Reliability and precision

Explore how split-half correlation, test length, internal consistency and the standard error of measurement are related.
Two halves of the test
Half A = odd-numbered items. Half B = even-numbered items. You can change the scores for both halves to observe how the correlation and corrected reliability change.
Full test: 35 items · Half A: 18 odd-numbered items · Half B: 17 even-numbered items.
OrderMitad AMitad BTotal
Relationship between the two halves
Each point represents a respondent. A stronger association between the scores on the two halves produces a higher correlation.
X-axis: Half A (odd-numbered) · Y-axis: Half B (even-numbered)
Correlation between halves
Correlation between halves
Corrected reliability
Spearman–Brown correction
rSB = 2r12 1 + r12
rSB: estimate of full-test reliability corrected using Spearman–Brown.
r12: correlation between the scores obtained on the two halves of the test.
2: length factor used because the full test is twice as long as each half.
Test characteristics
Set the reliability and length of the original test. Then shorten or lengthen the test to observe the expected effect on reliability.
Initial reliability rold
0.70
0.100.95
Original length Noriginal
40 items
10100
Change in test length
0 items
1 item No change 120 items
Original items40
Change in items0
New length40
Length factor n1.00
Note: Here n is a general test-length factor. A value of n = 2 means the test has twice as many items as the original. This is different from the “2” in the Spearman–Brown split-half formula.
Spearman–Brown prophecy
rnew
=
n rold1 + (n−1)rold
n
=
NnewNoriginal
n
=
rnew
=
rnew = 0.700
rnew: expected reliability of the test at the new length.
rold: reliability of the original test.
n: length factor, calculated as Nnew/Noriginal.
Noriginal: number of items in the original test.
Nnew: total number of items after shortening or lengthening the test.
Relative length and expected reliability
The point indicates the selected length. Values of n below 1 represent a shorter test; n = 1 retains the original length; values above 1 represent a longer test.
Shortening the test reduces expected reliability. Lengthening it increases expected reliability, although the gains become progressively smaller.
Simulated dichotomous test
Click any response to switch between 0 and 1. The matrix contains 20 respondents and 6 items.
Components by item
Itempqpq
Spread of total scores
Each bar represents a respondent's total score. Changing one response affects both the item and the score distribution.
KR-20 result
k
Σpq
Mean total score
Total-score variance
Σpq / SX2
KR-20
KR-20 = k k−1 × (1 − Σ piqi SX2 )
KR-20: internal-consistency coefficient for dichotomous items.
k: number of items in the test.
pi: proportion of correct responses to item i.
qi = 1 − pi: proportion of incorrect responses to item i.
piqi: variance of dichotomous item i.
Σpiqi: sum of the item variances.
SX2: variance of the total test score.
Test with polytomous items
Each item can take integer values from 0 to 5. Enter the desired value directly in each cell.
Item variances
ItemVariance Si2
Cronbach's alpha result
Number of items k
6
Σ Si2
—
Total-score variance SX2
—
Cronbach's alpha
—
α = k k−1 × (1 − Σ Si2 SX2 )
α: Cronbach's alpha coefficient.
k: number of items in the test.
Si2: variance of item i.
ΣSi2: sum of the item variances.
SX2: variance of the total score.
When items are dichotomous (0/1), KR-20 and Cronbach's alpha produce the same result.
Respondents
The same SEM estimate applies to all scores. Click a row to view a respondent.
Order Score X X − SEM X + SEM
Controls
Reliability rxx
0.100.99
Standard deviation SD
220
SEM = SD√(1−rxx)
Precision of the observed score
Higher reliability → lower SEM → greater score precision.
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