Unit 2 Beta

Probability, Random Variables, and Probability Distributions: the one-page sheet

2.1 Tabular and Graphical Representations for the Distributions of Two Categorical Variables

A two-way table counts two categorical variables together. To compare groups, turn each group's counts into percents and show them as segmented or side-by-side bars, or a mosaic plot. If the percents differ between groups, the variables are associated; association in a survey does not show cause.

A two-way table shows the counts for each pairing of categories of two categorical variables, with totals.
Compare groups with percents of each group, never with raw counts when group sizes differ.
A segmented bar chart has one bar per group, each split into the other variable's categories and reaching 100%. A side-by-side bar chart puts the bars next to each other; a mosaic plot also makes each bar's width show the group's size.
Two variables are associated when the percents differ from group to group. Equal percents mean no association.
In a survey, association is not cause and effect.

a two-way table counts each pairing of categories, with row and column totals counts cannot be compared directly, so each row is turned into percents of its own total a segmented bar chart puts the groups side by side on the same scale the variables are associated in these data

two-way table: A table of counts for two categorical variables: the categories of one variable label the rows, the other the columns, and each cell counts the individuals in that pairing of categories, with row and column totals.
segmented bar chart: A segmented bar chart has one bar for each group, split into the categories of the other variable by percent of that group, so every bar reaches 100%. A side-by-side bar chart puts those percents as separate bars next to each other; a mosaic plot is a segmented bar chart whose bar widths show the group sizes.
association: Two categorical variables are associated when the distribution of one differs across the categories of the other, for example when the percent who ride the bus differs between class years. Equal percents in every group mean no association.

2.2 Summary Statistics for Two Categorical Variables

From one two-way table you can find three kinds of share. Joint: a cell over the grand total. Marginal: a row or column total over the grand total. Conditional: a cell over the total of its group. Compare groups with conditional relative frequencies; if they differ, the variables are associated.

Joint relative frequency: one cell over the grand total.
Marginal relative frequency: a row or column total over the grand total. The marginal distribution of a variable lists them for every category.
Conditional relative frequency: one cell over its row or column total. Read "of those who ..." to find the group, and divide by that group's total.
Conditional relative frequencies for one group add to 1. If they are the same for every group, there is no association.

a joint relative frequency: the share of everyone in that pairing (45 of 250 are first visits who hiked, 0.18) a marginal relative frequency: the share of everyone in one category (105 of 250 hiked, 0.42) a conditional relative frequency: the share within one group (45 of the 100 first visits hiked, 0.45) if they differ, the variables are associated

joint relative frequency: A joint relative frequency is the count in one cell of a two-way table divided by the grand total: the share of all individuals who are in both categories, such as first visit and hiking.
marginal relative frequency: A marginal relative frequency is a row or column total divided by the grand total: the share of all individuals in one category of one variable. Listed for every category, they form the marginal distribution of that variable.
conditional relative frequency: A conditional relative frequency is a cell count divided by the total of its row or column: the share of one group that falls in a category, such as the share of first-time visitors who hiked. The shares for one group form its conditional distribution and add to 1.

2.3 Estimating Probabilities Using Simulation

To estimate a probability by simulation, model the chance process with a device that has the right chances, define one trial and what to record, repeat many trials, and use the share of trials with the result. More trials give a steadier estimate (the law of large numbers). A rare simulated result casts doubt on the claim behind the model.

A simulation imitates a chance process with a model that has the same chances. Each repetition is a trial.
The probability of a result is the share of times it happens in a very long run of trials. A simulation estimates it with the share of its trials.
Describe a simulation so someone else could run it: the device and what each value means, what one trial is, what you record, how many trials.
The law of large numbers: as the number of trials grows, the share of trials with the result gets closer to the true probability.
A result that happens rarely in a simulation of a claim gives reason to doubt the claim.

imitate it with a model that has the same chances (random numbers, a coin, a die) one trial: here, the number of seniors among 4 random picks the share of trials with the result estimates its probability (2 of 100 gave 4 seniors, about 0.02) by the law of large numbers the estimate settles near the true probability

simulation: A simulation imitates a chance process using a model with the same chances, such as random digits, a coin or a die, and repeats it many times to estimate how often a result happens.
trial: One repetition of a chance process or of its simulation, such as one random pick of 4 council members or one set of 10 simulated free throws.
law of large numbers: The law of large numbers says that as a chance process is repeated more and more times, the proportion of trials with a given result gets closer and closer to its true probability.
probability: The probability of a result is the proportion of times it would happen in a very long series of repetitions of a chance process. It is a number from 0 (never happens) to 1 (always happens), and a simulation estimates it with the share of its trials.

