Bocconi Statistics 30001 · exam prep

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227 formulas from the Bocconi Statistics 30001 course, each rendered cleanly with a plain-English explanation and a breakdown of what every symbol stands for. Built to help you prepare efficiently and drill until it sticks.

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Descriptive statistics & frequencies44

What it means

Share of observations in category k: divide its count by the total number of observations. Always between 0 and 1.

Components

$n$ sample size
$k, j$ category/class/variable indices
$f_{k}$ absolute frequency
$p_{k}$ relative frequency

What it means

The same share expressed on a 0-100 scale; just multiply the relative frequency by 100.

Components

$n$ sample size
$k, j$ category/class/variable indices
$f_{k}$ absolute frequency
$p_{k}$ relative frequency

What it means

Counting every category once must return the whole sample: a useful check that no observation was lost or double-counted.

Components

$n$ sample size
$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$f_{k}$ absolute frequency
$\sum$ sum

What it means

Shares of all categories must add up to the whole (1). If they don't, a frequency is wrong.

Components

$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$p_{k}$ relative frequency
$\sum$ sum

What it means

Length of a grouped interval: upper endpoint minus lower endpoint. Needed whenever classes have unequal size.

Components

$k, j$ category/class/variable indices
$L_{k}, U_{k}$ lower and upper class endpoints
$w_{k}$ class width

What it means

Height of a histogram bar: relative frequency spread over the class width, so unequal classes stay comparable by area.

Components

$k, j$ category/class/variable indices
$w_{k}$ class width
$p_{k}$ relative frequency
$c_{k}$ frequency density

What it means

Inverting the density: the area of a histogram bar (height x width) is the class's relative frequency.

Components

$k, j$ category/class/variable indices
$w_{k}$ class width
$p_{k}$ relative frequency
$c_{k}$ frequency density

What it means

Adds up shares from the first class to class k, i.e. the proportion of data at or below that class.

Components

$k, j$ category/class/variable indices
$p_{k}$ relative frequency
$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$\sum$ sum

What it means

Function giving the fraction of observations less than or equal to x; the empirical version of a CDF, non-decreasing from 0 to 1.

Components

$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$X, Y$ random variables

What it means

The balance point of the data: add all values and divide by how many there are.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$\sum$ sum

What it means

When data are grouped, weight each class value by its count instead of listing all observations.

Components

$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$f_{k}$ absolute frequency
$x ˉ, y ˉ$ observed sample means
$\sum$ sum

What it means

Same weighted average written with shares, so it is directly a weighted mean with weights summing to 1.

Components

$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$p_{k}$ relative frequency
$x ˉ, y ˉ$ observed sample means
$\sum$ sum

What it means

Deviations above and below the mean cancel exactly, which is why we square them to measure spread.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$\sum$ sum

What it means

Rearranged mean: knowing the mean and n gives the total. Handy for combining or updating groups.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$\sum$ sum

What it means

Crudest spread measure: distance between largest and smallest value. Very sensitive to outliers.

Components

$x_{min}, x_{ma x}$ minimum and maximum observed values

What it means

Spread of the middle 50% of the data. Robust to outliers, unlike the range.

Components

$Q_{1}, Q_{2}, Q_{3}$ first quartile, median, third quartile
$I QR$ interquartile range

What it means

Boxplot rule: points below this threshold are flagged as (mild) outliers.

Components

$Q_{1}, Q_{2}, Q_{3}$ first quartile, median, third quartile
$I QR$ interquartile range

What it means

Boxplot rule: points above this threshold are flagged as (mild) outliers.

Components

$Q_{1}, Q_{2}, Q_{3}$ first quartile, median, third quartile
$I QR$ interquartile range

What it means

Average squared distance from the population mean, computed over all N units.

Components

$N$ population size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$\sum$ sum

What it means

Computationally faster form: mean of squares minus square of the mean.

Components

$N$ population size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$\sum$ sum

What it means

Average squared deviation using n-1 (Bessel's correction) so the estimator is unbiased for the population variance.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$s^{2}, S^{2}$ sample variance / sample variance estimator
$s, S, σ$ sample/sample-statistic/population standard deviations
$\sum$ sum

What it means

Same value from raw sums: sum of squares minus n times the squared mean, divided by n-1.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$s^{2}, S^{2}$ sample variance / sample variance estimator
$s, S, σ$ sample/sample-statistic/population standard deviations
$\sum$ sum

What it means

Square root of the population variance, back in the original units of the variable.

Components

$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations

What it means

Square root of the sample variance; the usual reported measure of spread.

Components

$s^{2}, S^{2}$ sample variance / sample variance estimator
$s, S, σ$ sample/sample-statistic/population standard deviations

What it means

Relative spread: standard deviation per unit of mean. Lets you compare variability across different scales or units.

Components

$x ˉ, y ˉ$ observed sample means
$s, S, σ$ sample/sample-statistic/population standard deviations
$C V$ coefficient of variation

What it means

Average product of paired deviations over the population: positive when the two variables tend to move together.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$μ, μ_{X}, μ_{Y}$ population means
$s_{X Y}, σ_{X Y}$ sample and population covariance
$\sum$ sum

What it means

Mean of the products minus product of the means; faster from raw sums.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$μ, μ_{X}, μ_{Y}$ population means
$s_{X Y}, σ_{X Y}$ sample and population covariance
$\sum$ sum

What it means

Sample version with n-1 in the denominator, so it is unbiased for the population covariance.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$s_{X Y}, σ_{X Y}$ sample and population covariance
$\sum$ sum

What it means

Raw-sum form of sample covariance, convenient when you are given sums from a table.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$s_{X Y}, σ_{X Y}$ sample and population covariance
$\sum$ sum

What it means

Covariance rescaled by both standard deviations, giving a unit-free measure of linear association in [-1,1].

Components

$s, S, σ$ sample/sample-statistic/population standard deviations
$s_{X Y}, σ_{X Y}$ sample and population covariance
$r_{X Y}, ρ_{X Y}$ sample and population correlation

What it means

Sample version of the correlation coefficient; measures strength and direction of a linear relationship only.

Components

$s_{X Y}, σ_{X Y}$ sample and population covariance
$r_{X Y}, ρ_{X Y}$ sample and population correlation

What it means

Correlation can never leave [-1,1]; the extremes mean perfectly linear relationships, 0 means no linear association.

Components

$r_{X Y}, ρ_{X Y}$ sample and population correlation

What it means

The most frequent value (or class): pick the category with the highest absolute frequency. A distribution can have more than one mode.

