Statistical Tables Compiled by Alan J. Terry School of Science and Sport University of the West of Scotland Paisley, Scotland
Contents Table 1: Cumulative binomial probabilities Page 1 Table 2: Cumulative Poisson probabilities Page 6 Table 3: Cumulative standard normal probabilities Page 12 Table 4: Inverse standard normal distribution Page 15 Table 5: Upper percentage points of the t distribution Page 17 Table 6: Control chart limits for sample mean Page 19 Table 7: Control chart limits for sample range Page 21 Table 8: Factor to estimate population standard deviation from mean of sample ranges Page 22 Table 9: Control chart limits for sample standard deviation Page 23 Table 10: Derivation of single sampling plans Page 24 Table 11: Construction of OC curves for single sampling plans Page 26 Table 12: Probability distributions Page 27 Table 13: Various formulae Page 29
Table 1 Cumulative Distribution Function for the Binomial Distribution The function tabulated is the cumulative distribution function (cdf) for the Binomial distribution, for different values of n (the number of trials) and p (the probability of success on any one trial). The value is: x ( ) n F (x) = P (X x) = p r (1 p) n r r r=0 1
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Table 2 Cumulative Distribution Function for the Poisson Distribution The function tabulated is the cumulative distribution function (cdf) for the Poisson distribution, for different values of m (the mean). The value is: F (x) = P (X x) = x e m m r r=0 r! 6
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Table 3 Cumulative Distribution Function for the Standard Normal Distribution The function tabulated is Φ(u), which is the cumulative distribution function (cdf) for the standard normal distribution, U N(0, 1 2 ). The values are produced from the following formula: Φ(u) = P (U u) = u e t2 /2 2π dt Note that Φ(u) = P (U u) is the area under the probability density function (pdf) for U to the left of u, as illustrated in the figure below. Note also that, if X N(µ, σ 2 ), then the variable U = (X µ)/σ follows the standard normal distribution N(0, 1 2 ). The tables for the cdf of U are shown on the next two pages. 12
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Table 4 The Inverse of the Standard Normal Distribution The function tabulated is Φ 1. Note that we have u = Φ 1 (q) = Φ 1 (1 p), as illustrated in the figure below. The table is shown on the next page. 15
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Table 5 Upper Percentage Points of the t Distribution The table gives the values of t ν (α). There are the 100α percentage points of the t distribution with ν degrees of freedom. The value t ν (α) is the solution to: α = Γ [(ν + 1)/2] νπγ(n/2) t ν(α) ) (1 + x2 dx ν The value t ν (α) is that number such that the area under the probability density function of the t distribution with ν degrees of freedom to the right of the number is α. This is illustrated in the figure below. The table is shown on the next page. 17
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Table 6 Control Chart Limits for Sample Mean, x The tabulated values can be used to derive warning and action limits for an X bar control chart. Let x be the mean of the sample means. If using the mean of the sample ranges, R, to create the X bar chart, then: warning limits = x ± A 0.025 R action limits = x ± A 0.001 R If using the mean of the sample standard deviations, s, to create the X bar chart, then: The table is shown on the next page. warning limits = x ± A 0.025 s action limits = x ± A 0.001 s 19
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Table 7 Control Chart Limits for Sample Range, R The tabulated values can be used to derive warning and action limits for a Range control chart, or R chart. Let R be the mean of the sample ranges. Then: The table is shown below. lower action limit = D 0.999 R lower warning limit = D 0.975 R upper warning limit = D 0.025 R upper action limit = D 0.001 R 21
Table 8 Factor to estimate population standard deviation, σ, from mean of sample ranges, R An estimate of the population standard deviation, σ, may be found as follows: estimate of σ = R d 2 where R is the mean of the sample ranges, and where d 2 is a factor that depends on the sample size as in the table below. Sample size d 2 2 1.128 3 1.693 4 2.059 5 2.326 6 2.534 7 2.704 8 2.847 9 2.970 10 3.078 11 3.173 12 3.256 22
Table 9 Control Chart Limits for Sample Standard Deviation, s The tabulated values can be used to derive warning and action limits for a standard deviation control chart, or s chart. Let s be the mean of the sample standard deviations. Then: lower action limit = D 0.999 s lower warning limit = D 0.975 s upper warning limit = D 0.025 s upper action limit = D 0.001 s The table below shows values for the factors D 0.999, D 0.975, D 0.025, and D 0.001. Note that the table can also be used to produce an estimate for the mean of the sample ranges, R, using the following formula: estimate of R = sd 2 23
Table 10 Derivation of Single Sampling Plans The table on the next page can be used to construct a single sampling plan with Operating Characteristic curve passing through (or very close to) the following two points: where (p 1, 1 α) and (p 2, β) p 1 = Acceptable Quality Level (AQL) p 2 = Limiting Quality (LQ) α = Producer s Risk β = Consumer s Risk To use the table, first identify the appropriate column, that is, the column with the specified values for α and β. In this column, select the row with value closest to (or just less than) p 2 /p 1. The acceptance number is c for that row, whilst the sample size n is found by reading off the value for np 1 for that row and dividing this by p 1. If this produces a value for n that is not a whole number, then round up to the next whole number and take this as the value for n. 24