2.4 Introduction to Probability

List the sample space. If its outcomes are equally likely, the probability of an event is its number of outcomes over the total. Probabilities run from 0 to 1 and the outcomes' probabilities add to 1. The complement rule, P(A^c) = 1 − P(A) , is often the fastest route, especially for "at least one".

The sample space is the list of every possible outcome. An event is a set of outcomes, such as "sum is 8".
When outcomes are equally likely, $P(A)$ $= \frac{\text{number of outcomes in } A}{\text{number of outcomes}}$ .
Every probability is from 0 to 1, and the probabilities of all the outcomes add to 1.
The complement $A^c$ is "A does not happen": $P(A^c) = 1 - P(A)$ . It is the quick route for "at least one".

the sample space (36 pairs for two dice) the probability of an event is the number of its outcomes over the total: P(sum is 8) = 5/36 use the complement rule: P(A) = 1 − P(A^c) counting outcomes fails; use the long-run shares instead

sample space: The sample space of a chance process is the list of all its possible outcomes. When those outcomes are equally likely, the probability of an event is its number of outcomes divided by the number in the sample space.
event: An event is a collection of outcomes from the sample space, such as "the sum of two dice is 8". Its probability is the chance that the outcome falls in that collection.
complement: The complement of an event A, written <math class="stat-m"><semantics><mrow><msup><mi>A</mi><mi>c</mi></msup></mrow><annotation encoding="application/x-tex">A^c</annotation></semantics></math>, is the event that A does not happen. The complement rule: <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msup><mi>A</mi><mi>c</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A^c) = 1 - P(A)</annotation></semantics></math>.

2.5 Mutually Exclusive Events

The joint probability P(A∩B) is the chance both events happen; in a table it is one cell over the grand total. Events are mutually exclusive when that joint probability is 0. Only then can you find P(A or B) by adding P(A) and P(B).

The joint probability $P(A \cap B)$ is the probability that A and B both happen.
Events are mutually exclusive (disjoint) when they cannot happen together: $P(A \cap B) = 0$ .
For mutually exclusive events, P(A or B) = P(A) + P(B). If they overlap, plain adding double counts.
Two events whose probabilities add to more than 1 cannot be mutually exclusive.

the joint event A∩B ("A and B") is those shared outcomes, and its probability is above 0 they are mutually exclusive: P(A∩B) = 0 the probability of one or the other is the sum: P(A) + P(B) the shared outcomes are counted twice and the answer is too big

joint probability: The joint event "A and B", written <math class="stat-m"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \cap B</annotation></semantics></math> (the intersection of A and B), happens when both A and B happen. Its probability is the joint probability <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A \cap B)</annotation></semantics></math>; in a two-way table it is one cell over the grand total.
mutually exclusive: Two events are mutually exclusive, or disjoint, when they have no outcomes in common, so they cannot happen together: <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P(A \cap B) = 0</annotation></semantics></math>. Then the probability that one or the other happens is <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A) + P(B)</annotation></semantics></math>.

2.6 Conditional Probability

Given B, the outcomes in B are the new whole, so P(A|B) = (P(A∩B))/(P(B)) . Rearranged, the multiplication rule P(A∩B) = P(A)⋅P(B|A) is what a tree diagram does along each path. Adding paths gives a total probability, and dividing reverses a condition, which usually changes the answer.

A conditional probability $P(A \mid B)$ is the probability of A given that B happened: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ . In a table, divide by the total of B's row or column.
The multiplication rule: $P(A \cap B) = P(A) \cdot P(B \mid A)$ . Drawing without replacement changes the second probability.
A tree diagram shows stages: first-stage probabilities, then conditional ones. Multiply along a path; add paths.
$P(A \mid B)$ and $P(B \mid A)$ are usually different. Name the group after the bar.

only the outcomes in B are possible now, so B becomes the new total is the conditional probability: P(A|B) = (P(A∩B))/(P(B)) the multiplication rule: P(A∩B) = P(A)⋅P(B|A) , which is multiplying along a tree's path you get its total probability, and dividing one path by it reverses the condition

conditional probability: The conditional probability of A given B, written <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>|</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A \mid B)</annotation></semantics></math>, is the probability that A happens when we know B has happened: <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>|</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">P(A \mid B) = \frac{P(A \cap B)}{P(B)}</annotation></semantics></math>. In a two-way table it is a cell divided by the total of the row or column for B.
multiplication rule: The general multiplication rule finds the probability that two events both happen: <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo>|</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A \cap B) = P(A) \cdot P(B \mid A)</annotation></semantics></math>. For two draws without replacement, the second probability is the one given the first draw.
tree diagram: A tree diagram shows a chance process in stages: branches for the first stage with their probabilities, then branches for the second stage with probabilities given the first. Multiplying along a path gives the probability of that path; adding paths gives the probability of an event.

