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Representation of data

Box plots

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Further kinematics

Vector methods with projectiles

Variable acceleration in one dimension

Differentiating vectors

Integrating vectors

Application of forces

Static particles

Modelling with statics

Friction with static particles

Statics of rigid bodies

Dynamics and inclined planes

Connected particles

Projectiles

Horizontal projection

Projectiles: Horizontal and vertical components

Projection at any angle

Equation of the trajectory of a projectile

Forces and friction

Resolving forces

Inclined planes

The coefficient of friction

Moments

Resultant moments

Equilibrium

Centre of mass

Tilting

Laminas

Forces and motion

Force diagrams

Forces as vectors

Forces and acceleration

Motion in two dimensions

Connected particles

Pulleys

Variable acceleration

Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

Using integration to find velocity and displacement

Constant acceleration formulae

Constant acceleration

Displacement-time graphs

Velocity-time graphs

Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

Modelling assumptions

Quantities and units in mechanics

Working with vectors

Normal distribution

The normal distribution

The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

Conditional probability

Set notation

Conditional probability

Conditional probability in Venn diagrams

Probability formulae

Conditional probability: Tree diagrams

Correlation and hypothesis testing

Exponential models

Measuring correlation

Hypothesis testing for zero correlation

Hypothesis testing I

Hypothesis testing

Hypothesis testing: Finding critical values

One-tailed tests

Two-tailed tests

Binomial distribution

Probability distributions

The binomial distribution

Cumulative probabilities

Probability

Calculating probabilities

Venn diagrams

Probability: Mutually exclusive and independent events

Probability: Tree diagrams

Representation of data

Outliers

Box plots

Cumulative frequency

Histograms

Comparing data sets

Correlation

Linear regression

Measures of location and spread

Measures of central tendency: Mean, median, mode

Measures of spread: Range

Variance and standard deviation

Data collection

Population and samples

Sampling techniques

Types of data

Vectors II

3D coordinates

Vectors in 3D

Solving geometric problems

Applications to mechanics: 3D vectors

Numerical methods

Locating roots

Iteration

Staircase and cobweb diagrams

Newton-Raphson method

Numerical methods: Applications to modelling

Integration II

Integrating standard functions

Integrating f(ax + b)

Integration with trigonometric identities

Reverse chain rule

Integration by substitution

Integration by parts

Integration using partial fractions

Finding areas using integration

The trapezium rule

Solving differential equations

Modelling with differential equations

Differentiation II

Differentiating sin x and cos x

Differentiating exponentials and logarithmic functions

The chain rule

The product rule

The quotient rule

Differentiating inverse functions

Differentiating trigonometric functions

Parametric differentiation

Implicit differentiation

Second derivatives: Concave and convex functions

Connected rates of change

Trigonometry and modelling

Addition formulae

Using the angle addition formulae

Double angle formulae

Solving trigonometric equations: Double angle formula

Simplifying a cos x + b sin x

Proving trigonometric identities

Modelling with trigonometric functions

Trigonometric functions

Secant, cosecant and cotangent

Graphs of sec x, cosec x and cot x

Solving equations with sec x, cosec x and cot x

Further trigonometric identities

Inverse trigonometric functions

Radians

Radian measure

Arc length

Areas of sectors and segments

Solving trigonometric equations

Small angle approximations

Sequences and series

Sequences revision

Arithmetic sequences

Arithmetic series

Geometric sequences

Geometric series

Sum to infinity

Sigma notation

Modelling with sequences and series

Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

Binomial expansion with partial fractions

Binomial expansion of compound expressions

Parametric equations

Using trigonometric identities

Sketching parametric curves

Points of intersection of parametric curves

Modelling with parametric equations

Functions

Functions and mappings

Composite functions

Inverse functions

The modulus function

y = |f(x)| and y = f(|x|)

