MATHEMATICS ELEMENTARY STATISTICAL TABLES. F D J Dunstan, A B J Nix, J F Reynolds, R J Rowlands.

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1 MATHEMATICS ELEMENTARY STATISTICAL TABLES F D J Dunstan, A B J Nix, J F Reynolds, R J Rowlands

3 MATHEMATICS Elementary Statistical Tables F D J Dunstan, A B J Nix, J F Reynolds, R J Rowlands

4 PREFACE This set of tables has been designed by RND Publications in collaboration with the Associated Examining Board for use in Advanced level and university courses in Statistics. Each table is receded by a brief exlanation of its contents and we hoe that, in general, the lay-out is sufficiently familiar to enable the tables to be used satisfactorily without further exlanation. We have resisted the temtation to include excessive material on the use of tables and we leave this to the textbooks. Furthermore, since the tables will be used in examinations, many of the formulae which are exected to be known by the candidates have also been omitted. We have tried to maintain a degree of consistency in resentation. To this end, we have tabulated the distribution function in Tables, and. In Tables,, and, the ercentage oints are tabulated since these distributions are used in many different ways. In Tables,, and, the uer tail critical values are tabulated since the corresonding distributions are used almost exclusively in hyothesis testing. In Tables and, for ease of resentation, as soon as the value of the distribution function reaches unity, all succeeding ones are omitted. Thus if, in using these tables, a blank is obtained as the required robability, this should be interreted as unity. In Tables, and where non-arametric discrete statistics are tabulated, the values given should be included within the critical region. Furthermore, as exlained in the headings, exact significance levels cannot in general be obtained using these statistics. The critical values given are those which ensure a significance level as close as ossible to the stated levels. If, in using these tables, a blank is obtained as the required critical value, this means that the nearest achievable significance level to the stated level is %. The corresonding critical value is omitted on the grounds that such a test has no ractical value. Mathematics Elementary Statistical Tables Published by WJEC CBAC Ltd Western Avenue, Cardiff CF YX Published

5 CONTENTS TABLE PAGE The Binomial Distribution Function The Poisson Distribution Function The Normal Distribution Function Percentage Points of the Normal Distribution Control Chart Limits for Samle Range Percentage Points of the χ -Distribution Percentage Points of the Student s t-distribution Percentage Points of the F-Distribution Critical Values of the Product Moment Correlation Coefficient Critical Values of the Searman Rank Correlation Coefficient The Fisher z-transformation The Inverse Fisher z-transformation Critical Values of the Wilcoxon Signed Rank Statistic Critical Values of the Mann-Whitney Statistic Random Digits The Negative Exonential Function Turn over.

6 TABLE BINOMIAL DISTRIBUTION FUNCTION The table gives the robability of obtaining at most x successes in a sequence of n indeendent trials, each of which has a robability of success, i.e. x n P (X x) = ( r ( ) n r r r = where X denotes the number of successes. ( x x n = n = n = n = n = n = n =

7 BINOMIAL DISTRIBUTION FUNCTION x n = n = n = n = n = x Turn over.

8 BINOMIAL DISTRIBUTION FUNCTION x n = n = n = x

9 BINOMIAL DISTRIBUTION FUNCTION x n = x n = Turn over.

10 x n = n = BINOMIAL DISTRIBUTION FUNCTION x

11 TABLE POISSON DISTRIBUTION FUNCTION The table gives the robability that a Poisson random variable X with mean m is less than or equal to x, i.e. x P (X x) = m r e m r! r = m x m x m x m x m x m x Turn over.

12 TABLE NORMAL DISTRIBUTION FUNCTION The table gives the robability that a normally distributed random variable Z with zero mean and unit variance is less than or equal to z. z z

13 TABLE PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION The table gives the values of z satisfying P (Z z) = where Z is a normally distributed random variable with zero mean and unit variance. z TABLE CONTROL CHART LIMITS FOR SAMPLE RANGE The table gives (i) values of k satisying σ = ke(w), where E(W) may be estimated by W, (ii) values of D α satisfying P(W D α σ) = α, (iii) values of D' α satisfying P(W D' α E(W)) = α, where W is the range of a random samle of size n from a normal distribution with standard deviation σ. n k D. D. D'. D' Turn over.