2.7 Independent Events and Unions of Events

Events are independent when P(A|B) = P(A) ; then the probability of both is the product P(A)⋅P(B) . Check independence with the numbers, never by assumption alone. The probability of A or B is P(A) + P(B) − P(A∩B) , subtracting the overlap that would be counted twice; for mutually exclusive events the overlap is 0.

Events are independent when knowing one happened does not change the probability of the other: $P(A \mid B) = P(A)$ . Then, and only then, $P(A \cap B) = P(A) \cdot P(B)$ .
To check independence, compare $P(A \mid B)$ with $P(A)$ , or $P(A \cap B)$ with $P(A) \cdot P(B)$ , using the numbers.
The union $A \cup B$ is "A or B or both". The general addition rule: $P(A \cup B)$ $= P(A) + P(B) - P(A \cap B)$ .
A Venn diagram shows events as overlapping circles; fill the overlap first.
Mutually exclusive is not independent: events that cannot happen together affect each other completely.

A and B are independent: P(A|B) = P(A) for independent events, P(A∩B) = P(A)⋅P(B) the outcomes in both are counted twice the general addition rule: P(A∪B) = P(A) + P(B) − P(A∩B)

independent events: Two events are independent when knowing that one happened does not change the probability of the other: <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>|</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A \mid B) = P(A)</annotation></semantics></math>. For independent events, <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A \cap B) = P(A) \cdot P(B)</annotation></semantics></math>. Events that are not independent are dependent.
union: The union of A and B, written <math class="stat-m"><semantics><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \cup B</annotation></semantics></math>, is the event that A or B or both happen. The general addition rule gives its probability: <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A \cup B)</annotation></semantics></math><wbr><math class="stat-m"><semantics><mrow><mo form="infix">=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= P(A) + P(B) - P(A \cap B)</annotation></semantics></math>.
Venn diagram: A Venn diagram draws the sample space as a rectangle and events as circles inside it. Overlapping circles show outcomes in both events; the regions can hold counts or probabilities.

2.8 Introduction to Random Variables and Probability Distributions

A random variable is a number from a chance process, defined in context. Its probability distribution lists each value with its probability; the probabilities are from 0 to 1 and add to 1. Probabilities of events like "at least 3" or "fewer than 3" come from adding the right values, and cumulative probabilities add everything up to a boundary.

A random variable is a number that comes from a chance process. Define it in context: "Let X = the number of people in a randomly chosen car."
A probability distribution gives every value of the random variable and its probability, as a table or a probability histogram. The probabilities are from 0 to 1 and add to 1.
A discrete random variable has separate values (counts); a continuous one can take any value in an interval.
A cumulative probability adds up to a boundary: $P(X \le 3)$ . "Fewer than 3" is $X \le 2$ ; "at least 3" is $X \ge 3$ .

that number is a random variable, named with a capital letter and defined in context the probability distribution, a table or a probability histogram the probabilities are from 0 to 1 and add to 1, which finds a missing value a cumulative probability such as P(X ≤ 3) ; translate "fewer than 3" as X ≤ 2

random variable: A random variable is a numerical value that depends on the outcome of a chance process, such as the number of people in a randomly chosen car. It is named with a capital letter and defined in context.
probability distribution: The probability distribution of a random variable gives each of its possible values and the probability of each, as a table, a formula or a probability histogram. The probabilities are from 0 to 1 and add to 1.
cumulative probability: A cumulative probability is the probability that a random variable is at most a given value, such as <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X \le 3)</annotation></semantics></math>, found by adding the probabilities of every value up to and including that value.

2.9 Parameters of Random Variables

The mean of a discrete random variable weights each value by its probability: μ_X = ∑ x_ip_i . It is the average over many repetitions. The standard deviation σ_X = √(∑ (x_i − μ_X)²p_i) is a typical distance from that mean. Write the expression with the numbers substituted, then interpret both in context.