Components

$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$f_{k}$ absolute frequency
$s, S, σ$ sample/sample-statistic/population standard deviations

What it means

Sort the data: with an odd n the median is the middle observation, with an even n it is the average of the two central ones.

Components

$n$ sample size
$i$ observation index
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability
$S E, se$ standard error / estimated standard error

What it means

For a frequency distribution, the median is the smallest value whose cumulative relative frequency reaches 0.5.

Components

$n$ sample size
$i$ observation index
$k, j$ category/class/variable indices
$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$F$ F test statistic in regression

What it means

Inside the class that first passes 50%: start at its lower endpoint and move forward by the missing share divided by the frequency density.

Components

$k, j$ category/class/variable indices
$L_{k}, U_{k}$ lower and upper class endpoints
$c_{k}$ frequency density
$F$ F test statistic in regression

What it means

Same rule as the median, but with thresholds 0.25 and 0.75 on the cumulative relative frequency.

Components

$n$ sample size
$i$ observation index
$k, j$ category/class/variable indices
$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$Q_{1}, Q_{2}, Q_{3}$ first quartile, median, third quartile
$F$ F test statistic in regression

What it means

Interpolate inside the class that first crosses the 25% (or 75%) threshold, using the frequency density of that class.

Components

$k, j$ category/class/variable indices
$L_{k}, U_{k}$ lower and upper class endpoints
$c_{k}$ frequency density
$F$ F test statistic in regression

What it means

Position method on sorted raw data: if the position is not an integer, interpolate between the two neighbouring observations.

Components

$n$ sample size
$Q_{1}, Q_{2}, Q_{3}$ first quartile, median, third quartile
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability

What it means

The q-th percentile is the value below which q% of the distribution lies. The median is P_50.

Components

$X, Y$ random variables

What it means

General definition of a quantile: the smallest value whose cumulative distribution function reaches the level alpha.

Components

$n$ sample size
$i$ observation index
$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$X, Y$ random variables
$p$ population proportion / Bernoulli success probability
$F$ F test statistic in regression
$α$ significance level / Type I error probability

What it means

The five values that summarise shape and spread and that a box plot draws.

Components

$n$ sample size
$i$ observation index
$x_{min}, x_{ma x}$ minimum and maximum observed values
$Q_{1}, Q_{2}, Q_{3}$ first quartile, median, third quartile

What it means

With classed data the exact values are unknown, so each class is represented by its midpoint and weighted by its relative frequency.

Components

$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$L_{k}, U_{k}$ lower and upper class endpoints
$p_{k}$ relative frequency
$x ˉ, y ˉ$ observed sample means
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability
$\sum$ sum

What it means

When building K equal-width classes, divide the range of the data by the number of classes.

Components

$n$ sample size
$i$ observation index
$K, J$ numbers of categories/classes or explanatory variables
$x_{min}, x_{ma x}$ minimum and maximum observed values

Random variables & distributions52

What it means

The PMF: probability that a discrete random variable takes exactly the value x.

Components

$X, Y$ random variables
$P (\cdot)$ probability
$P_{X} (x)$ probability mass function

What it means

Every probability lies between 0 (impossible) and 1 (certain).

Components

$P_{X} (x)$ probability mass function

What it means

The PMF must exhaust all possible outcomes, so its values add to 1.

Components

$P_{X} (x)$ probability mass function
$\sum$ sum

What it means

Cumulative distribution function: probability of being at or below x. Non-decreasing, from 0 to 1.

Components

$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$X, Y$ random variables
$P (\cdot)$ probability

What it means

Long-run average of X: each value weighted by its probability.

Components

$μ, μ_{X}, μ_{Y}$ population means
$X, Y$ random variables
$P_{X} (x)$ probability mass function
$E (X)$ expected value
$\sum$ sum

What it means

Expected squared distance from the mean, weighted by probabilities.

Components

$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$P_{X} (x)$ probability mass function
$V a r (X)$ variance
$\sum$ sum

What it means

Variance equals the mean of the square minus the square of the mean. Fastest route in most exercises.

Components

$μ, μ_{X}, μ_{Y}$ population means
$X, Y$ random variables
$E (X)$ expected value
$V a r (X)$ variance

What it means

A single trial with two outcomes, success (1) with probability p and failure (0).

Components

$X, Y$ random variables
$p$ population proportion / Bernoulli success probability
$\sim$ is distributed as

What it means

Compact one-line PMF: it returns p at x=1 and 1-p at x=0.

Components

$P_{X} (x)$ probability mass function
$p$ population proportion / Bernoulli success probability

What it means

The expected value of a 0/1 variable is exactly the success probability.

Components

$X, Y$ random variables
$E (X)$ expected value
$p$ population proportion / Bernoulli success probability

What it means

Variance p(1-p); maximal at p=0.5 (most uncertainty), zero at p=0 or 1.

Components

$X, Y$ random variables
$V a r (X)$ variance
$p$ population proportion / Bernoulli success probability

What it means

A density can never be negative, though it may exceed 1 (it is a density, not a probability).

Components

$f_{X} (x)$ probability density function

What it means

The whole area under a density curve equals 1.

Components

$f_{X} (x)$ probability density function
$\int$ integral

What it means

For continuous variables, probability is area under the density between the two limits.

Components

$X, Y$ random variables
$P (\cdot)$ probability
$f_{X} (x)$ probability density function
$\int$ integral

What it means

Accumulated area under the density up to x.

Components

$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$f_{X} (x)$ probability density function
$\int$ integral

What it means

Interval probability as a difference of CDF values; the standard way to use normal tables.

Components

$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$X, Y$ random variables
$P (\cdot)$ probability

What it means

Mean of a continuous variable: integral of x weighted by its density.

Components

$X, Y$ random variables
$f_{X} (x)$ probability density function
$E (X)$ expected value
$\int$ integral

What it means

Integral version of the average squared deviation from the mean.

Components

$μ, μ_{X}, μ_{Y}$ population means
$X, Y$ random variables
$f_{X} (x)$ probability density function
$V a r (X)$ variance
$\int$ integral

What it means

Notation for a bell-shaped variable fully described by its mean and variance.

Components

$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$\sim$ is distributed as

What it means

The Gaussian bell curve: symmetric around mu, with spread controlled by sigma.

Components

$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$f_{X} (x)$ probability density function

What it means

Converts any normal variable into a standard normal by centring at the mean and scaling by the standard deviation.

Components

$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$Z, z$ standard normal random variable / observed z-score

What it means

The reference normal with mean 0 and variance 1, the one tabulated in exam tables.

