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==Randomization==
By [[randomization]], that is to say the run sequence of the experimental units is determined randomly. For example, if there are 3 levels of the primary factor with each level to be run 2 times, then there are 6! (where ! denotes [[factorial]]) possible run sequences (or ways to order the experimental trials). Because of the [[replication (statistics)|replication]], the number of unique orderings is 90 (since 90 = 6!/(2!*2!*2!)). An example of an unrandomized design would be to always run 2 replications for the first level, then 2 for the second level, and finally 2 for the third level. To randomize the runs, one way would be to put 6 slips of paper in a box with 2 having level 1, 2 having level 2, and 2 having level 3. Before each run, one of the slips would be drawn blindly from the box and the level selected would be used for the next run of the experiment.
In practice, the randomization is typically performed by a computer program. However, the randomization can also be generated from [[random number table]]s or by some [[random number generation#Physical methods|physical mechanism]] (e.g., drawing the slips of paper).
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* ''k'' = number of factors (= 1 for these designs)
* ''L'' = number of levels
* ''n'' = number of replications
and the total [[sample size]] (number of runs) is ''N'' = ''k'' × ''L'' × ''n''. Balance dictates that the number of replications be the same at each level of the factor (this will maximize the sensitivity of subsequent statistical ''t''- (or ''F''-) tests).
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* ''L'' = 4 levels of that single factor (called "1", "2", "3", and "4")
* ''n'' = 3 replications per level
* ''N'' = 4 levels × 3 replications per level = 12 runs
===Sample randomized sequence of trials===
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* ''Y''<sub>i,j</sub> being any observation for which ''X''<sub>1</sub> = ''i'' (''i'' and ''j'' denote the level of the factor and the replication within the level of the factor, respectively)
* μ (or mu) is the general [[___location parameter]]
* ''T''<sub>i</sub> is the effect of having treatment level ''i''
==Estimates and statistical tests==
===Estimating and testing model factor levels===
* Estimate for μ : <math>\bar{Y}</math> = the [[average]] of all the data
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