A-level and IB · Mathematics

A-level Maths: statistics and mechanics

A-level applied maths flashcards: sampling, probability, binomial and normal models, hypothesis tests, SUVAT, forces, friction, moments and projectiles.

40 cards6 topicsAbout 15 minutes a roundUK A-level Mathematics applied content (statistics and mechanics), similar to an introductory statistics course plus first-year kinematics

Independent study material written by StudyBrix, not produced or reviewed by any exam board or by the International Baccalaureate Organization. Check your own specification or subject guide.

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About this deck

This deck covers the applied third of A-level Mathematics: statistical sampling, data presentation and summary measures, probability, the binomial and normal distributions, hypothesis testing, kinematics, Newton's laws, friction, connected particles, moments and projectiles. It suits students in England preparing for the applied paper or papers, and anyone comparing A-level applied maths with Canadian Grade 12 or first-year statistics and physics.

Exam boards (AQA, Edexcel, OCR) cover the same core applied content but split it across papers differently, choose different large data sets for the statistics section, and may state a particular value of g. Check your board's specification, its large data set and its formula booklet, and use the value of g your paper gives.

Before starting you should know the pure content on algebra, calculus and vectors, because mechanics leans on differentiation and integration and statistics uses algebraic rearranging. Draw a diagram for every mechanics card: forces with arrows, the positive direction marked, and the particle or rigid body labelled. For statistics cards, write hypotheses in symbols with the parameter named.

Common mistakes: writing hypotheses about the sample result instead of the population parameter; forgetting to halve the significance level in a two-tailed test; mixing up mass and weight; and resolving the wrong component of weight on a slope (the component along the slope is mg sin theta). In conclusions, state results in context and avoid saying a hypothesis has been proved.

Before you start: A-level pure maths alongside (or Grade 12 functions and calculus), plus basic GCSE statistics and probability.

All 40 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Sampling and data

Simple random sample: A sample chosen so every possible group of the chosen size has the same chance of selection, often by numbering the population and using random numbers.Example: Number 500 students from 1 to 500 and use a random number generator to pick 40 distinct numbers.
Stratified sampling: The population is split into groups (strata) and a random sample is taken from each, in proportion to the group's size.Example: 300 Year 12 and 200 Year 13 students, sample of 50: choose 30 from Year 12 and 20 from Year 13.
Systematic sampling: Choosing every kth member from an ordered list after a random start, where k is population size divided by sample size.Example: List of 600, sample of 30: k = 20. Pick a random start from 1 to 20, say 7, then take 7, 27, 47, ...
Opportunity sampling: Taking whoever is available and willing at the time. Quick and cheap but unlikely to represent the population.Example: Surveying the first 20 people through the school gate misses students who arrive by car or are absent.
Quota sampling: An interviewer fills set numbers of people from each group but chooses who to ask, so the selection is not random.Example: Interview 25 men and 25 women in a shopping centre, stopping each quota when it is full.
Outlier rule using IQR: A common rule labels a value an outlier if it lies more than 1.5 times the interquartile range below Q1 or above Q3.Example: Q1 = 12, Q3 = 20, IQR = 8: outliers are below 12 - 12 = 0 or above 20 + 12 = 32.
Standard deviation from summary statistics: Use variance = (sum of x^2)/n - (mean)^2, then take the square root.Example: n = 5, sum x = 30, sum x^2 = 200: mean = 6, variance = 40 - 36 = 4, standard deviation = 2.
Product moment correlation coefficient: A value r from -1 to 1 measuring the strength and direction of a linear relationship. It does not show cause and effect.Example: r = -0.85 shows strong negative linear correlation, such as hours of screen time against test score in a sample.
Interpolation and extrapolation: Using a regression line inside the range of the data (interpolation) is usually reliable; predicting outside it (extrapolation) may not be.Example: y = 2.4 + 0.8x fitted for x from 2 to 15: x = 10 gives 10.4, reasonable; x = 40 would be extrapolation.

Probability

Mutually exclusive events: Events that cannot both happen on one trial, so P(A and B) = 0 and P(A or B) = P(A) + P(B).Example: One card is drawn: P(king or queen) = 4/52 + 4/52 = 2/13.
Independent events: Events where one happening does not change the probability of the other, so P(A and B) = P(A) * P(B).Example: P(A) = 0.3, P(B) = 0.5 and P(A and B) = 0.15 = 0.3 * 0.5, so A and B are independent.
Conditional probability: The probability of A given that B has happened: P(A | B) = P(A and B)/P(B).Example: P(A and B) = 0.12 and P(B) = 0.4 give P(A | B) = 0.3.
General addition rule: P(A or B) = P(A) + P(B) - P(A and B), subtracting the overlap so it is not counted twice.Example: P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2: P(A or B) = 0.7.
Discrete random variable: A variable taking separate values, each with a probability; all the probabilities must add to 1.Example: P(X = x) = kx for x = 1, 2, 3, 4: k(1 + 2 + 3 + 4) = 1, so k = 1/10.