Components

$Z, z$ standard normal random variable / observed z-score
$\sim$ is distributed as

What it means

Going back from z-scale to original units, used to convert critical values into real-world thresholds.

Components

$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$Z, z$ standard normal random variable / observed z-score

What it means

How many standard deviations an observed value sits above (or below) the mean.

Components

$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$Z, z$ standard normal random variable / observed z-score

What it means

Turns a z-score back into the original measurement scale, e.g. for percentiles.

Components

$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$Z, z$ standard normal random variable / observed z-score

What it means

Joint PMF: probability that both variables take the given pair of values simultaneously.

Components

$X, Y$ random variables
$P (\cdot)$ probability
$P_{X} (x)$ probability mass function

What it means

Expected product of the two centred variables; sign tells the direction of linear co-movement.

Components

$μ, μ_{X}, μ_{Y}$ population means
$s_{X Y}, σ_{X Y}$ sample and population covariance
$X, Y$ random variables
$E (X)$ expected value

What it means

Mean of the product minus product of the means. Almost always the quickest computation.

Components

$s_{X Y}, σ_{X Y}$ sample and population covariance
$X, Y$ random variables
$E (X)$ expected value

What it means

Standardized covariance: linear association on a unit-free [-1,1] scale.

Components

$s, S, σ$ sample/sample-statistic/population standard deviations
$s_{X Y}, σ_{X Y}$ sample and population covariance
$r_{X Y}, ρ_{X Y}$ sample and population correlation
$X, Y$ random variables

What it means

Independence means the joint PMF factorizes into the two marginal PMFs, for every pair.

Components

$P_{X} (x)$ probability mass function

What it means

Continuous analogue: the joint density is the product of the marginal densities.

Components

$f_{X} (x)$ probability density function

What it means

Independence implies zero covariance. Careful: zero covariance alone does NOT imply independence.

Components

$s_{X Y}, σ_{X Y}$ sample and population covariance
$X, Y$ random variables

What it means

Expectation is linear: it passes through sums and constants regardless of dependence.

Components

$μ, μ_{X}, μ_{Y}$ population means
$E (X)$ expected value

What it means

Variance of a combination adds the scaled variances plus twice the covariance term.

Components

$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$s_{X Y}, σ_{X Y}$ sample and population covariance
$V a r (X)$ variance

What it means

Recovering covariance when you are told the correlation and both standard deviations.

Components

$s, S, σ$ sample/sample-statistic/population standard deviations
$s_{X Y}, σ_{X Y}$ sample and population covariance
$r_{X Y}, ρ_{X Y}$ sample and population correlation

What it means

With independence the covariance term vanishes, leaving only the scaled variances.

Components

$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$V a r (X)$ variance

What it means

Number of successes in m independent Bernoulli trials, each with success probability p.

Components

$i$ observation index
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$p$ population proportion / Bernoulli success probability
$\sim$ is distributed as

What it means

Probability of exactly x successes: count the arrangements, then multiply by the probability of each specific sequence.

Components

$n$ sample size
$i$ observation index
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$p$ population proportion / Bernoulli success probability

What it means

Expected number of successes: trials times success probability.

Components

$X, Y$ random variables
$E (X)$ expected value
$p$ population proportion / Bernoulli success probability

What it means

Variance of the count of successes; it is largest when p = 0.5.

Components

$X, Y$ random variables
$V a r (X)$ variance
$p$ population proportion / Bernoulli success probability

What it means

A continuous uniform random variable spreads probability evenly over the interval [a, b].

Components

$X, Y$ random variables
$f_{X} (x)$ probability density function

What it means

Cumulative probability grows linearly across the interval.

Components

$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$X, Y$ random variables
$F$ F test statistic in regression

What it means

The mean sits at the centre of the interval; the variance depends only on its length.

Components

$X, Y$ random variables
$E (X)$ expected value
$V a r (X)$ variance

What it means

To get a percentile of a normal variable, take the standard normal quantile and rescale it by the mean and standard deviation.

Components

$i$ observation index
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability
$Z, z$ standard normal random variable / observed z-score

What it means

The standard normal is symmetric around 0, so opposite-tail quantiles are mirror images.

Components

$p$ population proportion / Bernoulli success probability
$Z, z$ standard normal random variable / observed z-score

What it means

For any random variable X and constants a, b, the expected value of the linear transformation Y=a+bX equals a plus b times the expected value of X. Used to derive expected profit/cost from a random quantity.

Components

$X, Y$ random variables
$E (X)$ expected value

What it means

Adding a constant a does not change variability; scaling by b multiplies the variance by b squared (and the standard deviation by |b|). Used together with the mean formula to find the distribution of a rescaled random variable, e.g. profit as a function of units sold.

Components

$i$ observation index
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$V a r (X)$ variance

What it means

Any linear transformation of a normally distributed random variable is itself normally distributed, with mean and variance updated by the linear transformation rules. This is the basis for standardization, since Z=(X-\mu)/\sigma is a special case.

Components

$N$ population size
$i$ observation index
$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$\sim$ is distributed as

What it means

If X and Y have a bivariate (jointly) normal distribution, any linear combination aX+bY is also normally distributed, with mean and variance given by the usual linear-combination formulas. This allows computing probabilities (e.g. via pnorm) for sums or differences of correlated normal variables.

Components

$N$ population size
$i$ observation index
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$s_{X Y}, σ_{X Y}$ sample and population covariance
$X, Y$ random variables
$\sim$ is distributed as

What it means

When X_1,...,X_n are independent and identically distributed as N(mu, sigma^2), their sum is exactly (not just approximately) normal, with mean n times mu and variance n times sigma squared. This is stronger than the CLT, which only gives an approximation for large n.

Components

$n$ sample size
$N$ population size
$i$ observation index
$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$\sum$ sum

What it means

By the Central Limit Theorem, for a large number of i.i.d. observations with mean mu and variance sigma squared, the sum S can be approximated by a normal distribution with mean n mu and variance n sigma squared, regardless of the shape of the original distribution. Used to compute probabilities on totals (e.g. total demand across many customers) via pnorm/qnorm.

Components

$n$ sample size
$N$ population size
$i$ observation index
$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$p$ population proportion / Bernoulli success probability

What it means

S counts the number of successes out of n independent Bernoulli(p) trials; for sufficiently large n (typically n>30, or np(1-p)>5 when p is extreme) its distribution can be approximated by a normal with mean np and variance np(1-p). This complements the normal approximation of the sample proportion already used for confidence intervals and tests.

Components

$n$ sample size
$N$ population size
$i$ observation index
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$p$ population proportion / Bernoulli success probability
$S E, se$ standard error / estimated standard error
$\sum$ sum

Sampling & point estimation38

What it means

Total of the n sampled random variables; the building block of the sample mean.

