,
k,
so

p_{1} + p_{2} + … +
p_{k}
= 100%.

The number of outcomes of each type has a multinomial distribution.
In particular, the probability that the n trials result in
n_{1}
outcomes of type 1, n_{2} outcomes of type 2,
… , and
n_{k} outcomes of type k is

n!/(n_{1}! × n_{2}! ×
… × n_{k}!) ×
p_{1}^{n1} ×
p_{2}^{n2} ×
… ×
p_{k}^{nk},

if n_{1}, … , n_{k} are
nonnegative integers that sum to n; the chance is zero otherwise.

Multiplication rule.

The chance that events A and B both occur (i.e.,
that event AB occurs), is the
conditional probability that A occurs given that B
occurs, times the unconditional probability that B occurs.

Multiplicity in hypothesis tests.

In hypothesis testing,
if more than one hypothesis is tested, the actual
significance level of
the combined tests is not equal to the nominal
significance level of the individual tests.
See also false discovery rate.

Multivariate Data.

A set of measurements of two or more variables per individual.
See bivariate.

Mutually Exclusive.

See disjoint events or
disjoint sets.

N

Nearly normal distribution.

A population of numbers (a list of numbers) is said to have a nearly normal
distribution if the histogram of its values in
standard units nearly
follows a normal curve.
More precisely, suppose that the mean of the
list is μ and the standard deviation
of the list is SD.
Then the list is nearly normally distributed if, for every two numbers
a < b, the fraction of numbers in the list that are
between a and b is approximately equal to the area under the normal
curve between (a − μ)/SD and
(a − μ)/SD.

Negative Binomial Distribution.

Consider a sequence of independent trials with the same
probability p of success in each trial. The number of trials up to and including
the rth success has the negative Binomial distribution with parameters n
and r. If the random variable N has the negative
binomial distribution with parameters n and r, then
P(N=k) =
_{k−1}C_{r}−1 × p^{r} ×
(1−p)^{k−r},

for k = r, r+1, r+2, …, and zero for k
< r, because there must be at least r trials to have r
successes. The negative binomial distribution is derived as follows: for the rth
success to occur on the kth trial, there must have been r−1 successes in
the first k−1 trials, and the kth trial must result in success. The
chance of the former is the chance of r−1 successes in k−1
independent trials with the same probability of success in each
trial, which, according to the Binomial distribution with
parameters n=k−1 and p, has probability
_{k−1}C_{r}−1 ×
p^{r−}1 × (1−p)^{k−r}.

The chance of the latter event is p, by assumption. Because the trials are
independent, we can find the chance that both
events
occur by multiplying their chances together, which gives the expression for P(N=k)
above.

No causation without manipulation.

A slogan attributed to Paul
Holland. If the conditions were not deliberately manipulated (for example, if the
situation is an observational study rather than an
experiment),
it is unwise to conclude that there is any causal relationship between the outcome and the
conditions. See post hoc ergo propter hoc
and cum hoc ergo propter hoc.

Nonlinear Association.

The relationship between two variables is nonlinear if a change in one is associated
with a change in the other that is depends on the value of the first; that is, if the
change in the second is not simply proportional to the change in the first, independent of
the value of the first variable.

Nonresponse.

In surveys, it is rare that everyone who is ``invited'' to participate (everyone whose
phone number is called, everyone who is mailed a questionnaire, everyone an interviewer
tries to stop on the street…) in fact responds. The difference between the
"invited" sample sought, and that obtained, is the nonresponse.

Nonresponse bias.

In a survey, those who respond may differ from those who do not, in ways that are
related to the effect one is trying to measure. For example, a telephone survey of how
many hours people work is likely to miss people who are working late, and are therefore
not at home to answer the phone. When that happens, the survey may suffer from nonresponse
bias. Nonresponse bias makes the result of a survey differ systematically
from the truth.

Nonresponse rate.

The fraction of nonresponders in a survey:
the number of nonresponders divided by the number of people invited to participate
(the number sent questionnaires, the number of interview attempts, etc.)
If the nonresponse rate is appreciable, the survey suffer from large
nonresponse bias.

