Glossary

The terms the chapters and tools use, each with a short definition and a link to the section that explains it.

Showing 90 of 90 terms.

68-95-99.7 rule

About 68 %, 95 % and 99.7 % of a normal distribution lie within 1, 2 and 3 standard deviations of the mean. Chapter 8 · The standard normal and z-scores: The 68-95-99.7 rule

Addition rule

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\): add the two probabilities and take the overlap away once, since it was counted twice. Chapter 1 · Probability and events: The addition rule

Base rate

How common something is before any evidence. The rarer the condition, the less a positive result means, however good the test. Chapter 2 · Conditional probability and Bayes: Natural frequencies

Bayes' rule

Turns \(P(B \mid A)\) into \(P(A \mid B)\): \(P(A \mid B) = \tfrac{P(B \mid A)\, P(A)}{P(B)}\). It answers what a positive test result means. Chapter 2 · Conditional probability and Bayes: Bayes' rule

Bessel's correction

Dividing the squared deviations by \(n - 1\) rather than \(n\), which makes \(s^2\) an unbiased estimate of \(\sigma^2\). Chapter 11 · Student's t and confidence intervals: Bias and Bessel's n − 1

Binomial coefficient

\(\binom{n}{k} = \tfrac{n!}{k!\,(n - k)!}\): the number of ways to choose which \(k\) of \(n\) trials are the successes. In Python math.comb, in Excel COMBIN. Chapter 5 · The binomial distribution: Deriving the formula

Binomial distribution

The number of successes in \(n\) independent trials with the same probability of success \(p\): \(B(n, p)\), with mean \(np\) and variance \(np(1 - p)\). Chapter 5 · The binomial distribution: Counting successes

Central limit theorem (CLT)

For independent draws from almost any population with a finite \(\sigma\), the sample mean is approximately \(N(\mu, \tfrac{\sigma}{\sqrt{n}})\) once \(n\) is large enough. The data themselves do not become normal. Chapter 10 · Sampling distributions and the CLT: The central limit theorem

Chi-square distribution

The sum of the squares of \(k\) independent standard normals, \(\chi^2_k\): the building block of Student's t, and of tests of a variance. The chi-square distribution

Complement

Every outcome that is not in an event \(A\), written \(A^c\). Its probability is \(1 - P(A)\): the easy way into "at least one" questions. Chapter 1 · Probability and events: The complement rule

Conditional probability

The probability of \(A\) once \(B\) is known to have happened, \(P(A \mid B) = \tfrac{P(A \cap B)}{P(B)}\). It is a different question from \(P(B \mid A)\). Chapter 2 · Conditional probability and Bayes: Conditional probability

Confidence interval

An estimate plus and minus a margin of error, \(\bar{x} \pm t^* \tfrac{s}{\sqrt{n}}\), built so that a chosen share of such intervals contain the true value. Chapter 11 · Student's t and confidence intervals: z and t intervals

Confidence level

The share of intervals built by the procedure that contain the true value, such as 95 %. It describes the procedure, not any one interval. Chapter 11 · Student's t and confidence intervals: The t* table

Continuity correction

The half added when a normal curve approximates counts: the bar at \(k\) runs from \(k - \tfrac{1}{2}\) to \(k + \tfrac{1}{2}\), so \(P(X \le k)\) uses the curve up to \(k + \tfrac{1}{2}\). Chapter 8 · The standard normal and z-scores: The normal approximation to the binomial

Continuous random variable

A random variable that can take any value in an interval, such as a time or a weight. A single value has probability 0. Chapter 3 · Random variables: From outcomes to numbers

Continuous uniform distribution

Any value between \(a\) and \(b\) equally likely, with a flat density of \(\tfrac{1}{b - a}\); a probability is a length divided by the width. Chapter 4 · The uniform distribution: The continuous uniform

Coverage

The share of intervals, over many samples, that contain the true value. A 95 % procedure should cover it about 95 times in 100. Chapter 11 · Student's t and confidence intervals: Reading an interval correctly

Critical value

\(z^*\) or \(t^*\): the value that leaves \(\tfrac{1 - C}{2}\) in each tail for a confidence level \(C\). Chapter 11 · Student's t and confidence intervals: z and t intervals

Cumulative distribution function (CDF)

\(F(x) = P(X \le x)\): the probability of a value at most \(x\). It answers every "at most" question. In scipy, cdf. Chapter 3 · Random variables: The cumulative distribution function

Degrees of freedom

The t distribution's parameter \(\nu\), \(n - 1\) for a mean: one degree of freedom goes into estimating \(\bar{x}\). Chapter 11 · Student's t and confidence intervals: The t statistic