Components

$s, S, σ$ sample/sample-statistic/population standard deviations

What it means

Sample average, seen as a random variable before the data are observed.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$X, Y$ random variables

What it means

On average the sum is n times the population mean.

Components

$n$ sample size
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$E (X)$ expected value

What it means

With independent draws, variances add, so the sum's variance grows linearly in n.

Components

$n$ sample size
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$V a r (X)$ variance

What it means

The sample mean is an unbiased estimator of the population mean.

Components

$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$X, Y$ random variables
$E (X)$ expected value

What it means

Averaging shrinks variance by a factor n: larger samples give more stable means.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$V a r (X)$ variance

What it means

Standard error: typical sampling error of the mean, falling with the square root of n.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$E (X)$ expected value
$S E, se$ standard error / estimated standard error

What it means

If the population is normal, the sample mean is exactly normal with variance sigma squared over n.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$\sim$ is distributed as

What it means

Central Limit Theorem: for large n the sample mean is approximately normal whatever the population shape (rule of thumb n>=30).

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$\sim$ is distributed as
$\sim a pp r o x$ approximately distributed as

What it means

Average of 0/1 indicators, i.e. the fraction of successes in the sample.

Components

$n$ sample size
$P^, p^$ sample proportion random variable / observed proportion

What it means

The sample proportion is unbiased for the population proportion p.

Components

$E (X)$ expected value
$p$ population proportion / Bernoulli success probability
$P^, p^$ sample proportion random variable / observed proportion

What it means

Bernoulli variance divided by n; smallest for p near 0 or 1.

Components

$n$ sample size
$V a r (X)$ variance
$p$ population proportion / Bernoulli success probability
$P^, p^$ sample proportion random variable / observed proportion

What it means

Standard error of a proportion, used when the true p is known or hypothesized.

Components

$n$ sample size
$E (X)$ expected value
$p$ population proportion / Bernoulli success probability
$P^, p^$ sample proportion random variable / observed proportion
$S E, se$ standard error / estimated standard error

What it means

Plug-in standard error using the observed proportion; used for confidence intervals.

Components

$n$ sample size
$p$ population proportion / Bernoulli success probability
$P^, p^$ sample proportion random variable / observed proportion
$S E, se$ standard error / estimated standard error

What it means

An estimator is a rule (a function of the random sample) with its own sampling distribution.

Components

$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate

What it means

The single number obtained by applying the estimator to the observed data.

Components

$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate

What it means

Unbiasedness: on average across samples the estimator hits the true parameter.

Components

$E (X)$ expected value
$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate

What it means

Systematic error of an estimator: expected value minus the true parameter.

Components

$E (X)$ expected value
$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate
$B ia s$ estimator bias

What it means

Bias may exist for small n but vanishes as the sample grows.

Components

$n$ sample size
$E (X)$ expected value
$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate

What it means

Overall accuracy measure: expected squared distance between estimator and parameter.

Components

$E (X)$ expected value
$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate
$M S E$ mean squared error
$S E, se$ standard error / estimated standard error

What it means

MSE splits into variance plus squared bias, the trade-off behind estimator choice.

Components

$E (X)$ expected value
$V a r (X)$ variance
$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate
$B ia s$ estimator bias
$M S E$ mean squared error
$S E, se$ standard error / estimated standard error

What it means

Dividing by n-1 makes the variance estimator unbiased.

Components

$n$ sample size
$i$ observation index
$X ˉ, Y ˉ$ sample means as random variables
$s^{2}, S^{2}$ sample variance / sample variance estimator
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$\sum$ sum

What it means

Confirms the n-1 version is unbiased for the population variance.

Components

$s^{2}, S^{2}$ sample variance / sample variance estimator
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$E (X)$ expected value

What it means

Dividing by n instead; simpler but biased downward.

Components

$i$ observation index
$X ˉ, Y ˉ$ sample means as random variables
$s^{2}, S^{2}$ sample variance / sample variance estimator
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$\sum$ sum

What it means

Shows the downward bias factor (n-1)/n, negligible for large n.

Components

$n$ sample size
$s^{2}, S^{2}$ sample variance / sample variance estimator
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$E (X)$ expected value

What it means

Standard error with sigma replaced by s; this is why the t distribution appears.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$S E, se$ standard error / estimated standard error

What it means

Solve for n to reach a desired precision (standard error) of the mean.

Components

$n$ sample size
$s, S, σ$ sample/sample-statistic/population standard deviations

What it means

Estimator for the gap between two population means from two independent samples.

Components

$X ˉ, Y ˉ$ sample means as random variables
$X, Y$ random variables

What it means

Unbiased: the expected difference of sample means is the difference of population means.

Components

$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$X, Y$ random variables
$E (X)$ expected value

What it means

With independent samples variances add, each divided by its own sample size.

Components

$X ˉ, Y ˉ$ sample means as random variables
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$V a r (X)$ variance

What it means

Weighted average of the two sample variances, used when the populations share a common variance.

Components

$s^{2}, S^{2}$ sample variance / sample variance estimator

What it means

For matched data, work with the within-pair difference instead of two separate samples.

Components

$i$ observation index

What it means

The mean of the differences equals the difference of the means.

Components

$X ˉ, Y ˉ$ sample means as random variables
$X, Y$ random variables

What it means

Pairing subtracts twice the covariance: positive correlation makes the paired design more precise.

Components

$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$s_{X Y}, σ_{X Y}$ sample and population covariance

What it means

Standard error of the mean difference: a one-sample problem on the D values.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$s, S, σ$ sample/sample-statistic/population standard deviations
$S E, se$ standard error / estimated standard error

What it means

Estimator for the gap between two population proportions.

Components

$P^, p^$ sample proportion random variable / observed proportion

What it means

Sum of the two independent proportion variances.

Components

$p$ population proportion / Bernoulli success probability

What it means

For a large enough sample the sample proportion is approximately normal, which is what makes z intervals and z tests for proportions valid.

Components

$n$ sample size
$N$ population size
$i$ observation index
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability
$P^, p^$ sample proportion random variable / observed proportion
$\sim$ is distributed as

Confidence intervals17

What it means

A random interval that covers the unknown parameter with probability 1-alpha across repeated samples.

Components

$P (\cdot)$ probability
$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate
$α$ significance level / Type I error probability

What it means

Every interval in this course has the same shape: an estimate plus/minus a margin of error.

Components

$M E$ margin of error

What it means

Half-width of the interval: a critical value (reliability factor) times the standard error.