Normal approximation.

The normal approximation to data is to approximate areas under the
histogram
of data, transformed into standard units, by the
corresponding areas under the normal curve.

Many probability distributions can be approximated by a normal distribution, in the
sense that the area
under the probability histogram is close to the area under a corresponding part of the
normal curve. To find the corresponding part of the normal curve, the range must be
converted to standard units, by subtracting the expected value
and dividing by the standard error.
For example, the area under the binomial
probability histogram for n = 50 and p =
30% between 9.5 and 17.5 is 74.2%. To use the normal approximation, we transform
the endpoints to standard units, by subtracting the
expected value (for the Binomial
random variable, n×p = 15
for these values of n and p) and dividing the
result by the standard error
(for a Binomial,
(n × p ×
(1−p))^{1/2}
= 3.24 for these values of n and p).
The area normal approximation is the area under the normal curve between
(9.5 − 15)/3.24 = −1.697 and (17.5 − 15)/3.24 = 0.772; that area is 73.5%, slightly
smaller than the corresponding area under the binomial histogram. See also the
continuity
correction.
The tool on this page
illustrates the normal approximation to the
binomial
probability histogram.
Note that the approximation gets worse when p gets close to 0 or 1, and
that the approximation improves as n increases.

Normal curve.

The normal curve is the familiar
"bell curve:," illustrated on
this page.
The mathematical expression for the normal curve is
y = (2×pi)^{−½}e^{−x2/2},
where pi is the ratio of the circumference of a circle to its diameter
(3.14159265…),
and e is the base
of the natural logarithm (2.71828…).
The normal curve is symmetric around the point x=0, and
positive for every value of x. The area under the normal curve is unity, and the
SD of the normal curve, suitably defined, is also unity. Many (but not most)
histograms, converted into
standard units,
approximately follow the normal curve.

Normal distribution.

A random variable X has a normal distribution with mean m and
standard error s if for every pair of numbers a ≤
b, the chance that a < (X−m)/s < b is
P(a < (X−m)/s < b) = area under the normal curve between a
and b.
If there are numbers m and s such that X has a normal
distribution with mean m and standard error s, then X is said to have
a normal distribution or to be normally distributed. If X has a normal
distribution with mean m=0 and standard error s=1, then X is said
to have a standard normal distribution. The notation X~N(m,s^{2}) means that
X has a normal distribution with mean m and
standard error s; for example, X~N(0,1), means X has a standard normal distribution.

NOT, !, Negation, Logical Negation.

The negation of a logical proposition p,
!p, is a proposition that is the logical opposite of
p.
That is, if p is true, !p is false, and
if p is false, !p is true. Negation takes
precedence over other logical operations.
Other common symbols for the negation operator include ¬, − and ˜.

Null hypothesis.

In hypothesis testing, the hypothesis we wish to falsify
on the basis of the data. The null hypothesis is typically that something is not present,
that there is no effect, or that there is no difference between treatment and control.

O

Observational Study.

C.f. controlled experiment.

Odds.

The odds in favor of an event is the ratio
of the probability that the event occurs to the
probability that the
event does not occur. For example, suppose an experiment can result in any of n
possible outcomes, all equally likely, and that k of the outcomes result in a
"win" and n−k result in a "loss." Then the chance of
winning is k/n; the chance of not winning is
(n−k)/n;
and the odds in favor of winning are
(k/n)/((n−k)/n)
= k/(n−k), which is the number of favorable outcomes divided by the
number of unfavorable outcomes. Note that odds are not synonymous with probability, but
the two can be converted back and forth. If the odds in favor of an event are q,
then the probability of the event is q/(1+q). If the probability of an
event is p, the odds in favor of the event are p/(1−p) and the
odds against the event are (1−p)/p.

Onesided Test.

