Bocconi Statistics 30001 · exam prep

Bocconi Statistics 30001:
every formula, one search away.

279 entries from the Bocconi Statistics 30001 course — every formula plus the R and UBStats commands you need at the computer — each with a plain-English explanation and a breakdown of what every symbol or argument stands for. Built to help you prepare efficiently and drill until it sticks.

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Descriptive statistics & frequencies32

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

Random variables & distributions8

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

Showing 40 / 279

Descriptive statistics & frequencies32

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

Random variables & distributions8

Discrete probability function

Probability bounds

Sum of probabilities

CDF

Discrete expected value

Discrete variance

Variance shortcut

Bernoulli distribution