Components

$M E$ margin of error

What it means

When sigma is known the standardized sample mean is exactly standard normal.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$Z, z$ standard normal random variable / observed z-score
$\sim$ is distributed as

What it means

Interval for mu with known sigma; uses the normal critical value (1.96 for 95%).

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$z_{α}, z_{α /2}$ standard-normal critical values
$α$ significance level / Type I error probability

What it means

Margin of error for the mean with known sigma; shrinks like 1/sqrt(n).

Components

$n$ sample size
$s, S, σ$ sample/sample-statistic/population standard deviations
$z_{α}, z_{α /2}$ standard-normal critical values
$M E$ margin of error
$α$ significance level / Type I error probability

What it means

Full width is twice the margin of error: quadrupling n halves the width.

Components

$n$ sample size
$s, S, σ$ sample/sample-statistic/population standard deviations
$z_{α}, z_{α /2}$ standard-normal critical values
$α$ significance level / Type I error probability

What it means

Choose n to guarantee a target margin of error; always round up.

Components

$n$ sample size
$s, S, σ$ sample/sample-statistic/population standard deviations
$z_{α}, z_{α /2}$ standard-normal critical values
$M E$ margin of error
$α$ significance level / Type I error probability

What it means

With sigma unknown, replacing it by S gives a t distribution with n-1 degrees of freedom.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$T, t_{o b s}$ Student t statistic / observed t statistic
$t_{df, α}$ Student t critical value
$\sim$ is distributed as

What it means

The usual interval for mu in practice: sample mean plus/minus t critical value times s/sqrt(n).

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$t_{df, α}$ Student t critical value
$α$ significance level / Type I error probability

What it means

Large-sample interval for p using the plug-in standard error.

Components

$n$ sample size
$p$ population proportion / Bernoulli success probability
$z_{α}, z_{α /2}$ standard-normal critical values
$P^, p^$ sample proportion random variable / observed proportion
$α$ significance level / Type I error probability

What it means

p(1-p) is largest at p=0.5, giving the worst-case value 0.25 used for conservative planning.

Components

$p$ population proportion / Bernoulli success probability

What it means

Worst-case sample size for a proportion when no prior guess for p is available.

Components

$n$ sample size
$z_{α}, z_{α /2}$ standard-normal critical values
$M E$ margin of error
$α$ significance level / Type I error probability

What it means

Interval for the difference of two means when both variances are known: normal critical value.

Components

$X ˉ, Y ˉ$ sample means as random variables
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$z_{α}, z_{α /2}$ standard-normal critical values
$α$ significance level / Type I error probability

What it means

Two-sample t interval assuming equal variances, with n_X+n_Y-2 degrees of freedom.

Components

$X ˉ, Y ˉ$ sample means as random variables
$s^{2}, S^{2}$ sample variance / sample variance estimator
$X, Y$ random variables
$t_{df, α}$ Student t critical value
$α$ significance level / Type I error probability

What it means

Paired-data interval: a one-sample t interval applied to the differences.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$s, S, σ$ sample/sample-statistic/population standard deviations
$t_{df, α}$ Student t critical value
$α$ significance level / Type I error probability

What it means

Large-sample interval for the difference of two proportions, standard errors added under independence.

Components

$p$ population proportion / Bernoulli success probability
$z_{α}, z_{α /2}$ standard-normal critical values
$P^, p^$ sample proportion random variable / observed proportion
$α$ significance level / Type I error probability

Hypothesis testing31

What it means

Rejecting a true null; its probability is the significance level alpha you choose in advance.

Components

$P (\cdot)$ probability
$α$ significance level / Type I error probability
$H_{0}, H_{1}$ null and alternative hypotheses

What it means

Failing to reject a false null; its probability beta depends on the true parameter value and n.

Components

$P (\cdot)$ probability
$β$ Type II error probability
$H_{0}, H_{1}$ null and alternative hypotheses

What it means

Probability of correctly detecting a false null. Power rises with n, effect size and alpha.

Components

$P (\cdot)$ probability
$β$ Type II error probability
$1 - β$ statistical power
$H_{0}, H_{1}$ null and alternative hypotheses

What it means

The null hypothesis fixes the parameter at a specific value used to build the test statistic.

Components

$θ, θ_{0}$ generic population parameter / null value
$H_{0}, H_{1}$ null and alternative hypotheses

What it means

One-sided alternative claiming the parameter is larger; rejection in the right tail only.

Components

$θ, θ_{0}$ generic population parameter / null value
$H_{0}, H_{1}$ null and alternative hypotheses

What it means

One-sided alternative claiming the parameter is smaller; rejection in the left tail only.

Components

$θ, θ_{0}$ generic population parameter / null value
$H_{0}, H_{1}$ null and alternative hypotheses

What it means

Two-sided alternative: any departure counts, so alpha is split across both tails.

Components

$θ, θ_{0}$ generic population parameter / null value
$H_{0}, H_{1}$ null and alternative hypotheses
$\neq =$ not equal to

What it means

Test statistic for the mean with known sigma; standard normal under the null.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$Z, z$ standard normal random variable / observed z-score

What it means

The number actually computed from the data and compared with the critical value.

Components

$n$ sample size
$x ˉ, y ˉ$ observed sample means
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$Z, z$ standard normal random variable / observed z-score

What it means

Reject the null in a right-tailed test when the observed z exceeds the upper critical value.

Components

$Z, z$ standard normal random variable / observed z-score
$z_{α}, z_{α /2}$ standard-normal critical values
$α$ significance level / Type I error probability

What it means

Reject in a left-tailed test when the observed z falls below the negative critical value.

Components

$Z, z$ standard normal random variable / observed z-score
$z_{α}, z_{α /2}$ standard-normal critical values
$α$ significance level / Type I error probability

What it means

Reject in a two-sided test when the magnitude of z exceeds the alpha/2 critical value.

Components

$Z, z$ standard normal random variable / observed z-score
$z_{α}, z_{α /2}$ standard-normal critical values
$α$ significance level / Type I error probability

What it means

Probability, under the null, of a value as extreme or more extreme on the right.

Components

$P (\cdot)$ probability
$Z, z$ standard normal random variable / observed z-score

What it means

Same idea on the left tail; reject when the p-value is below alpha.

Components

$P (\cdot)$ probability
$Z, z$ standard normal random variable / observed z-score

What it means

Doubling the one-tail probability because departures in both directions count.

Components

$P (\cdot)$ probability
$Z, z$ standard normal random variable / observed z-score

What it means

Test statistic for the mean with unknown sigma; follows a t with n-1 degrees of freedom.