Distributions and hypothesis tests

Binomial distribution: Models the number of successes in n independent trials with a fixed success probability p. Written X ~ B(n, p).Example: X ~ B(10, 0.3): P(X = 2) = 45 * 0.3^2 * 0.7^8, about 0.2335.
Normal distribution: A symmetric bell-shaped continuous model with mean mu and standard deviation sigma; about 68%, 95% and 99.7% of values lie within 1, 2 and 3 sigma.Example: X ~ N(50, 4^2): P(X < 58) = P(Z < 2), about 0.9772.
Standardizing a normal variable: Convert X to Z = (X - mu)/sigma so that tables or the standard normal function can be used.Example: mu = 50, sigma = 8, X = 62: z = 12/8 = 1.5.
Normal approximation to the binomial: When n is large and p is not too near 0 or 1, B(n, p) is close to N(np, np(1 - p)). A continuity correction of 0.5 is applied.Example: B(100, 0.5) is about N(50, 25). P(X <= 45) is approximated by P(Y < 45.5).
Null and alternative hypotheses: H0 states the default value of a population parameter; H1 states the change the test is looking for (greater, less, or different).Example: Testing a coin for bias toward heads: H0: p = 0.5, H1: p > 0.5.
Significance level: The probability threshold for rejecting H0; it is the largest acceptable chance of rejecting H0 when it is in fact true.Example: At the 5% level, reject H0 only if the observed result has probability 0.05 or less under H0.
Critical region: The set of test statistic values that would lead to rejecting H0; its boundary is the critical value.Example: X ~ B(20, 0.5), upper tail at 5%: P(X >= 15) = 0.0207 and P(X >= 14) = 0.0577, so the region is X >= 15.
p-value: The probability, assuming H0 is true, of a result at least as extreme as the one observed. Reject H0 if it is at or below the significance level.Example: 14 heads in 20 tosses: P(X >= 14) = 0.0577 > 0.05, so there is not enough evidence of bias.
Two-tailed test: A test where H1 says the parameter has changed in either direction, so the significance level is split between both tails.Example: At 5%, each tail gets 2.5%. H0: p = 0.5 vs H1: p is not 0.5.
Testing a normal population mean: If X ~ N(mu, sigma^2), the sample mean of n values follows N(mu, sigma^2/n), which is used as the test distribution.Example: mu = 100, sigma = 15, n = 25: the sample mean has standard deviation 3, so a mean of 106 gives z = 2.

Kinematics

SUVAT equations: For constant acceleration: v = u + at, s = ut + at^2/2, v^2 = u^2 + 2as, s = (u + v)t/2 and s = vt - at^2/2.Example: u = 4 m/s, a = 2 m/s^2, t = 3 s: v = 10 m/s and s = 12 + 9 = 21 m.
Displacement versus distance: Displacement is a vector measured from the start point; distance is the total path length and is never negative.Example: Walk 5 m east then 3 m west: distance 8 m, displacement 2 m east.
Velocity-time graph: The gradient gives acceleration and the area under the graph gives displacement.Example: 0 to 12 m/s in 4 s, then 12 m/s for 6 s: displacement = 24 + 72 = 96 m.
Variable acceleration: When acceleration is not constant, use calculus: v = ds/dt, a = dv/dt, and integrate to go the other way.Example: s = t^3 - 6t^2 + 9t gives v = 3t^2 - 12t + 9, which is 0 at t = 1 and t = 3 (the particle is momentarily at rest).

Forces and Newton's laws

Newton's second law: The resultant force on a body equals its mass times its acceleration, F = ma, with the acceleration in the direction of the resultant.Example: A 2 kg mass with a resultant force of 10 N accelerates at 5 m/s^2.
Newton's third law: When body A exerts a force on body B, B exerts an equal and opposite force on A. The pair act on different bodies.Example: A book pushes down on a table; the table pushes up on the book with a force of the same size.
Weight: The gravitational force on a mass, W = mg, acting vertically downward. Use the value of g stated in the paper.Example: With g = 9.8 m/s^2 a 3 kg mass has a weight of 29.4 N.
Connected particles over a pulley: Two masses joined by a light inextensible string over a smooth pulley share the same acceleration and the string tension is the same throughout.Example: 5 kg and 3 kg hanging: a = 2g/8 = 2.45 m/s^2 and T = 3(9.8 + 2.45) = 36.75 N.
Limiting friction: Friction can grow up to a maximum of mu times the normal reaction R; at that point the body is on the point of slipping or already sliding.Example: A 4 kg block on a rough horizontal floor with mu = 0.3: maximum friction = 0.3 * 39.2 = 11.76 N.
Resolving on an inclined plane: Split weight into mg sin theta along the slope and mg cos theta at right angles to it; the normal reaction balances the second part.Example: A particle on a smooth 30 degree slope accelerates at g sin 30 = 4.9 m/s^2.
Modelling assumptions: Simplifications such as treating a body as a particle, a string as light and inextensible, a pulley as smooth, or a rod as rigid.Example: Treating a car as a particle ignores its size and rotation, so the model cannot predict tipping.
Force vectors in i, j form: Forces written as ai + bj can be added component by component; the resultant's size is found with Pythagoras.Example: (3i + 4j) N + (2i - j) N = (5i + 3j) N, magnitude sqrt(34), about 5.83 N.

Moments and projectiles

Moment of a force: The turning effect of a force about a point: force times the perpendicular distance from the point to its line of action.Example: 20 N acting 0.5 m from a hinge, at right angles to the door, gives a moment of 10 Nm.
Equilibrium of a uniform beam: For a rigid body at rest, the resultant force is zero and the total moment about every point is zero.Example: 4 m beam, weight 100 N, supports at both ends, extra 60 N load 1 m from A: moments about A give 4R(B) = 200 + 60, so R(B) = 65 N and R(A) = 95 N.
Projectile motion: Treat horizontal and vertical motion separately: horizontal velocity stays constant and vertical motion has acceleration g downward.Example: 20 m/s at 30 degrees: time of flight = 2 * 10/9.8, about 2.04 s; range = 400 sin 60/9.8, about 35.3 m.
Greatest height of a projectile: Reached when the vertical velocity is zero, giving height = (vertical launch speed)^2/(2g).Example: Vertical launch speed 10 m/s: height = 100/19.6, about 5.10 m.