Coefficient Alpha and the Internal Structure of Tests
Psychometrika, 1951 · paper · Mathematics and statistics
Shows that coefficient alpha, generalizing the Kuder-Richardson formula, equals the mean of all possible split-half coefficients and estimates the correlation between two random item samples. It examines what alpha indicates about test structure and implications for designing tests.
6 findings
Substance "When m is 1.1 … the values of k5 are as follows: for r = .20, .62; for r = .60, .70; for r = 1.00, .999." Eq. (6) gives about .95 at r = .60, not .70.
- p. 6 Paragraph after Table 2
for r = .60, .70 ; for r = 1.00, .999.
- p. 6 Paragraph after Table 2
for r = .60, .70
- p. 6 text after Table 2
for r = .20, .62; for r = .60, .70; for r = 1.00, .999
Only the authors can say what is right here.
Judge's note: Page 6 image: Eq. (6) is k5 = (2mr − m² + 1)(1 + 2mr + m²)/[2mr(2mr − m² + 3)], and the text prints .70 for r = .60. I recomputed with m = 1.1. The equation reproduces .62 at r = .20 and .999 at r = 1.00, but gives .954 at r = .60. The printed .70 also breaks the rising sequence. The authors should confirm the intended value.
Second check: another model agreed
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Substance "In a test of only five items, α reaches .86" (and "φ̄ is .54"): for a perfect scale with p = .50, .58, .71, .80, .89, the computation gives α ≈ .886.
- p. 32 First lines
only five items, a reaches .86
- p. 31 Last paragraph
the p, being .50, .58, .71, .80, and .89
- p. 32 Second paragraph
is .54
Only the authors can say what is right here.
Judge's note: I recomputed the perfect-scale α using item variances pq and covariances p_hard(1 - p_easy). The highest between-level φ comes out at .85, matching the printed value, which supports the method. The five-item α comes out about .886 rather than the printed .86. The printed .54 for φ̄ is consistent with .86 through Eq. (44), so both numbers rest on the same computation. The authors should confirm the five-item figures.
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Reference or label Table 1 footnote reads "‖After Mosier (28)", but reference 28 is Loevinger, "The technic of homogeneous tests…"; Mosier's "A short cut in the estimation of split-halves coefficients" is reference 29.
- p. 5 Table 1 footnote
After Mosier (28)
- p. 37 Reference 28
Loevinger, J. The technic of homogeneous tests
- p. 37 References 28, 29
29. Mosier, C.I. A short cut in the estimation of split-halves coefficients
- p. 5 Table 1 footnotes
‖After Mosier (28)
Judge's note: Page 5 image: the footnote reads (28). The text layer's (18) is an OCR error. Page 37 image: 28 = Loevinger (1948, Psychol. Bull.) and 29 = Mosier (1941). This mirrors C012, where Loevinger is cited as (29), so the two numbers appear to have been swapped.
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Reference or label "This finding agrees with Kelley's earlier analysis of much the same question (2, 3)." Refs 2 and 3 are Brown (1910) and Brownell (1933); Kelley's works are refs 23–25.
- p. 6 Section II, last paragraph
agrees with Kelley's earlier analysis of much the same question (2, 3).
- p. 37 References 23-25
Kelley, T. L. The reliability coefficient.
- p. 6 Section II, last paragraph
agrees with Kelley's earlier analysis of much the same question (2, 3)
- p. 6 Section II, last paragraph
This finding agrees with Kelley's earlier analysis of much the same question (2, 3).
Only the authors can say what is right here.
Judge's note: The page 6 crop shows (2, 3). The page 36 reference list gives 2 = Brown, W. (1910) and 3 = Brownell, W. A. (1933). Kelley is refs 23, 24 and 25 on page 37. The sentence names Kelley, so the pointer does not match the author. It may be a misprint for (23), but only the authors can say which entry was intended.
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Reference or label Table 6's second pattern is labelled "new group factors in each set of 5 items", but its entries ("4 each") and the text ("15 factors in four items each") use 4 items per group factor.
- p. 20 Table 6, second pattern
new group factors in each set of 5 items:
- p. 21 Text
the 60-item test containing 15 factors in four items each
- p. 20 Table 6, second pattern
new group factors in each set of 5 items
- p. 21 Second paragraph
60-item test containing 15 factors in four items each
Judge's note: The page 22 image shows the pattern label and k range carried over from Table 5, while the tabulated term counts require 4 items per factor. Recomputing Vt with 4-item groups reproduces the printed 500 (n = 20) and 3900 (n = 60), and the between-halves total of 960. The text on page 21 agrees with 4 items per factor.
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Reference or label "Examining Loevinger's proposed coefficient of homogeneity (29)": ref 29 is Mosier, "A short cut in the estimation of split-halves coefficients"; Loevinger's works are refs 27 and 28.
- p. 31 First line
Examining Loevinger's proposed coefficient of homogeneity (29),
- p. 37 Reference 29
Mosier, C.I. A short cut in the estimation of split-halves coefficients.