Components

$n$ sample size
$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$T, t_{o b s}$ Student t statistic / observed t statistic
$t_{df, α}$ Student t critical value
$\sim$ is distributed as

What it means

Computed t value using the sample standard deviation, compared with a t critical value.

Components

$n$ sample size
$x ˉ, y ˉ$ observed sample means
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$T, t_{o b s}$ Student t statistic / observed t statistic

What it means

Proportion test: uses the null value p0 (not the sample p) inside the standard error.

Components

$n$ sample size
$p$ population proportion / Bernoulli success probability
$Z, z$ standard normal random variable / observed z-score
$P^, p^$ sample proportion random variable / observed proportion

What it means

Combined success rate over both samples, valid under the null of equal proportions.

Components

$p$ population proportion / Bernoulli success probability
$P^, p^$ sample proportion random variable / observed proportion

What it means

Two-proportion z test with the pooled proportion in the standard error.

Components

$p$ population proportion / Bernoulli success probability
$Z, z$ standard normal random variable / observed z-score
$P^, p^$ sample proportion random variable / observed proportion

What it means

Expected count in each category under the null distribution: n times the hypothesized share.

Components

$k, j$ category/class/variable indices
$O_{k}, E_{k}$ observed and expected category frequencies

What it means

Adds squared, relatively-weighted gaps between observed and expected counts. Large values contradict the null.

Components

$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$χ^{2}, χ^^{2}$ chi-square random variable / observed statistic
$O_{k}, E_{k}$ observed and expected category frequencies
$\sum$ sum

What it means

Under the null the statistic is approximately chi-square with K-1 degrees of freedom; the test is always right-tailed.

Components

$K, J$ numbers of categories/classes or explanatory variables
$χ^{2}, χ^^{2}$ chi-square random variable / observed statistic
$\sim$ is distributed as

What it means

Expected cell count under independence: row total times column total divided by n.

Components

$n$ sample size
$k, j$ category/class/variable indices
$O_{k j}, E_{k j}$ observed and expected contingency-table cell frequencies
$R_{k}, C_{j}$ row and column totals

What it means

Same chi-square logic summed over all table cells to test independence of two categorical variables.

Components

$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$χ^{2}, χ^^{2}$ chi-square random variable / observed statistic
$O_{k j}, E_{k j}$ observed and expected contingency-table cell frequencies
$\sum$ sum

What it means

Degrees of freedom for a K x J contingency table.

Components

$K, J$ numbers of categories/classes or explanatory variables
$df$ degrees of freedom

What it means

With a big sample you can replace the unknown sigma by s and still use standard normal critical values instead of the t.

Components

$n$ sample size
$N$ population size
$i$ observation index
$x ˉ, y ˉ$ observed sample means
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$Z, z$ standard normal random variable / observed z-score
$\sim$ is distributed as

What it means

A two-tailed test at level alpha and the corresponding 1-alpha confidence interval always give the same answer.

Components

$n$ sample size
$i$ observation index
$k, j$ category/class/variable indices
$p$ population proportion / Bernoulli success probability
$t_{df, α}$ Student t critical value
$θ, θ_{0}$ generic population parameter / null value
$α$ significance level / Type I error probability
$H_{0}, H_{1}$ null and alternative hypotheses

What it means

When the null difference d0 is not zero you cannot pool the proportions: estimate each standard error separately.

Components

$n$ sample size
$N$ population size
$i$ observation index
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$p$ population proportion / Bernoulli success probability
$Z, z$ standard normal random variable / observed z-score
$P^, p^$ sample proportion random variable / observed proportion

What it means

Gives the critical value and p-value rule for the chi-square goodness-of-fit test: the test is always right-tailed since large discrepancies between observed and expected frequencies produce large chi-square values. Requires E_k \ge 5 for the chi-square approximation to hold.

Components

$i$ observation index
$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$p$ population proportion / Bernoulli success probability
$t_{df, α}$ Student t critical value
$χ^{2}, χ^^{2}$ chi-square random variable / observed statistic
$α$ significance level / Type I error probability
$H_{0}, H_{1}$ null and alternative hypotheses

What it means

Critical value and p-value rule for the chi-square test of independence in a contingency table with K rows and J columns. As with the goodness-of-fit test the rejection region is always in the right tail of the chi-square distribution.

Components

$i$ observation index
$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$p$ population proportion / Bernoulli success probability
$t_{df, α}$ Student t critical value
$χ^{2}, χ^^{2}$ chi-square random variable / observed statistic
$α$ significance level / Type I error probability
$H_{0}, H_{1}$ null and alternative hypotheses

Regression45

What it means

Population model: Y is a linear function of x plus a random error term.

Components

$X, Y$ random variables
$β_{0}$ population regression intercept
$β_{1}, \dots, β_{K}$ population regression coefficients
$ε_{i}$ regression error

What it means

The model written for a single observation i, each with its own error.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$β_{0}$ population regression intercept
$β_{1}, \dots, β_{K}$ population regression coefficients
$ε_{i}$ regression error

What it means

Fitted line from the data, giving predicted values for any x.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients
$y^_{i}$ fitted/predicted value

What it means

What the model missed for observation i: observed minus fitted value.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$e_{i}$ residual
$y^_{i}$ fitted/predicted value

What it means

Ordinary Least Squares picks the line minimizing the sum of squared residuals.

Components

$n$ sample size
$i$ observation index
$e_{i}$ residual
$\sum$ sum

What it means

Unexplained variation: the total squared vertical distance from points to the fitted line.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$e_{i}$ residual
$y^_{i}$ fitted/predicted value
$S S E$ sum of squared errors
$\sum$ sum

What it means

OLS slope as a ratio of the x-y co-movement to the variation in x.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients
$\sum$ sum

What it means

Equivalent form: sample covariance divided by the variance of x.

Components

$s_{X Y}, σ_{X Y}$ sample and population covariance
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients

What it means

Slope equals the correlation rescaled by the ratio of standard deviations.

Components

$r_{X Y}, ρ_{X Y}$ sample and population correlation
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients

What it means

The fitted line always passes through the point of means, which pins down the intercept.

Components

$x ˉ, y ˉ$ observed sample means
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients

What it means

Total variation in Y around its mean, the benchmark to be explained.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$S S T$ total sum of squares
$\sum$ sum

What it means

Variation captured by the model: spread of fitted values around the mean of Y.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$y^_{i}$ fitted/predicted value
$S S R$ regression sum of squares
$\sum$ sum

What it means

Total variation splits exactly into explained plus unexplained parts.