14 TABLE PERCENTAGE POINTS OF THE χ -DISTRIBUTION The table gives the values of x satisfying P (X x) = where X is a χ random variable with v degrees of freedom. x v

15 TABLE PERCENTAGE POINTS OF THE STUDENT S t-distribution The table gives the values of x satisfying P (X x) = where X is a random variable having the Student s t-distribution with v degrees of freedom. x v..... v Turn over.

16 TABLE PERCENTAGE POINTS OF THE F- DISTRIBUTION The tables give the values of x satisfying P (X x) = where X is a random variable having the F-distribution with v degrees of freedom in the numerator and v degrees of freedom in the denominator. The table below corresonds to =. and should be used for one-tail tests at significance level.% or two-tail tests at significance level %. v v The table below corresonds to =. and should be used for one-tail tests at significance level % or two-tail tests at significance level %. v v

17 PERCENTAGE POINTS OF THE F-DISTRIBUTION x = F (v, v ) The relationshi F (v, v ) = /F (v, v ) can be used to find the ercentage oints in the lower tail. The table below corresonds to =. and should be used for one-tail tests at significance level.% or two-tail tests at significance level %. v v The table below corresonds to =. and should be used for one-tail tests at significance level % or two-tail tests at significance level %. v v Turn over.

18 TABLE CRITICAL VALUES OF THE PRODUCT MOMENT CORRELATION COEFFICIENT The table gives the critical values, for different significance levels, of the samle roduct moment correlation coefficient r based on n indeendent airs of observations from a bivariate normal distribution with correlation coefficient =. One tail Two tail n % %.% %.% % % % % %

19 TABLE CRITICAL VALUES OF THE SPEARMAN RANK CORRELATION COEFFICIENT The table gives the critical values, for different significance levels, of the Searman rank correlation coefficient r s for various samle sizes n. It should be noted that, since r s is discrete, exact significance levels cannot in general be achieved. The critical values given are those whose significance levels are nearest to the stated values. One tail % %.% %.% Two tail % % % % % n Turn over.

20 TABLE THE FISHER z-transformation The table gives the values of the function z(r) = tanh _ r. For r <, the relationshi z(r) = z( r) may be used. r r TABLE THE INVERSE FISHER z-transformation The table gives the values of the function r(z) = tanh z. For z <, the relationshi r(z) = r( z) may be used. z

21 TABLE CRITICAL VALUES OF THE WILCOXON SIGNED RANK STATISTIC The table gives the uer tail critical values w c of the statistic n W = U i R i i = where R i denotes the rank of the magnitude of the ith. observation in a samle of size n and U i = or according as to whether this observation is ositive or negative. The lower tail critical values are given by n(n + ) w c.since W is discrete, exact significance levels cannot in general be achieved. The critical values given are those whose significance levels are nearest to those stated. One tail Two tail n % %.% %.% % % % % % Turn over.

22 TABLE CRITICAL VALUES OF THE MANN-WHITNEY STATISTIC The table gives the uer tail critical values u c of the statistic m n U = Z ij i = j = where Z ij = if X i < Y j and Z ij = if X i > Y j given the indeendent samles X, X,... X m and Y, Y,... Y n. The lower tail values are given by mn u c. One tail.% Two tail % n m One tail % Two tail % n m

23 CRITICAL VALUES OF THE MANN-WHITNEY STATISTIC Since U is discrete, exact significance levels cannot in general be achieved. The critical values given are those whose significance levels are nearest to those stated. n m One tail.% Two tail % One tail % Two tail % n m Turn over.

24 TABLE RANDOM DIGITS The table gives random digits, from to, arranged for convenience in blocks of.

25 TABLE NEGATIVE EXPONENTIAL FUNCTION The table gives the values of the function f(x) = e x. SUBTRACT x

26 MATHEMATICS ELEMENTARY STATISTICAL TABLES F D J Dunstan, A B J Nix, J F Reynolds, R J Rowlands

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