The mean or expected value of a discrete random variable is $\mu_X = E(X) = \sum x_i p_i$ : each value times its probability, added up. It is the long-run average and need not be a possible value.
The standard deviation is $\sigma_X$ $= \sqrt{\sum (x_i - \mu_X)^2 p_i}$ : a typical distance of the values from the mean, in the units of X.
Show the expression, the substitution and the answer: $1(0.20) + 2(0.35) + \dots$ $= 2.5$ .
Interpret in context: "Over many orders, the average number of tacos per order is about 2.5."

a plain average of the values is wrong; each value must count by its probability the mean (expected value) μ_X = ∑ x_ip_i , the long-run average the variance; its square root is the standard deviation σ_X the mean is the average over many repetitions; the SD is how far values typically land from it

expected value: The mean, or expected value, of a discrete random variable X is <math class="stat-m"><semantics><mrow><msub><mi>μ</mi><mi>X</mi></msub><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo largeop="true" movablelimits="false">∑</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mu_X = E(X) = \sum x_i p_i</annotation></semantics></math>: each value times its probability, added up. It is the average value of X over very many repetitions of the chance process.
standard deviation of a random variable: The standard deviation of a discrete random variable X is <math class="stat-m"><semantics><mrow><msub><mi>σ</mi><mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_X</annotation></semantics></math><wbr><math class="stat-m"><semantics><mrow><mo form="infix">=</mo><msqrt><mrow><mo largeop="true" movablelimits="false">∑</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>X</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><msub><mi>p</mi><mi>i</mi></msub></mrow></msqrt></mrow><annotation encoding="application/x-tex">= \sqrt{\sum (x_i - \mu_X)^2 p_i}</annotation></semantics></math>. It measures how far the values of X typically are from the mean, in the units of X. Its square, before the square root, is the variance.

2.10 The Binomial Distribution

When a yes-or-no trial repeats n times independently with the same success probability p, the number of successes is binomial: P(X = k) = (nCk)p^k(1 − p)^(n − k) , with mean np and standard deviation √(np(1 − p)) . Name the distribution and its parameters, translate the event into an inequality, and use the complement for "at least" questions. A tiny probability under a claim casts doubt on the claim.

A binomial setting: two outcomes per trial, independent trials, a fixed number n of trials, the same success probability p on every trial.
The binomial distribution: $P(X = k)$ $= \binom{n}{k}p^k(1 - p)^{n - k}$ for $k = 0, 1, \dots, n$ .
Show work: name the distribution and parameters, the boundary and the direction: "X is binomial, $n = 10$ , $p = 0.3$ ; $P(X \ge 4)$ $= 1 - P(X \le 3)$ $= 0.350$ ."
Mean $\mu_X = np$ and standard deviation $\sigma_X = \sqrt{np(1 - p)}$ .
Translate carefully: "fewer than 3" is $P(X \le 2)$ , "at least 4" is $P(X \ge 4)$ .

the count of successes X is binomial with parameters n and p and there are (nCk) such orders, so P(X = k) = (nCk)p^k(1 − p)^(n − k) add them, or use the complement: P(X ≥ 4) = 1 − P(X ≤ 3) the result gives reason to doubt the claim

binomial setting: A binomial setting is a chance process with a fixed number n of trials, each with two outcomes (success or failure), trials independent of each other, and the same probability of success p on every trial.
binomial distribution: In a binomial setting, the binomial random variable X counts the successes in the n trials. Its distribution, the binomial distribution with parameters n and p, gives <math class="stat-m"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X = k)</annotation></semantics></math><wbr><math class="stat-m"><semantics><mrow><mo form="infix">=</mo><mrow><mo>(</mo><mfrac linethickness="0"><mi>n</mi><mi>k</mi></mfrac><mo>)</mo></mrow><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">= \binom{n}{k}p^k(1 - p)^{n - k}</annotation></semantics></math> for <math class="stat-m"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k = 0, 1, \dots, n</annotation></semantics></math>.
mean of a binomial distribution: A binomial random variable with parameters n and p has mean <math class="stat-m"><semantics><mrow><msub><mi>μ</mi><mi>X</mi></msub><mo>=</mo><mi>n</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">\mu_X = np</annotation></semantics></math> and standard deviation <math class="stat-m"><semantics><mrow><msub><mi>σ</mi><mi>X</mi></msub><mo>=</mo><msqrt><mrow><mi>n</mi><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sigma_X = \sqrt{np(1 - p)}</annotation></semantics></math>.