Combining transformations

Solving modulus equations

Modulus inequalities

Algebraic fractions

Operations with algebraic fractions

Partial fractions

Repeated factors

Algebraic division

Proof II

Proof by contradiction

Criticising proofs

Vectors I

Vectors

Representing vectors

Magnitude and direction of a vector

Position vectors and displacement vectors

Solving geometric problems

Modelling with vectors

Integration I

Integrating x^n

Indefinite integrals

Finding functions by integrating

Definite integrals

Area under a curve

Area under the x-axis

Area between a curve and a line

Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

Differentiating functions with multiple terms

Gradients, tangents and normals

Increasing and decreasing functions

Second order derivatives

Stationary points

Sketching gradient functions

Modelling with differentiation

Exponentials and logarithms

Exponential functions

y = e^x

Exponential modelling

Logarithms

Laws of logarithms

Solving equations using logarithms

Natural logarithms

Logarithms and non-linear data

Harder exponential modelling

Trigonometric equations

Angles in all four quadrants

Exact trigonometric values

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Simple trigonometric equations

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Equations and identities

Trigonometry

The cosine rule

The sine rule

Area of a triangle

Solving triangle problems

Trig graphs: sin x, cos x, tan x

Transforming trigonometric graphs

Binomial expansion I

Pascal's triangle

Factorial notation

Binomial expansion

Solving binomial problems

Binomial estimation

Circles

Midpoints and perpendicular bisectors

Equation of a circle

Intersection of straight lines and circles

Using tangent and chord properties

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Intersection of two circles

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y = mx + c

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Length and area

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Transformations

Translating graphs

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Transforming graphs

Sketching graphs

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Quartic graphs

Reciprocal graphs

Points of intersection

Using the discriminant

Direct and inverse proportion

Polynomials

Functions

Expanding brackets

Factorising

Simplifying algebraic fractions

Polynomial division

The factor theorem

Equations and inequalities

Linear simultaneous equations

Quadratic simultaneous equations

Set notation and interval notation

Linear inequalities

Quadratic inequalities

Inequalities on graphs

Inequality regions

Quadratics

Solving quadratic equations

Completing the square

Quadratic graphs

Identifying the discriminant

Solving disguised quadratics

Indices and surds

Laws of indices

Negative and fractional indices

Surds

Rationalising the denominator

Proof I

Mathematical structure and arguments

Disproof by counterexample

Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Bilal

Summary

Box plots

In a nutshell

Box plots are diagrams which represent the spread and location of data. Box plots can be used to efficiently compare two or more data sets.

Features of a box plot

Before drawing a box plot, certain features of a data set need to be defined and calculated. This includes the quartiles, median, maximum, minimum and outliers of a given data set. The table below defines and shows the formulae to calculate each of these features where applicable for a discrete data set with $n$ entries.

feature	Definition	Formula
Lower quartile ( $Q_1$ )	The value under which $25\%$ of data points are found.	$\dfrac{n+1}{4}^{\text{th}}$ point
Median ( $Q_2$ )	The value under which $50\%$ of the data points are found.	$\dfrac{n+1}{2}^{\text{th}}$ point
Upper quartile ( $Q_3$ )	The value under which $75\%$ of the data points are found.	$\dfrac{3(n+1)}{4}^{\text{th}}$ point
Interquartile range ( $IQR$ )	The size of the range which contains the middle $50\%$ of the data points.	$Q_3 - Q_1$
Outliers	Extreme values which lie outside the normal trend of a data set which is determined by the constant, $k$ . The constant $k$ will be given or implied in a question.	Greater than $Q_3 + k(IQR)$
Outliers		Less than $Q_1 - k(IQR)$
Maximum	Highest value of data set which is not an outlier or at the boundary of an outlier.
Minimum	Lowest value of data set which is not an outlier or at the boundary of an outlier.

Note: When calculating quartiles or the median, if the data point calculated ends in $.5$ then the value of the given quartile will be halfway between the data points above and below it. If the data point calculated is any other decimal number, round up to the next data point.

Example 1

Olivia is collecting data about the ages of her friends' siblings. The answers she collects are $1,1,2,2,3,4,5,5,5,5,7,7,7,7,8$ . Given that there are no outliers, find the maximum, minimum, quartiles and median of ages collected.