Density

Probability per unit of the measurement, the height of a PDF. It is not a probability and can be greater than 1; only areas under it are probabilities. Chapter 4 · The uniform distribution: A density can be greater than 1

Discrete random variable

A random variable whose values can be listed one by one, such as a count. Discrete means countable, not finite. Chapter 3 · Random variables: From outcomes to numbers

Discrete uniform distribution

Every whole number from \(a\) to \(b\) equally likely, each with probability \(\tfrac{1}{b - a + 1}\): a die, a bookseller's daily sales. In scipy, randint(a, b + 1). Chapter 4 · The uniform distribution: The discrete uniform

Estimator

A rule for estimating a population's number from a sample, such as "average the sample". Being computed from a random sample, it is a random variable. Chapter 11 · Student's t and confidence intervals: Point estimates

Event

A set of outcomes: something that either happens or does not once the experiment is over, such as "an even number". Chapter 1 · Probability and events: Experiments, outcomes and events

Excess kurtosis

A measure of how heavy a distribution's tails are, set so that a normal scores 0. The simulated share's daily returns score far above it. Chapter 7 · The normal distribution: When the normal is the wrong tool

Expected value

The mean of a random variable, \(E[X]\) or \(\mu\): the values weighted by their probabilities, and the long-run average over many repetitions. Chapter 3 · Random variables: Expected value

Experiment

Anything with an uncertain result that could, in principle, be repeated: a roll of a die, a customer deciding whether to buy. Chapter 1 · Probability and events: Experiments, outcomes and events

Exponential distribution

The waiting time between events that arrive independently at a steady rate \(\lambda\): mean \(\tfrac{1}{\lambda}\), and \(P(T > t) = e^{-\lambda t}\). Chapter 9 · The exponential distribution: Waiting times

False positive

A positive result for someone without the condition, or an alert on a genuine payment. When the condition is rare, false positives can outnumber true ones. Chapter 2 · Conditional probability and Bayes: Natural frequencies

Fat tails

More extreme values than a normal distribution allows, as in share returns. A normal model then understates the crashes. Chapter 7 · The normal distribution: When the normal is the wrong tool

Galton board

Balls falling through rows of pegs, bouncing left or right at each: the bins they land in fill up to a binomial distribution. Galton board

Histogram

Bars that count how many values fall in each bin, drawn as densities so that a fitted curve can be laid over them. Data-fit lab

Hypothesis test

A decision whether data are consistent with a claim, such as a stated mean. A claimed mean outside the 95 % interval would be rejected at the 5 % level. The Hypothesis Testing Lab covers it. Chapter 11 · Student's t and confidence intervals: From intervals to tests

Independent events

Events where knowing that one happened does not change the chance of the other, so \(P(A \cap B) = P(A)\, P(B)\). Not the same as mutually exclusive. Chapter 2 · Conditional probability and Bayes: Independent events

Intersection

The event "\(A\) and \(B\)", \(A \cap B\): both happen. Chapter 1 · Probability and events: The addition rule

Law of large numbers

When an experiment is repeated independently, the relative frequency of an event settles towards its probability. It says nothing about the next trial. Chapter 1 · Probability and events: The law of large numbers

Law of total probability

The probability of \(B\) is the sum over the branches that lead to it: \(P(B) = P(A)\, P(B \mid A) + P(A^c)\, P(B \mid A^c)\). Chapter 2 · Conditional probability and Bayes: Tree diagrams and total probability

Linearity of expectation

\(E[aX + b] = a\,E[X] + b\), and the mean of a sum is the sum of the means. A shift moves the mean but not the spread. Chapter 3 · Random variables: Linearity of expectation

Log return

The change in a price on a log scale, \(\ln(P_t / P_{t-1})\). Log returns add up over time; a 50 % fall is \(\ln 0.5\), not −0.5. Chapter 7 · The normal distribution: The simulated share's daily log returns

Margin of error

Half the width of a confidence interval: the critical value times the standard error. More confidence widens it; more data narrows it. Chapter 11 · Student's t and confidence intervals: The three levers

Median

The middle of a distribution, its 50th percentile. For a skewed distribution such as the exponential it lies below the mean. Chapter 9 · The exponential distribution: Waiting times

Memorylessness

The exponential's property that time already waited says nothing about the time still to wait: \(P(T > s + t \mid T > s) = P(T > t)\). Chapter 9 · The exponential distribution: Memorylessness