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Table 11 Construction of OC Curves for Single Sampling Plans The table below shows values of np for which the probability of acceptance of c or fewer defectives in a sample of n is P (A). To find the proportion of defectives, p, corresponding to an acceptance probability P (A) in a single sampling plan with sample size n and acceptance number c, divide the table entry by n. 26
Table 12 Probability Distributions Hypergeometric distribution This is a discrete distribution. It can be used to represent the exact probability for the number of defectives in a sample taken from a single batch. If a batch has N items, of which m are defective, and a sample of size n is taken from the batch by simple random sampling, then the number of defectives in the sample, X, is a hypergeometric random variable. We represent this by writing X hyper(n, m, n). Observe that X has the following frequency function: ( m )( N m ) k n k P (X = k) = ( N, max{0, n + m N} k min{m, n} n) The mean and variance of X are as follows: mean of X is nm N, variance of X is ( nm N ) ( 1 m N ) ( ) N n N 1 Computations involving the hypergeometric distribution are very involved. Hence the hypergeometric distribution is typically approximated by other distributions when these approximations are valid. Binomial distribution This is a discrete distribution. It can be used to represent the probability for the number of defectives in a sample taken from a continuous process. If there are n independent trials, with probability p of success on each trial, then the total number of successes, X, is a binomial random variable. We represent this by writing X B(N, p). Observe that X has the following frequency function: P (X = k) = ( n k ) p k (1 p) n k, k = 0, 1, 2,..., n Here we could have that n is the size of a sample taken from a continuous process, with p equal to the probability of any particular item being defective, and with X equal to the total number of defectives in the sample. The mean and variance of X are as follows: mean of X is np, variance of X is np(1 p) The binomial distribution can be used to approximate the hypergeometric distribution. To be specific, X hyper(n, m, n) can be approximated by Y B(n, p) where p = m/n. The approximation is reasonable provided that the probability of success on each trial is not measurably altered by sampling. 27
Poisson distribution This is a discrete distribution. It can be used to represent the probability for the number of times an event can happen, when there is no limit to how many times the event could happen. The notation X Po(λ) is used to indicate that X is a Poisson random variable with parameter λ. Here λ is the average rate, or simply the rate. The frequency function for X is: P (X = k) = λk k! e λ, k = 0, 1, 2,... The mean and variance of X are as follows: mean of X is λ, variance of X is λ The Poisson distribution can be used to approximate the binomial distribution. To be specific, X B(n, p) can be approximated by Y Po(λ) where λ = np. The approximation is reasonable provided that n is large and p is small. A rule of thumb is that the approximation is good if p 0.1 and np < 5. Normal distribution This is a continuous distribution. It can be used to represent the probability that a measurement, on a continous scale, lies within a certain interval of values. It arises as a model for sample means due to the central limit theorem. The notation X N(µ, σ 2 ) is used to indicate that X is a normal random variable with mean µ and variance σ 2. The density function for X is: ( ) 1 f(x) = σ e (x µ)2 /(2σ 2 ) 2π The normal distribution can be used to approximate the binomial distribution. To be specific, X B(n, p) can be approximated by Y N(µ, σ 2 ) where µ = np and σ 2 = np(1 p). The approximation is reasonable provided that n is large and p is not very small. A rule of thumb is that the approximation is good if p 0.1 and np > 5. In using the approximation, a continuity correction should be applied. 28
Table 13 Various formulae Sample mean If x 1, x 2,..., x n are the values in a sample of size n, then: ( ) n 1 sample mean = x = x i n i=1 Sample variance and sample standard deviation If x 1, x 2,..., x n are the values in a sample of size n, then: sample variance = s 2 = ( ) 1 n 1 n x 2 i i=1 ( ) ( 1 n n i=1 ) 2 x i The sample standard deviation, s, is the (non-negative) square root of the sample variance, s 2. C p index and C pk index For a process with standard deviation σ, and for which USL and LSL refer to the Upper Specification Limit and Lower Specification Limit, respectively, then the C p index is: USL LSL C p = 6σ Moreover, the C pk index is: { USL µ C pk = min, 3σ where µ is the process mean. } µ LSL 3σ Confidence intervals If x 1, x 2,..., x n are the values in a sample of size n, and if the values are independent realisations from the same normal population, then a 100(1 α)% confidence interval for the population mean µ is x ± (s/ n) t n 1 (α/2), where x is the sample mean, s is the sample standard deviation, and t n 1 (α/2) is the upper percentage point of the t distribution with n 1 degrees of freedom that corresponds to an upper tail probability of α/2. 29