- p. 31 p. 327, first line
Examining Loevinger's proposed coefficient of homogeneity (29)
- p. 37 Reference 29
29. Mosier, C.I.
Judge's note: Page 31 crop shows (29). On page 37, 27 and 28 are Loevinger and 29 is Mosier. This is the counterpart of C001 (Mosier cited as 28) and looks like a swap of 28 and 29. It is a separate pointer error in a separate place.
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7 minor (typos and the like; most checked by the judge only)
Wording "Where α is .715 for the morale scale, the mean of parallel splits is .748". Table 4 gives .737 as the mean of all 6 parallel splits; .748 is the mean of the 5 splits with 1.1 > σb/σa > .9.
- p. 15 Paragraph on Tables 3 and 4
the mean of parallel splits is .748
- p. 14 Table 4
.681-.780 .737
- p. 14 Table 4
.681-.780 .737 5 ,712-.780
- p. 14 Table 4, Parallel (Type II) row
6 .681-.780 .737
Only the authors can say what is right here.
Judge's note: Page 14 Table 4: All Splits mean .715 (126 splits) and .717 (82 restricted splits); Parallel (Type II) mean .737 (6 splits) and .748 (5 restricted splits). The page 15 text compares the restricted-subset parallel mean (.748) with the all-splits alpha (.715) and does not say it is using the subset. The like-for-like comparisons would be .737 vs .715, or .748 vs .717. Both values appear in the table, so this is an unlabelled choice rather than a wrong number.
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Wording "The range for the morale scale is greater (.20)", but Table 4 gives .609–.797, a range of .188 (≈.19).
- p. 15 First paragraph
The range for the morale scale is greater (.20)
- p. 14 Table 4, All Splits
.609-.797
Judge's note: Page 14 Table 4, All Splits: range .609–.797. Page 15 text gives the range as (.20). The parallel mechanical-test figure, .08, matches its table (.860 − .779 = .081). The morale value is loosely rounded and does not change the point being made.
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Wording Summary point 3 says α computed for separate blocks and combined "gives a satisfactory estimate of first-factor concentration"; the body (p. 322, Eq. 43) presents the same procedure as "a superior coefficient of equivalence for the 'lumpy' test", i.e., a parallel-forms coefficient.
- p. 36 Summary point 3
gives a satisfactory estimate of first-factor concentration
- p. 26 Paragraph before Eq. (43)
a suggestion for obtaining a superior coefficient of equivalence for the "lumpy" test
Judge's note: I compared page 26 and Figure 3 on page 27 with page 36. The summary describes Eq. (43) differently from the body's own description of what it estimates.
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Wording Text says "Figure 4 shows the relation of φij to pi and pj for three levels of correlation: r_tet = .30, r_tet = .50, and r_tet = .80", but each panel of Figure 4 also plots and labels an r_tet = 1.00 curve.
- p. 30 Section 5 text
for three levels of correlation
- p. 29 Figure 4
Relation of ¢ij to Pl and pj for Several Levels of Correlation.
- p. 29 Figure 4 curve labels
r_tet = 1.00
Judge's note: Page 29 image: each of the three panels (pi = .20, .50, .80) has labelled curves for r_tet = 1.00, .80, .50 and .30. Page 30 text says 'three levels'. The 1.00 curve may be meant as a limiting reference, so this is a minor description mismatch.
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Wording "If our second test had been a further measure of pitch, placing the subject above or below the 75th percentile, then the two tests would have placed him in one of four categories." Cuts at the mean and at the 75th percentile of one dimension give three ordered categories.
- p. 32 Information-theory paragraph
then the two tests would have placed him in one of four categories
Only the authors can say what is right here.
Judge's note: Page 32 image confirms the wording. On a single dimension, being above the 75th percentile implies being above the mean. The cuts therefore give three regions: below the mean, mean to P75, and above P75. A fourth pattern would need inconsistent responses, which the passage does not invoke. The authors should say what was intended. This affects the illustration only.
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Typo Table 7, row r_ij(tet), column p_j = .80 reads ".80", although the caption says the tetrachoric correlation is held constant and the text gives it as .30.
- p. 30 Table 7, row r_ij(tet)
.30 .30 .30 .30 .30 .30 .80 .30 .30
Judge's note: The text layer reads .80 in that cell. In the scan the 3/8 glyphs are hard to tell apart; the same row's H value is also degraded (".~4" for .34). The symmetric φ = .17 confirms that .30 was intended. This is at most a typographic slip.
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Typo "For any single item, R must be greater than p_i or q_j, whichever is greater": the subscripts are mixed for a single item and should both be i.
- p. 31 Paragraph after eq. 48
R must be greater than p
- p. 31 First paragraph
For any single item, R must be greater than p~ or qj, whichever is greater
Judge's note: The crop of page 31 shows p_i and q_j. For one item the comparison should be with that item's own p_i and q_i.
Second check: none (judge only)