Components

$S S E$ sum of squared errors
$S S R$ regression sum of squares
$S S T$ total sum of squares

What it means

Fraction of the variation of Y explained by the model; between 0 and 1.

Components

$S S E$ sum of squared errors
$S S R$ regression sum of squares
$S S T$ total sum of squares
$R^{2}, R ˉ^{2}$ coefficient of determination / adjusted R²

What it means

In simple regression only, R squared is the squared correlation between x and y.

Components

$r_{X Y}, ρ_{X Y}$ sample and population correlation
$R^{2}, R ˉ^{2}$ coefficient of determination / adjusted R²

What it means

Model assumption: errors average out to zero, so the line is correctly centred.

Components

$i$ observation index
$E (X)$ expected value
$ε_{i}$ regression error

What it means

Homoskedasticity: the error spread is the same at every value of x.

Components

$i$ observation index
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$V a r (X)$ variance
$ε_{i}$ regression error

What it means

Normality of errors is what justifies the t and F inference on the coefficients.

Components

$i$ observation index
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$ε_{i}$ regression error
$\sim$ is distributed as

What it means

Combining the assumptions: for a given x, Y is normal around the regression line.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$β_{0}$ population regression intercept
$β_{1}, \dots, β_{K}$ population regression coefficients
$ε_{i}$ regression error
$\sim$ is distributed as

What it means

Slope precision improves with less noise and with more spread-out x values.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$V a r (X)$ variance
$β_{1}, \dots, β_{K}$ population regression coefficients
$ε_{i}$ regression error
$\sum$ sum

What it means

Estimator of the error variance in simple regression; n-2 because two coefficients were estimated.

Components

$n$ sample size
$M S E$ mean squared error
$S S E$ sum of squared errors

What it means

Typical size of a prediction error, in the units of Y.

Components

$s, S, σ$ sample/sample-statistic/population standard deviations
$M S E$ mean squared error
$ε_{i}$ regression error

What it means

Estimated standard error of the slope, the denominator of its t test.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$s, S, σ$ sample/sample-statistic/population standard deviations
$S E, se$ standard error / estimated standard error
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients
$ε_{i}$ regression error
$\sum$ sum

What it means

Confidence interval for the true slope with n-2 degrees of freedom.

Components

$n$ sample size
$t_{df, α}$ Student t critical value
$S E, se$ standard error / estimated standard error
$α$ significance level / Type I error probability
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients

What it means

Tests whether the slope differs from a hypothesized value; with 0 it tests significance of x.

Components

$T, t_{o b s}$ Student t statistic / observed t statistic
$S E, se$ standard error / estimated standard error
$β$ Type II error probability
$1 - β$ statistical power
$β_{1}, \dots, β_{K}$ population regression coefficients
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients

What it means

Point prediction from the fitted line at a chosen x value.

Components

$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients
$y^_{i}$ fitted/predicted value

What it means

Uncertainty about the average Y at x; smallest at the mean of x.

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$s, S, σ$ sample/sample-statistic/population standard deviations
$ε_{i}$ regression error
$\sum$ sum

What it means

Wider than for a mean because it also includes the individual error term (the extra 1).

Components

$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$s, S, σ$ sample/sample-statistic/population standard deviations
$ε_{i}$ regression error
$\sum$ sum

What it means

Model with several explanatory variables; each coefficient is a partial effect holding the others fixed.

Components

$X, Y$ random variables
$β_{0}$ population regression intercept
$β_{1}, \dots, β_{K}$ population regression coefficients
$ε_{i}$ regression error

What it means

Penalizes extra regressors, so it can fall when a useless variable is added.

Components

$n$ sample size
$K, J$ numbers of categories/classes or explanatory variables
$R^{2}, R ˉ^{2}$ coefficient of determination / adjusted R²

What it means

Error variance estimator with n-K-1 degrees of freedom for K regressors.

Components

$n$ sample size
$K, J$ numbers of categories/classes or explanatory variables
$M S E$ mean squared error
$S S E$ sum of squared errors

What it means

t test for one coefficient in multiple regression, other variables held constant.

Components

$n$ sample size
$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$T, t_{o b s}$ Student t statistic / observed t statistic
$t_{df, α}$ Student t critical value
$S E, se$ standard error / estimated standard error
$β_{1}, \dots, β_{K}$ population regression coefficients
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients
$\sim$ is distributed as

What it means

Confidence interval for one coefficient in multiple regression.

Components

$n$ sample size
$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$t_{df, α}$ Student t critical value
$S E, se$ standard error / estimated standard error
$α$ significance level / Type I error probability
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients

What it means

Tests all slopes jointly zero: explained variance per regressor over unexplained variance per degree of freedom.

Components

$n$ sample size
$K, J$ numbers of categories/classes or explanatory variables
$F$ F test statistic in regression
$S S E$ sum of squared errors
$S S R$ regression sum of squares

What it means

Equivalent F expressed through R squared, handy when only R squared is reported.

Components

$n$ sample size
$K, J$ numbers of categories/classes or explanatory variables
$F$ F test statistic in regression
$R^{2}, R ˉ^{2}$ coefficient of determination / adjusted R²

What it means

Binary indicator coding a qualitative condition; its coefficient shifts the intercept.

Components

$I$ dummy/indicator variable

What it means

With three categories you include two dummies; the omitted one is the baseline.

Components

$X, Y$ random variables
$β_{0}$ population regression intercept
$β_{1}, \dots, β_{K}$ population regression coefficients
$ε_{i}$ regression error
$I$ dummy/indicator variable

What it means

Adding a squared term allows a curved (non-linear) effect while staying linear in the parameters.

Components

$X, Y$ random variables
$β_{0}$ population regression intercept
$β_{1}, \dots, β_{K}$ population regression coefficients
$ε_{i}$ regression error
$I$ dummy/indicator variable

What it means

A further OLS assumption: errors of different observations are uncorrelated, so no observation's error predicts another's.

Components

$n$ sample size
$i$ observation index
$k, j$ category/class/variable indices
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability
$ε_{i}$ regression error
$\neq =$ not equal to

What it means

Rule of thumb in multiple regression: strongly correlated explanatory variables inflate coefficient standard errors and make them unreliable.

Components

$i$ observation index
$k, j$ category/class/variable indices
$X, Y$ random variables

What it means

Confidence interval for the expected value of Y at a given value x_g of the explanatory variable in simple linear regression. It quantifies uncertainty about the average response, and is narrower than the interval for an individual prediction.

