Choose an appropriate method for presenting raw statistical data, and analyze the advantages and disadvantages of each representation.
Create and analyze stem-and-leaf diagrams, box plots, histograms, and cumulative frequency graphs (including back-to-back stem-and-leaf diagrams), and interpret the data represented in each type of graph.
Comprehend and apply various measures of central tendency (mean, median, mode) and dispersion (range, interquartile range, standard deviation).
Utilize a cumulative frequency graph to estimate medians, quartiles, percentiles, the proportion of a distribution above (or below) a given value, or between two values.
Compute and apply the mean and standard deviation of a dataset (including grouped data) using the raw data, given totals Σx and Σx2, or coded totals Σ(x-a) and Σ(x-a)2. Apply these calculations to solve problems involving up to two datasets.
Permutations and Combinations
Comprehend the concepts of permutation and combination, and solve basic problems involving selections.
Resolve problems related to the arrangement of objects in a line, including scenarios involving:
Repetition (e.g. determining the number of ways to arrange the letters of the word 'NEEDLESS').
Restrictions (e.g. calculating the number of ways several people can stand in a line if two particular people must or must not stand next to each other) [problems may involve scenarios such as people sitting in two (or more) rows].
Probability
Calculate probabilities in simple cases by enumerating equiprobable elementary events or by using permutations or combinations [for example, finding the total score when two fair dice are thrown, or drawing balls at random from a bag containing balls of different colors].
Apply addition and multiplication of probabilities, as necessary, in simple cases [the explicit use of the general formula P(A \(\cup\) B) = P(A) + P(B) - P(A ∩ B) is not mandatory].
Comprehend the concepts of exclusive and independent events, including determining whether events A and B are independent by comparing the values of P(A ∩ B) and P(A) x P(B).
Compute and apply conditional probabilities in simple cases [such as situations that can be represented by a sample space of equiprobable elementary events or a tree diagram]. In simple cases, the formula P(A|B) = \(\frac{\text{P(A ∩ B)}}{\text{P(B)}}\) may be necessary.
Discrete Random Variables
Create a probability distribution table for a given scenario involving a discrete random variable X, and determine both the expected value E(X) and variance Var(X).
Apply formulas for probabilities in binomial and geometric distributions, and identify practical situations where these distributions are appropriate models [including the notations B(n,p) and Geo(p). In the geometric distribution, Geo(p) represents the distribution in which P(X = r) = p(1 - p)r - 1 for r = 1, 2, 3, …].
Apply formulas for calculating the expected value and variance of the binomial distribution, as well as the expected value of the geometric distribution.
The Normal Distribution
Comprehend the application of a normal distribution to represent a continuous random variable, and utilize normal distribution tables. Sketches of normal curves to illustrate distributions or probabilities may be necessary.
Solve problems concerning a variable X, where X ~ N(\(\mu\), \(\sigma\)2), including:
Finding the value of P(X > x1), or a related probability, given the values of x1, \(\mu\), and \(\sigma\).
Finding a relationship between x1, \(\mu\), and \(\sigma\) given the value of P(X > x1) or a related probability.
[For calculations involving standardisation, full working should be shown, e.g., Z = \(\frac{X - \mu}{\sigma}\).
Remember the conditions for using the normal distribution as an approximation to the binomial distribution, and apply this approximation, with continuity correction, to solve problems [where n is sufficiently large to ensure that both np > 5 and nq > 5].
