I NTERPRETING C OMBINED E RRORS
I NTERPRETING C OMBINED E RRORS
Combined errors are errors caused by the interaction of two or more indepen- dent variables, each one causing a different problem. The system propagates the interaction of these variables; therefore, combined errors are also called propagation errors . By calculating statistical data about the sample before propagation and knowing the average and standard deviation requirements for the final product, the user can predict the outcome of the final product and make corrections for propagation errors throughout the process.
The value of an outcome formed by several variables (e.g., materials going into a batching process) is directly related to the average value of each variable. For instance, if a batching process uses two ingredients, A and B,
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S ECTION PLC Process Data Measurements C HAPTER 4 Applications
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and their average weights are A and B , then the final weight of a mix containing both materials would be A B + . This outcome is the addition of both A and because the operation to be performed is a blending, which B implies that the quantities are added. Thus, the final outcome is directly related to the equation that governs the process being performed. In real life, the actual equation of a process is very hard to obtain; it is usually only approximated.
Standard deviation specifies how each sample value relates to the mean. Accordingly, the standard deviation of an outcome product can predict how the value of the final product will be spread out about its mean in relation to each of its component variables. This information forecasts the variance of the final product value. In the previous blending example, the average weight outcome (W) is represented by:
W =+ A B
where A and are the average weights of the ingredient products. If the B distribution follows the normal (bell) curve, then:
• 68% of all samples lie within W ± 1 σ W , or ( A B +) ± 1 σ W
• 95% of all samples lie within W ± 2 σ W , or ( A + B ) ± 2 σ W
• 99% or all samples lie within W ± 3 σ W , or ( A B +) ± 3 σ W
where σ W is the standard deviation of the final product. However, to find the actual standard deviation, we must define the relation-
ship between ingredients A and B and σ W . To obtain an equation that allows two or more input variables, let’s define the function K as the equation governing the final product and/or process. After numerous sample observa- tions (n), the final product formula (K n ) will be a function of the amount of
ingredients A and B added during the sample observations—A n and B n . That
is: K n = KAB (,) n n
We can conclude that the most likely value for the function (the average value) is:
K n = KAB (,)
where the final outcome is a function of the two averages. We can define any deviation of a sample observation from the mean as ∆ K n , which is expressed
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S ECTION PLC Process Data Measurements C HAPTER 4 Applications
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as: ∆ K n = K n − KAB (,)
or ∆ K n = KAB (,) n n − KAB (,)
If the deviation from the mean is 0 ( ∆ K n = 0), implying that the value of the n th observation is the same as the mean, then we would have:
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