These are explanations of some technical terms found in the lessons. See also the Subject Index.
Absolute Value. The size or magnitude of a quantity, without reference to whether it is positive or negative. The absolute value of (x - m) is the same as that of (m - x). Both values express how far away the two are from each other, without specifying which is larger than the other. The sign for absolute value is two enclosing vertical lines, |. The expression |m - x| thus means "the absolute value of the quantity (m - x)."
We may add that the way to "get rid of minus signs" in computations is to take the absolute values of the quantities involved. The notion that one squares the quantities for this purpose is an ignorant misstatement (though one encountered in nearly every statistics textbook). One squares quantities in order to increase the statistical weight of their larger members, to apply a least-squares distance calculation to them, or for some other cogent reason.
Actual Value. The value of a variable which is actually found to occur, as distinct from its Expected Value. The conventional term is Observed Value, which however abbreviates as O, inviting confusion with zero (0). We will always write "actual value" in formulas as a capital A.
Alpha Level (a). The level at which we agree to conclude that something other than chance is at work in the observed data. The threshold of actionable significance. Often called the Action Level. In these Lessons we will mostly use a = 0.99. See also Beta Level and the discussion in Lesson 1.
Analysis of Variance. A technique for simultaneously comparing many data sets, to see if they come from the same population or are otherwise mutually compatible. The essence of the technique is that the variation within each sample establishes the background variability, and the variation between samples then reveals any significant differences. The alternative method for many data sets is a series of Chi Squared comparisons of each pair of sets. The labor involved when there are many data sets, as can happen in archaeological analysis, is prohibitive. See Lesson 14.
ANOVA = Analysis of Variance. We avoid acronyms in these pages, since the subject is already cute enough without them.
Arithmetic Triangle. See Pascal Triangle.
Average. See Mean (and compare Median, Mode).
Beta Level (b). The level at which it begins to be probable that something other than chance is at work in the observed data. The lower level of the gray zone; the "worry" threshold, used by some as an action (a) level for matters not involving honor, security, or large amounts of money. In these Lessons we will use b = 0.95. See also Alpha Level and the discussion in the Basics Lesson.
Binomial. Having two possible values, in general x and y. A coin toss may come down only heads (H) or tails (T), and is thus binomial. In algebra, the general binomial quantity is (x + y). There is a deep relation between the expansions of this expression, that is, the coefficients of (x + y)*n, and the results of coin tossing (or any other two-valued activity). See next entry.
Binomial Coefficient. The quantity (x + y) is binomial. If we multiply out the binomial expansion (x + y)² we get x² + 2xy + y². The coefficients (numerical multipliers) of successive terms in this result are 1, 2, 1.
The relation between coin tossing and the binomial coefficients is easily seen. If we toss a coin 2 times, the possible outcomes are four in number: 1 with all heads (HH), 2 with one head and one tail (HT and TH), and 1 with all tails (TT). There are thus 2 ways of getting the "one head, one tail" option, and only 1 way of getting the other options. Note that the sequence
1, 2, 1
occurs, the same as the series of binomial coefficients. A similar relation holds between the expansion of (x + y) to the nth power, or (x + y)*n, and the tossing of n coins. Thus for (x + y)³ we have the expansion x³ + 3x²y + 3xy² + y³, whose coefficients are
1, 3, 3, 1
and for 3 coins, we have 1 outcome with all heads (HHH), 3 outcomes with 2 heads and 1 tail (HHT, HTH, THH), 3 outcomes with 1 head and 2 tails (HTT, THT, TTH), and 1 outcome with all tails (TTT). Again, the series 1,3,3,1 appears.
For a graphic way of combining data from all binomial outcomes, see the Pascal Triangle.
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