IB Maths AA Topic 4 Definitions

This page contains our IB Maths AA definitions for topic 4. By learning each one of these definitions, you will fully cover the content for IB Maths AA 'Stats & Probability'.

All Topic 4 Sub-topics 4.1: Fundamentals Of Statistics 4.2: Visualising Statistics 4.3: Distribution Properties 4.4: Linear Statistics 4.5: Probability 4.6: Probability Distributions (HL)4.7: Bayes theorem (HL)4.8: Advanced distribution properties

bias

A tendency in the sampling or data collection process that favours certain outcomes or groups, so the results are not representative.

bivariate

Involving two numerical variables measured together as paired values, usually written as

(x,y)

class

A range of values used to group continuous data, usually written as an interval such as ' $40\leq x\lt 50$ '.

complement

The event that $A$ does not happen, written as $A'$ , with probability $P\left(A'\right)=1-P\left(A\right)$ .

conditional probability

The probability that one event happens given that another event has already happened, written as
$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ provided that $P(B) \neq 0$ .

continuous

Able to take any value within an interval.

correlation

A description of the direction and strength of the relationship between two variables, as seen in a scatter diagram.

cumulative

A running total found by adding successive values as you move through a table or list.

discrete

Taking separate, countable values, usually whole numbers.

dispersion

The extent to which data values vary about a central value, quantified using measures such as interquartile range, standard deviation, and variance.

event

A set of outcomes from the sample space.

exclusive

Describes events that cannot happen at the same time, so $P\left(A\cap B\right)=0$ and $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$ .

extrapolation

Using a regression line to predict a value that lies outside the range of the observed data, which is generally less reliable.

frequency

A table that shows how often each value (discrete) or each class interval (continuous) occurs.

gradient

The coefficient

a

y=ax+b

, giving the predicted change in

y

when

x

increases by

1

histogram

A diagram for grouped continuous data in which each class interval is shown by a touching bar, allowing the shape and spread of the distribution to be seen.

independent probability

Describes events or random variables where knowing one outcome gives no information about the other; for Poisson variables this condition allows their totals to remain Poisson.

interpolation

Estimating a value at an unknown point using known values at nearby points; in nearest neighbour interpolation, the estimate is taken from the closest site.

interquartile

Referring to the middle $50\%$ of the data, between the lower quartile $Q_1$ and upper quartile $Q_3$ .

interquartile range

Measures the spread of the middle

50\%

of an ordered data set, calculated as the difference between the upper and lower quartiles:

IQR=Q_3-Q_1

joint probability

Describing the probability that two events occur together, written as $P(A\cap B)$ .

mean

The expected value of a random variable; if $X \sim \mathrm{Poisson}(\lambda)$ , then $E(X) = \lambda$

median

The value

m

such that

\int_{-\infty}^{m} f(x)\,dx=\frac{1}{2}

, so half the total area under the density curve lies on each side.

modal

The class interval with the highest frequency in grouped data (used when class intervals are equal).

mode

The value of

x

where the probability density function has its greatest value.

negative correlation

Describing correlation where points tend to fall from left to right as

x

increases.

outcome

One possible result of a trial.

outlier

A value unusually far from the rest of the data; in this course, one that is below

Q_1-1.5\times IQR

or above

Q_3+1.5\times IQR

, where

IQR=Q_3-Q_1

partition

A collection of events (such as

A_1

A_2

A_3

) that are mutually exclusive and whose union is the whole sample space, so exactly one must occur.

Pearson's

Referring to the product-moment correlation coefficient

r

, a statistic that measures the strength and direction of a linear relationship and lies between

-1

and

1

percentiles

Values that split ordered data into $100$ equal parts; the $p$ th percentile has about $p$ % of the data below it and can be estimated from a cumulative frequency graph.

population

The full set of all possible values of interest, described by parameters such as the mean '

\mu

' and variance '

\sigma^2

positive correlation

Describing a correlation where the points in a scatter diagram tend to rise from left to right.

prior probability

The probability assigned to an event before any new information is taken into account, often written as $P(A)$ .

probability

Measures how likely an event is to happen, taking a value between $0$ and $1$ , where $0$ means impossible and $1$ means certain.

probability density function

A function

f(x)

used to describe a continuous random variable, where

f(x)\geq0

and

\int_{-\infty}^{\infty} f(x)\,dx=1

, so probabilities correspond to areas under its graph.

quartiles

Values that split ordered data into four equal parts; on a cumulative frequency graph with $n$ values, they are estimated at cumulative frequencies $\frac{n}{4}$ and $\frac{3n}{4}$ .

random

Describing an outcome that cannot be predicted with certainty in advance, even though long-run patterns may be modelled using probabilities.

range

The set of possible outputs a function can produce.

regression

A method for modelling a linear relationship with a line used to make predictions from bivariate data.

sample

A set of '

n

' observed values taken from a population and used to compute statistics such as '

\bar{x}

' and '

s_{n-1}^2

spread

How dispersed the data values are, described using measures of dispersion such as interquartile range, standard deviation, and variance.

standard deviation

The square root of the variance,

\sigma=\sqrt{\mathrm{Var}(X)}

, giving spread in the same units as the random variable.

theorem

A mathematical result that can be proved logically from definitions and earlier results, then used to solve problems.

tree

A connected graph with no cycles; if it has

n

vertices then it has exactly

n-1

edges.

updated probability

Describing a probability that has been revised after learning that another event has occurred, often written as $P(A|B)$ .

variance

Measures spread using the mean of squared distances from the mean, so values far from the mean have greater influence; measured in squared units and equal to the square of the standard deviation.

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