Components

$n$ sample size
$i$ observation index
$x ˉ, y ˉ$ observed sample means
$μ, μ_{X}, μ_{Y}$ population means
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability
$t_{df, α}$ Student t critical value
$α$ significance level / Type I error probability

What it means

Prediction interval for a single future/individual value of Y at x_g. It is wider than the confidence interval for the mean because it also accounts for the random deviation of an individual observation from the conditional mean.

Components

$n$ sample size
$i$ observation index
$x ˉ, y ˉ$ observed sample means
$s, S, σ$ sample/sample-statistic/population standard deviations
$X, Y$ random variables
$p$ population proportion / Bernoulli success probability
$t_{df, α}$ Student t critical value
$α$ significance level / Type I error probability

What it means

The leverage h_i measures how far x_i is from the mean of the explanatory variable relative to its spread; observations with high leverage have more potential influence on the fitted regression line.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$x ˉ, y ˉ$ observed sample means
$s, S, σ$ sample/sample-statistic/population standard deviations

What it means

This is the residual divided by its own estimated standard deviation, which depends on the leverage h_i of the observation. It is used in residual diagnostics (e.g., via rstandard()) to spot outliers more accurately than raw or simply-scaled residuals.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability
$S E, se$ standard error / estimated standard error
$e_{i}$ residual
$y^_{i}$ fitted/predicted value

What it means

The residual scaled by the overall residual standard error, used as a quick way to identify unusually large residuals when assessing model fit and checking for outliers.

Components

$n$ sample size
$i$ observation index
$x_{i}, y_{i}$ observed values for observation i
$s, S, σ$ sample/sample-statistic/population standard deviations
$p$ population proportion / Bernoulli success probability
$e_{i}$ residual
$y^_{i}$ fitted/predicted value

Symbol index

Every letter used in the sheet, and what it stands for.

$n$ sample size
$N$ population size
$i$ observation index
$k, j$ category/class/variable indices
$K, J$ numbers of categories/classes or explanatory variables
$x_{i}, y_{i}$ observed values for observation i
$x_{k}^{*}$ category value or representative class value
$x_{min}, x_{ma x}$ minimum and maximum observed values
$L_{k}, U_{k}$ lower and upper class endpoints
$w_{k}$ class width
$f_{k}$ absolute frequency
$p_{k}$ relative frequency
$c_{k}$ frequency density
$F_{k}, F (x), F_{X} (x)$ cumulative frequency / cumulative distribution functions
$Q_{1}, Q_{2}, Q_{3}$ first quartile, median, third quartile
$I QR$ interquartile range
$x ˉ, y ˉ$ observed sample means
$X ˉ, Y ˉ$ sample means as random variables
$μ, μ_{X}, μ_{Y}$ population means
$s^{2}, S^{2}$ sample variance / sample variance estimator
$σ^{2}$ population variance
$s, S, σ$ sample/sample-statistic/population standard deviations
$C V$ coefficient of variation
$s_{X Y}, σ_{X Y}$ sample and population covariance
$r_{X Y}, ρ_{X Y}$ sample and population correlation
$X, Y$ random variables
$P (\cdot)$ probability
$P_{X} (x)$ probability mass function
$f_{X} (x)$ probability density function
$E (X)$ expected value
$V a r (X)$ variance
$p$ population proportion / Bernoulli success probability
$Z, z$ standard normal random variable / observed z-score
$z_{α}, z_{α /2}$ standard-normal critical values
$T, t_{o b s}$ Student t statistic / observed t statistic
$t_{df, α}$ Student t critical value
$df$ degrees of freedom
$χ^{2}, χ^^{2}$ chi-square random variable / observed statistic
$F$ F test statistic in regression
$P^, p^$ sample proportion random variable / observed proportion
$θ, θ_{0}$ generic population parameter / null value
$θ^$ estimator or estimate
$B ia s$ estimator bias
$M S E$ mean squared error
$S E, se$ standard error / estimated standard error
$M E$ margin of error
$α$ significance level / Type I error probability
$β$ Type II error probability
$1 - β$ statistical power
$H_{0}, H_{1}$ null and alternative hypotheses
$O_{k}, E_{k}$ observed and expected category frequencies
$O_{k j}, E_{k j}$ observed and expected contingency-table cell frequencies
$R_{k}, C_{j}$ row and column totals
$β_{0}$ population regression intercept
$β_{1}, \dots, β_{K}$ population regression coefficients
$b_{0}, b_{1}, \dots, b_{K}$ estimated regression coefficients
$ε_{i}$ regression error
$e_{i}$ residual
$y^_{i}$ fitted/predicted value
$S S E$ sum of squared errors
$S S R$ regression sum of squares
$S S T$ total sum of squares
$R^{2}, R ˉ^{2}$ coefficient of determination / adjusted R²
$I$ dummy/indicator variable
$\sum$ sum
$\int$ integral
$\sim$ is distributed as
$\sim ii d$ independently and identically distributed as
$\sim a pp r o x$ approximately distributed as
$\neq =$ not equal to

Descriptive statistics & frequencies44

Relative frequency

Percentage frequency

Sum absolute frequencies

Sum relative frequencies

Class width

Frequency density

Relative frequency from density

Cumulative relative frequency

Cumulative frequency function

Arithmetic mean

Mean from frequency distribution

Mean using relative frequencies

Sum deviations from mean

Sum from mean

Range

Interquartile range

Lower outlier fence

Upper outlier fence

Population variance

Population variance, shortcut

Sample variance

Sample variance, shortcut

Population standard deviation

Sample standard deviation

Coefficient of variation

Population covariance

Population covariance, shortcut

Sample covariance

Sample covariance, shortcut

Population correlation

Sample correlation

Correlation bounds

Mode

Median, raw data

Median from cumulative frequencies

Median for grouped data

Quartiles from cumulative frequencies

Quartiles for grouped data

Quartile position, ordered data

Percentile

Quantile of order alpha

Five-number summary

Mean from class midpoints

Class width from range

Random variables & distributions52

Discrete probability function

Probability bounds

Sum of probabilities

CDF

Discrete expected value

Discrete variance

Variance shortcut

Bernoulli distribution

Bernoulli PMF

Bernoulli mean

Bernoulli variance

Density non-negativity

Total density

Continuous interval probability

Continuous CDF

Probability from CDF

Continuous expected value

Continuous variance

Normal distribution

Normal density

Standardization

Standard normal

Reverse standardization

Observed z-score

Original value from z

Joint discrete probability

Covariance of RVs

Covariance shortcut

Correlation of RVs

Independence, discrete

Independence, continuous

Independence covariance

Linear-combination mean

Linear-combination variance