Statistics for Environmental Engineers

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The equations for calculating the control limits are:


X chart Central line = X


Control limits = X ± к 1R R chart Central line = R


Upper control limit (UCL) = k2R


where X is the grand mean of sample means (the average of the X values used to construct the chart), R is the mean sample range (the average of the ranges [R] used to construct the chart), and n is the number of replicates used to compute the average and the range at each sampling interval. R is the absolute difference between the largest and smallest values in the subset of n measured values at a particular sampling interval.


Coefficients for Calculating Action Lines on X and Range Charts


n


k


u


2


1.880


3.267


3


1.023


2.575


4


0.729


2.282


5


0.577


2.115

Source: Johnson, R. A. (2000). Probability and Statistics for Engineers, 6th ed., Englewood Cliffs, NJ, Prentice-Hall.


The coefficients of k1 and k2 depend on the size of the subsample used to calculate X and R. A few values of k1 and k2 are given in Table 11.2. The term kjR is an unbiased estimate of the quantity 3aUfn, which is the half-length of a 99.7% confidence interval. Making more replicate measurements will reduce the width of the control lines.


The control charts in Figure 11.1 were constructed using values measured on two test specimens at each sampling time. The average of the two measurements, X1 and X2, is X; and the range R is the absolute difference of the two values. The average of the 15 pairs of X values is X = 4.08. The average of the absolute range values is R = 0.86. There are n = 2 observations used to calculate each X and R value. For the data in the Table 11.1 example, the action limits are:

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