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5. Probability & Statistics 1

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  1. 5.1 Representation of data

    1. 5.1.1Data representation

      • select a suitable way of presenting raw statistical data, and discuss advantages and/ or disadvantages that particular representations may have

    2. 5.1.2Charts and diagrams

      • draw and interpret stem-and-leaf diagrams, box-and-whisker plots, histograms and cumulative frequency graphs Including back-to-back stem-and-leaf diagrams.

    3. 5.1.3Averages and spread

      • understand and use different measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation) e.g. in comparing and contrasting sets of data.

    4. 5.1.4A cumulative frequency graph e

      • use a cumulative frequency graph e.g. to estimate medians, quartiles, percentiles, the proportion of a distribution above (or below) a given value, or between two values.

    5. 5.1.5And use the mean and standard

      • calculate and use the mean and standard deviation of a set of data (including grouped data) either from the data itself or from given totals x/ and x2/, or coded totals xa-^h/ and xa 2-^h/, and use such totals in solving problems which may involve up to two data sets.

  2. 5.2 Permutations and combinations

    1. 5.2.1Permutations and combinations

      • understand the terms permutation and combination, and solve simple problems involving selections

    2. 5.2.2Arrangement restrictions

      • solve problems about arrangements of objects in a line, including those involving - repetition (e.g. the number of ways of arranging the letters of the word 'NEEDLESS') - restriction (e.g. the number of ways several people can stand in a line if two particular people must, or must not, stand next to each other). Questions may include cases such as people sitting in two (or more) rows. Questions about objects arranged in a circle will not be included.

  3. 5.3 Probability

    1. 5.3.1Permutations and combinations

      • evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events, or by calculation using permutations or combinations e.g. the total score when two fair dice are thrown. e.g. drawing balls at random from a bag containing balls of different colours.

    2. 5.3.2Probability rules

      • use addition and multiplication of probabilities, as appropriate, in simple cases Explicit use of the general formula AB AB ABPP PP,+ -= +^^ ^^hh hh is not required.

    3. 5.3.3Exclusive and independent events

      • understand the meaning of exclusive and independent events, including determination of whether events A and B are independent by comparing the values of ABP +^h and ABPP #^^ hh

    4. 5.3.4Elementary probability

      • calculate and use conditional probabilities in simple cases. e.g. situations that can be represented by a sample space of equiprobable elementary events, or a tree diagram. The use of AB B ABP P P +=^ ^ ^h h h may be required in simple cases.

  4. 5.4 Discrete random variables

    1. 5.4.1Discrete random variables

      • draw up a probability distribution table relating to a given situation involving a discrete random variable X, and calculate E(X) and Var(X)

    2. 5.4.2Geometric distribution

      • use formulae for probabilities for the binomial and geometric distributions, and recognise practical situations where these distributions are suitable models Including the notations B(n, p) and Geo(p). Geo(p) denotes the distribution in which pr = p(1 - p)r - 1 for r = 1, 2, 3, ….

    3. 5.4.3Expectation and variance

      • use formulae for the expectation and variance of the binomial distribution and for the expectation of the geometric distribution. Proofs of formulae are not required.

  5. 5.5 The normal distribution

    1. 5.5.1Normal distribution

      • understand the use of a normal distribution to model a continuous random variable, and use normal distribution tables Sketches of normal curves to illustrate distributions or probabilities may be required.

    2. 5.5.2Problems concerning a variable X

      • solve problems concerning a variable X, where,XN 2+ nv_ i, including - finding the value of P 1Xx>_ i, or a related probability, given the values of x1, n, v. - finding a relationship between x1, n and v given the value of P 1Xx>_ i or a related probability For calculations involving standardisation, full details of the working should be shown, e.g. Z X v n = -_ i.

    3. 5.5.3Binomial distribution

      • recall conditions under which the normal distribution can be used as an approximation to the binomial distribution, and use this approximation, with a continuity correction, in solving problems. n sufficiently large to ensure that both np > 5 and nq > 5.