Multiplication rule

\(P(A \cap B) = P(A)\, P(B \mid A)\): the chance of the first event times the chance of the second given the first. Chapter 2 · Conditional probability and Bayes: The multiplication rule

Mutually exclusive events

Events that cannot happen together, such as a 1 and a 6 on the same roll, so \(P(A \cap B) = 0\). Chapter 1 · Probability and events: The addition rule

Natural frequencies

Probabilities written as counts of people out of a round number, such as 10,000, which makes Bayes' rule easy to follow. Chapter 2 · Conditional probability and Bayes: Natural frequencies

Normal approximation to the binomial

For large \(n\), \(B(n, p)\) is close to the normal with \(\mu = np\) and \(\sigma = \sqrt{np(1 - p)}\); a rule of thumb asks for \(np \ge 5\) and \(n(1 - p) \ge 5\). Chapter 8 · The standard normal and z-scores: The normal approximation to the binomial

Normal distribution

The bell-shaped model for measurements that cluster symmetrically around a mean, \(N(\mu, \sigma)\): \(\mu\) places it and \(\sigma\) sets its width. Chapter 7 · The normal distribution: Measurements, not counts

Number of samples

In a simulation, how many samples are drawn. It makes the histogram of means smoother, not narrower; its width is set by \(n\). Chapter 10 · Sampling distributions and the CLT: n and the number of samples

Off-by-one trap

Reading a count's boundary wrongly: "at least 11" includes 11, so it is \(1 - F(10)\), not \(1 - F(11)\). Chapter 5 · The binomial distribution: Words to probabilities, and the off-by-one trap

Outcome

One possible result of an experiment, such as the face a die shows. Chapter 1 · Probability and events: Experiments, outcomes and events

p-value

The probability, if a claim were true, of data at least as extreme as those seen. A small p-value casts doubt on the claim. Bulb reliability: From here to a hypothesis test

Percent point function (PPF)

The CDF turned around: the smallest value whose cumulative probability reaches \(q\). Also called the quantile function. In scipy, ppf. Chapter 3 · Random variables: Percentiles: the CDF turned around

Percentile

The value below which a given share of the distribution lies: the 90th percentile is the value that 90 % of outcomes do not exceed. Chapter 3 · Random variables: Percentiles: the CDF turned around

Point estimate

The single number an estimator gives for the sample at hand, such as \(\bar{x} = 53.6\) for the wildfires. Chapter 11 · Student's t and confidence intervals: Point estimates

Poisson distribution

The number of events in an interval when they arrive independently at a steady rate \(\lambda\), with no fixed number of trials. Its mean and variance are both \(\lambda\). Chapter 6 · The Poisson distribution: Counting events in an interval

Poisson limit of the binomial

When \(n\) is large and \(p\) small, the binomial \(B(n, p)\) is close to the Poisson with \(\lambda = np\): the model for rare events among many chances. Chapter 6 · The Poisson distribution: The Poisson as the limit of the binomial

Population

Every unit a question is about, such as every cup a machine will pour. Its mean and standard deviation are written \(\mu\) and \(\sigma\). Chapter 10 · Sampling distributions and the CLT: Population, sample and sampling distribution

Prevalence

The share of the people tested who have the condition: the base rate before any test result is known. Chapter 2 · Conditional probability and Bayes: Tree diagrams and total probability

Probability

A number from 0 to 1 saying how likely an event is: the share of equally likely outcomes it contains or, estimated from data, the share of past cases in which it happened. Chapter 1 · Probability and events: Probability as a share of equally likely outcomes

Probability density function (PDF)

For a continuous random variable, the curve \(f(x)\) whose area between two values is the probability of falling between them. In scipy, pdf. Chapter 3 · Random variables: Densities for continuous variables

Probability mass function (PMF)

For a discrete random variable, the probability of each value, \(f(k) = P(X = k)\). In scipy, pmf. Chapter 3 · Random variables: The probability mass function

Q-Q plot

Each sorted data value against the value a fitted distribution expects at the same rank. A good fit lies on the line; fat tails bend away from it at both ends. Data-fit lab

Random variable

A rule that gives each outcome of an experiment a number, written with a capital letter, \(X\). Chapter 3 · Random variables: From outcomes to numbers

Rate (λ)

The average number of events per interval, \(\lambda\). It scales with the interval: 4 calls an hour is 2 in half an hour. Chapter 6 · The Poisson distribution: Counting events in an interval

Relative frequency

The share of cases in which an event happened. It estimates the event's probability, and settles towards it as the cases grow in number. Chapter 1 · Probability and events: Probability from data

