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| = How do I derive an expected total for a subset of items based on the expected overall total? = | = How do I derive an expected total for a subset of items based on the expected overall total item score? = |
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| You can derive expected totals for subsets of items by taking fractions of the expected total of all T items IF each subscale has the same expected value and each subscale is weighted equally in computing the total scale score. Expressed more formally | You can derive expected totals for subsets of T items by taking fractions of the expected total of all T items ''if'' each item has the ''same'' expected value and each item is weighted ''equally'' in computing the total scale score (e.g. by simple summation). Expressed more formally: |
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| If each of T items has the same weight, W, and expected score, E(S), then it follows for 1 <= t <= T: | if each of T items comprising a total expected score has the ''same weight'', W, and ''expected score'', E(S), then it follows for 1 <= t <= T: |
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| E(total of all T items) = TW E(S) and | E(total of all T items) = TW E(S) and, since TW is a constant, |
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| For example if t=2 then do obtain the expected total for half the items you just need to half the total for all T items. | For example when t = 2 then it follows the expected total for half the items is equal to half the expected total of all T items. |
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| If neither the weight or expected score is different for each item score then it is not possible to derive the expected total score for each subset of items based on the expected overall item total by simple fractionation of the overall item expected total (as above). In this case it is necessary to know the individual item weights and item expected scores to derive the expected total for each subset of items. | If ''either'' the weight or expected score is different for ''any'' of the item scores then it is not possible to derive the expected total score for each subset of items based on the expected overall item total by simple fractionation of the overall item expected total (as above). In this case it is necessary to know the individual item weights and expected scores to derive the expected total for each subset of items. Expected scores may be obtained by taking averages based upon frequency distributions of individual item scores taken from large samples. For example suppose a scaled test score which can take values from 0 to 4 is distributed as follows: 0 (N=12) 1 (12) 2 (23) 3 (16) 4 (9). The median for these scores is 2 which can represent the overall average or expected score; One could also similarly work out a median for subsets such as 'above average' scores which is the median of scores above the overall median of 2. In this case the 'above average' median is 3 (= the (16+9+1)/2 = 13th lowest score above 2). Similarly 'below average' scores would take an expected value of 1 (= (12+12+1)/2 = average of the 13th and 14th lowest scores). |
How do I derive an expected total for a subset of items based on the expected overall total item score?
You can derive expected totals for subsets of T items by taking fractions of the expected total of all T items if each item has the same expected value and each item is weighted equally in computing the total scale score (e.g. by simple summation). Expressed more formally:
if each of T items comprising a total expected score has the same weight, W, and expected score, E(S), then it follows for 1 <= t <= T:
E(total of all T items) = TW E(S) and, since TW is a constant, E(total of a subset of t items) = [TW/t] E(S) = E(total of all T items)/t
For example when t = 2 then it follows the expected total for half the items is equal to half the expected total of all T items.
If either the weight or expected score is different for any of the item scores then it is not possible to derive the expected total score for each subset of items based on the expected overall item total by simple fractionation of the overall item expected total (as above). In this case it is necessary to know the individual item weights and expected scores to derive the expected total for each subset of items.
Expected scores may be obtained by taking averages based upon frequency distributions of individual item scores taken from large samples. For example suppose a scaled test score which can take values from 0 to 4 is distributed as follows: 0 (N=12) 1 (12) 2 (23) 3 (16) 4 (9). The median for these scores is 2 which can represent the overall average or expected score; One could also similarly work out a median for subsets such as 'above average' scores which is the median of scores above the overall median of 2. In this case the 'above average' median is 3 (= the (16+9+1)/2 = 13th lowest score above 2). Similarly 'below average' scores would take an expected value of 1 (= (12+12+1)/2 = average of the 13th and 14th lowest scores).