Lower quartile	Position of $Q_1 =\dfrac{n+1}{4} = \dfrac{15+1}{4}=4^{th}$ data point $4^{th}$ value $=\underline2$
Upper quartile	Position of $Q_3 =\dfrac{3(n+1)}{4} = \dfrac{3(15+1)}{4}=12^{th}$ data point $12^{th}$ value $=\underline7$
Median	$Q_2 = \dfrac{n+1}{2} = \dfrac{15+1}{2}= 8^{th}$ data point $8^{th}$ value $= \underline5$
Interquartile range	$IQR = UQ - LQ = 7-2=\underline5$
Maximum	$\underline8$
Minimum	$\underline1$

Drawing a box plot

Drawing a box plot involves using an appropriate scale to mark all the features of a data set as defined above.

procedure

$1.$	Draw an appropriate scale and label it.
$2.$	Mark $Q_1$ , $Q_2$ and $Q_3$ with vertical lines of an equal length. Use $Q_1$ and $Q_3$ to draw a box which is separated by $Q_2$ .
$3.$	Mark on the maximum and minimum with equally sized vertical lines and connect to the box with a line.
$4.$	Use a cross ( $\text{x}$ ) to denote any outliers beyond the minimum or maximum values.

Example 2

Draw a box plot to represent Olivia's data set.

Maths; Representation of data; KS5 Year 12; Box plots

Note: Sometimes outliers and quartiles will be given but not the maximum or minimum values of a data set. In this case, the effective maximum and minimum will respectively be the largest and smallest values which are not outliers.

Comparing box plots

Box plots can be used to compare the spread of data. Use the context of the question to compare the measure of location and the measure of spread. A measure of location is usually compared using the medians and spread is usually compared using the interquartile range and range.

Example 3

The two box plots below summarise the A-level marks in $2021$ and $2022$ in a given school. Without calculating individual values, compare the box plots and give your interpretation.

Compare the plots:

The median mark for $2022$ is slightly lower than the median mark for $2021$ . The interquartile range and range for $2022$ is less than the range for $2021$ .

Interpret the data:

In $2021$ , a larger proportion of students did better in their A-levels than in $2022$ . But, students in $2022$ performed much more similarly with a tighter spread of results.

Note: Sometimes the box plots will be on different scales. Ensure to compare them against the same scale.

Learn with Basics

Length:

Unit 1

Representing data: Graphs and charts

Unit 2

Box plots - Higher

Jump Ahead

Optional

Box plots

Videos

Summary

Exercises

Select Lesson

Exam Board

Further kinematics

Application of forces

Projectiles

Forces and friction

Moments

Forces and motion

Variable acceleration

Constant acceleration

Modelling in mechanics

Normal distribution

Conditional probability

Correlation and hypothesis testing

Hypothesis testing I

Binomial distribution

Probability

Representation of data

Measures of location and spread

Data collection

Vectors II

Numerical methods

Integration II

Differentiation II

Trigonometry and modelling

Trigonometric functions

Radians

Sequences and series

Binomial expansion II

Parametric equations

Functions

Algebraic fractions

Proof II

Vectors I

Integration I

Differentiation I

Exponentials and logarithms

Trigonometric equations

Trigonometry

Binomial expansion I

Circles

Straight line graphs

Transformations

Sketching graphs

Polynomials

Equations and inequalities

Quadratics

Indices and surds

Proof I

Explainer Video

Summary

Box plots

​​In a nutshell

Features of a box plot

feature

Definition

Formula

Example 1

Drawing a box plot

procedure

Example 2

Comparing box plots

Example 3

Learn with Basics

Unit 1

Representing data: Graphs and charts

Unit 2

Box plots - Higher

Jump Ahead

Unit 3

Box plots

Final Test

FAQs - Frequently Asked Questions

What is the lower quartile?

What is the upper quartile?

What are box plots?

In a nutshell

In a nutshell