Sample

The units actually measured, drawn from the population. Chapter 10 · Sampling distributions and the CLT: Population, sample and sampling distribution

Sample proportion

The share of a sample with a property, \(\hat{p}\): the estimate of a probability \(p\). Chapter 11 · Student's t and confidence intervals: Point estimates

Sample size

\(n\), the number of values in each sample: the CLT's knob, which narrows the sampling distribution. Chapter 10 · Sampling distributions and the CLT: n and the number of samples

Sample space

The set of all the outcomes of an experiment, written \(S\). Chapter 1 · Probability and events: Experiments, outcomes and events

Sample standard deviation

\(s\), the spread of a sample computed with \(n - 1\) in the denominator: the estimate of \(\sigma\). numpy std(ddof=1), Excel STDEV.S. Chapter 11 · Student's t and confidence intervals: Point estimates

Sampling distribution

The distribution of a statistic over all the samples of the same size that could have been drawn. Chapter 10 · Sampling distributions and the CLT: Population, sample and sampling distribution

Sensitivity

The share of people with the condition whom a test finds: the probability of a positive result given the condition. Chapter 2 · Conditional probability and Bayes: Tree diagrams and total probability

Skewness

A measure of a distribution's lopsidedness: positive with a long right tail, as for waiting times, and 0 when it is symmetric. Chapter 10 · Sampling distributions and the CLT: How large must n be?

Specificity

The share of people without the condition whom a test clears: the probability of a negative result given no condition. Chapter 2 · Conditional probability and Bayes: Tree diagrams and total probability

Square root of time

Over \(T\) independent periods the mean of a sum grows with \(T\) and its standard deviation with \(\sqrt{T}\): how a day's volatility becomes a year's. Chapter 10 · Sampling distributions and the CLT: Annualising returns: the square root of time

Standard deviation

The square root of the variance, \(\sigma\): the typical distance from the mean, in the units of \(X\). Chapter 3 · Random variables: Variance and standard deviation

Standard error

The standard deviation of a statistic's sampling distribution. For the mean, \(\tfrac{\sigma}{\sqrt{n}}\), estimated by \(\tfrac{s}{\sqrt{n}}\): four times the data halves it. Chapter 10 · Sampling distributions and the CLT: The standard error

Standard normal distribution

The normal with mean 0 and standard deviation 1, \(Z \sim N(0, 1)\). Its CDF is written \(\Phi\). Chapter 8 · The standard normal and z-scores: Standardising

Statistic

A number computed from a sample, such as the sample mean \(\bar{x}\). It changes from one sample to the next. Chapter 10 · Sampling distributions and the CLT: Population, sample and sampling distribution

Student's t distribution

The distribution of \(t = \tfrac{\bar{x} - \mu}{s / \sqrt{n}}\) for a roughly normal population: centred on 0 like the standard normal, with heavier tails that thin as the degrees of freedom grow. Chapter 11 · Student's t and confidence intervals: The t statistic

Survival function

\(1 - F(x) = P(X > x)\), computed directly so that small tail probabilities keep their precision. In scipy, sf; "at least \(k\)" for a count is sf(k - 1). Chapter 6 · The Poisson distribution: In code

Tree diagram

A drawing of a two-stage experiment whose branches carry conditional probabilities; multiplying along a path gives that path's probability. Chapter 2 · Conditional probability and Bayes: Tree diagrams and total probability

Trial

One of the repetitions a binomial counts over, each with two outcomes, success or failure. A success is simply the outcome being counted. Chapter 5 · The binomial distribution: Counting successes

Unbiased estimator

An estimator that, averaged over all possible samples, hits the true value. The sample mean is unbiased for the population mean. Chapter 11 · Student's t and confidence intervals: Bias and Bessel's n − 1

Uniform distribution

The model for "every value in a range equally likely", in a discrete and a continuous form. Chapter 4 · The uniform distribution: All values equally likely

Union

The event "\(A\) or \(B\)", \(A \cup B\): at least one of them happens. Chapter 1 · Probability and events: The addition rule

Variance

The expected squared distance from the mean, \(\operatorname{Var}(X) = E[(X - \mu)^2]\). Chapter 3 · Random variables: Variance and standard deviation

Venn diagram

A picture of events as overlapping regions, so the union, the intersection and the complement can be seen. Probability rules playground

z-score

How many standard deviations a value lies from the mean, \(z = \tfrac{x - \mu}{\sigma}\), with the distribution's own \(\mu\) and \(\sigma\). It puts unlike quantities on one scale. Chapter 8 · The standard normal and z-scores: Standardising