2.11 The Normal Distribution

A density curve models a distribution with areas as probabilities. Normal distributions N(μ, σ) are symmetric and bell-shaped, with about 68%, 95% and 99.7% of values within 1, 2 and 3 SDs. Standardize with z = (x − μ)/(σ) to find areas, or work back from an area to a boundary with x = μ + zσ . Always name the distribution, parameters, boundary and direction.

A density curve is on or above the axis with total area 1; the area over an interval is the probability of that interval.
A normal distribution $N(\mu, \sigma)$ is symmetric and bell-shaped, centered at $\mu$ with spread $\sigma$ . The standard normal is $N(0, 1)$ .
The empirical rule: about 68%, 95% and 99.7% within 1, 2 and 3 standard deviations of the mean.
Show work: name the distribution and parameters, the boundary and the direction, for example " $N(58, 4)$ , $P(W > 66)$ ", with a sketch.
Forward: $z = \frac{x - \mu}{\sigma}$ , then the area. Backward: the area, then z, then $x = \mu + z\sigma$ .

a normal curve N(μ, σ) can model it, and areas under the curve are probabilities about 68%, 95% and 99.7% of values lie within 1, 2 and 3 SDs (the empirical rule) any normal question becomes one about the standard normal N(0, 1) , answered by Table A or technology inverse normal gives the z, then x = μ + zσ gives the boundary

density curve: A density curve is a smooth curve that models a distribution: it is on or above the horizontal axis and has an area of exactly 1 under it. The area under the curve over an interval is the proportion of values, or the probability, in that interval.
normal distribution: A normal distribution, written <math class="stat-m"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mu, \sigma)</annotation></semantics></math>, has a symmetric, bell-shaped density curve centered at its mean <math class="stat-m"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>, with standard deviation <math class="stat-m"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> setting its spread. It models many roughly symmetric, mound-shaped distributions.
standard normal distribution: The standard normal distribution is the normal distribution with mean 0 and standard deviation 1, <math class="stat-m"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(0, 1)</annotation></semantics></math>. Standardizing a value, <math class="stat-m"><semantics><mrow><mi>z</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac></mrow><annotation encoding="application/x-tex">z = \frac{x - \mu}{\sigma}</annotation></semantics></math>, turns any normal question into a standard normal one, answered by Table A or technology.
empirical rule: The empirical rule (68-95-99.7 rule): in a normal distribution about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.

2.12 Sampling Distributions and the Central Limit Theorem

A statistic varies from sample to sample, and its sampling distribution describes how. For the sample mean, the center is μ and the spread is (σ)/(√n) , so larger samples give less variable means. By the Central Limit Theorem, for large n the sampling distribution of x-bar is approximately normal even from a skewed population, which lets you judge whether a sample mean is surprising.

A statistic varies from sample to sample. Its sampling distribution is the distribution of its values over all possible random samples of the same size.
Keep three distributions apart: the population, one sample's data, and the sampling distribution of a statistic.
For $\bar{x}$ : center $\mu_{\bar{x}} = \mu$ and spread $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ . Four times the sample size halves the spread. For $\hat{p}$ , the center is p.
The Central Limit Theorem: for large n (a common guide is 30 or more), the sampling distribution of $\bar{x}$ is approximately normal whatever the population's shape. A normal population gives normal $\bar{x}$ at any n.

a statistic such as x-bar or p-hat varies from sample to sample (sampling variability) its sampling distribution, with a center, a spread and a shape the center stays at the parameter, μ_(x-bar) = μ , and the spread shrinks, σ_(x-bar) = (σ)/(√n) the Central Limit Theorem makes the sampling distribution of x-bar approximately normal

sampling distribution: The sampling distribution of a statistic is the distribution of the values it takes in all possible random samples of the same size from the same population. Its spread shows sampling variability: how much the statistic changes from sample to sample. For the sample mean, <math class="stat-m"><semantics><mrow><msub><mi>μ</mi><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover></msub><mo>=</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu_{\bar{x}} = \mu</annotation></semantics></math> and <math class="stat-m"><semantics><mrow><msub><mi>σ</mi><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover></msub><mo>=</mo><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac></mrow><annotation encoding="application/x-tex">\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}</annotation></semantics></math>.
central limit theorem: The Central Limit Theorem says that when the sample size n is large, the sampling distribution of the sample mean is approximately normal, whatever the shape of the population. A common guide is n of 30 or more; if the population is normal, <math class="stat-m"><semantics><mrow><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{x}</annotation></semantics></math> is normal for any n.