C.f. twosided test.
An hypothesis test of the null hypothesis
that the value of a parameter, μ, is equal to
a null value, μ_{0}, designed to have power against either
the alternative hypothesis that μ < μ_{0}
or the alternative μ > μ_{0} (but not both).
For example, a significance level 5%, onesided
z test
of the null hypothesis that the mean of a population equals zero against the alternative
that it is greater than zero, would reject the null hypothesis for values of

or, , Disjunction, Logical Disjunction, ∨

An operation on two logical propositions.
If p and q are two propositions, (p  q)
is a proposition that is true if p is true or
if q is true (or both); otherwise, it is false. That is,
(p  q) is true unless both p and q
are false.
The operation  is sometimes represented by the symbol ∨ and sometimes by the
word or. C.f.
exclusive disjunction, XOR.

Ordinal Variable.

A variable whose possible values have a natural order, such as
{short, medium, long}, {cold, warm, hot}, or {0, 1, 2, 3, …}. In contrast, a variable
whose possible values are {straight, curly} or {Arizona, California, Montana, New York}
would not naturally be ordinal. Arithmetic with the possible values of an ordinal variable
does not necessarily make sense, but it does make sense to say that one possible value is
larger than another.

Outcome Space.

The outcome space is the set of all possible outcomes of a given
random experiment. The outcome space is often denoted
by the capital letter S.

Outlier.

An outlier is an observation that is many SD's from the
mean. It is sometimes tempting to discard outliers, but this is imprudent
unless the cause of the outlier can be identified, and the outlier is determined to be
spurious. Otherwise, discarding outliers can cause one to underestimate the true
variability of the measurement process.

P

Pvalue.

Suppose we have a family of hypothesis tests
of a null hypothesis that let us test the
hypothesis at any significance level
p between 0 and 100% we choose.
The P value of the null hypothesis
given the data is the smallest significance level p for which
any of the tests would have rejected the null hypothesis.

For example, let X be a test statistic,
and for p between 0 and 100%, let x_{p} be
the smallest number such that, under the null hypothesis,

P( X ≤ x ) ≥ p.

Then for any p between 0 and 100%, the rule

reject the null hypothesis if X < x_{p}

tests the null hypothesis at significance level p.
If we observed X = x, the Pvalue of
the null hypothesis given the data would be the smallest p such that
x < x_{p}.

Parameter.

A numerical property of a population, such as its
mean.

Partition.

A partition of an event A
is a collection of events
{A_{1}, A_{2}, A_{3}, … } such that the
events in the collection are disjoint, and their
union is A.
That is,
A_{j}A_{k} = {} unless j = k, and
A = A_{1} ∪ A_{2}
∪ A_{3} ∪
… .

If the event A is not specified, it is assumed to be the entire
outcome space S.

Payoff Matrix.

A way of representing what each player in a game wins or loses, as a function of his and
his opponent's strategies.

Percentile.

The pth percentile of a list is the smallest number such that at least
p%
of the numbers in the list are no larger than it.
The pth percentile of a random variable
is the smallest number such that the chance
that the random variable is no larger than it is at least p%.
C.f. quantile.

Permutation.

A permutation of a set is an arrangement of the elements of the set in some order. If
the set has n things in it, there are n!
different orderings of its elements. For the first element in an ordering,
there are n
possible choices, for the second, there remain n−1 possible choices, for the
third, there are n−2, etc., and for the nth element of the
ordering, there is a single choice remaining. By the fundamental rule of counting, the
total number of sequences is thus
n×(n−1)×(n−2)×…×1.
Similarly, the number of orderings of length k one
can form from n≥k things
is
n×(n−1)×(n−2)×…×(n−k+1) =
n!/(n−k)!. This
is denoted _{n}P_{k}, the number of permutations of
n
things taken k at a time. C.f.
combinations.

Placebo.

A "dummy" treatment that has no pharmacological
effect; e.g., a sugar pill.

Placebo effect.

The belief or knowledge that one is being treated can itself have an effect that
confounds with the real effect of the treatment.
Subjects given a placebo as a painkiller report statistically
significant reductions
in pain in randomized experiments that compare them with subjects who receive no treatment
at all. This very real psychological effect of a placebo, which has no direct biochemical
effect, is called the placebo effect. Administering a placebo to the
control group is thus important in experiments with human
subjects; this is the essence of a blind experiment.

Point of Averages.

In a scatterplot, the point whose coordinates are the
arithmetic means of the corresponding variables. For example, if the
variable X is plotted on the horizontal axis and the variable Y is plotted on the vertical
axis, the point of averages has coordinates (mean of X, mean of Y).

Poisson Distribution.

The Poisson distribution is a discrete
probability distribution that depends on one
parameter, m.
If X is a random variable with
the Poisson distribution with parameter m, then the probability that
X = k is
E^{−m} × m^{k}/k!,
k = 0, 1, 2, … ,

where E is the base of the natural logarithm and ! is the
factorial function.
For all other values of k, the probability is zero.

The expected value the Poisson distribution with parameter
m is m,
and the standard error of the Poisson distribution with parameter
m is m^{½}.

Population.

A collection of units being studied. Units can
be people, places, objects, epochs, drugs, procedures, or many other things. Much of
statistics is concerned with estimating numerical properties
(parameters)
of an entire population from a random sample of
units from the population.

Population Mean.

The mean of the numbers in a numerical population.
For example, the population mean of a box of numbered tickets is the mean of
the list comprised of all the numbers on all the tickets.
The population mean is a parameter. C.f.
sample mean.

Population Percentage.

The percentage of units in a population
that possess a specified property. For example, the percentage of a given collection of
registered voters who are registered as Republicans. If each unit that possesses the
property is labeled with "1," and each unit that does not possess the property
is labeled with "0," the population percentage is the same as the mean of that
list of zeros and ones; that is, the population percentage is the
population mean for a population of zeros and ones. The
population percentage is a parameter. C.f.
sample percentage.

Population Standard Deviation.

The standard deviation of the values of a variable for a population.
This is a parameter, not a
statistic.
C.f. sample standard deviation.

Post hoc ergo propter hoc.

"After this, therefore because of this." A fallacy of logic known since
classical times: inferring a causal relation from
correlation. Don't do this at home!

Power.

Refers to an hypothesis test. The power of a test against
a specific alternative hypothesis is the chance that the test
correctly rejects the null hypothesis when the
alternative hypothesis is true.

Premise, logical premise.

A premise is a proposition that is assumed to
be true as part of a logical argument.

Prima facie.

Latin for "at first glance." "On the face of it." Prima
facie evidence for something is information that at first glance supports the
conclusion. On closer examination, that might not be true; there could be another
explanation for the evidence.

Principle of insufficient reason (Laplace)

Laplace's principle of insufficient reason says that if
there is no reason to believe that the possible outcomes of an experiment are not
equally likely, one should assume that the
outcomes are equally likely.
This is an example of a fallacy called
appeal to ignorance.

Probability.

The probability of an event is a number between zero and 100%. The
meaning (interpretation) of probability is the subject of
theories
of probability, which differ in their interpretations. However, any rule for assigning
probabilities to events has to satisfy the
axioms of probability.

Probability density function.

The chance that a continuous random variable is in any range
of values can be calculated as the area under a curve over that range of values. The
curve is the probability density function of the random variable. That is, if X is
a continuous random variable, there is a function f(x) such that for every
pair of numbers a≤b,
P(a≤ X ≤b) = (area under f between
a and b);

f is the probability density function of X. For example, the probability
density function of a random variable with a standard normal
distribution is the normal curve.
Only continuous
random variables have probability density functions.

Probability Distribution.

The probability distribution of a random variable
specifies the chance that the variable takes a value in any subset of the real numbers.
(The subsets have to satisfy some technical conditions that are not important for this
course.) The probability distribution of a random variable
is completely characterized by the cumulative probability distribution
function; the terms sometimes are used synonymously.
The probability distribution
of a discrete
random variable can be
characterized by the chance that the random variable takes
each of its possible values. For example, the probability distribution of the total number
of spots S showing on the roll of two fair dice can be written as a table: