Can some one please tell me when i can use the Kurder-Richardson 21 Formula, KR20 (Coefficient Alpha) or spearman-Brown formula to calculate the reliability coefficient of a test.
Sifuna.
Dear James,
Please read the following text. For more details, see attached file.
KR20
Dichotomously scored items with a range of difficulty:
Multiple choice
Short answer
Fill in the blank
Formula:
KR20 = [n/(n - 1)] x [1 - (Σpq)/Var]
KR20 = estimated reliability of the full-length test
n = number of items
Var = variance of the whole test (standard deviation squared)
Σpq = sum the product of pq for all n items
p = proportion of people passing the item
q = proportion of people failing the item (or 1-p)
Coefficient Alpha
Items that have more than dichotomous, right-wrong
scores:
Likert scale (e.g rate 1 to 5)
Short answer
Partial credit
Formula:
Alpha = [n/(n - 1)] x [(Vart - ΣVari
)/Vart
]
Alpha = estimated reliability of the full-length test
n = number of items
Vart = variance of the whole test (standard deviation squared)
ΣVari = sum the variance for all n items
KR21
Used for dichotomously scored items that are all
about the same difficulty
Formula:
KR21 = [n/(n - 1)] x [1 - (M x (n - M) / (n x Var))]
KR21 = estimated reliability of the full-length test
n = number of items
Var = variance of the whole test (standard deviation squared)
M = mean score on the test
Limitations of KR20 and KR21
1. Single moment in time
2. Generalization across domains
3. Speededness
Lengthen test
Spearman-Brown Formula
rkk
= k(r11) / [1 + (k - 1)r11]
rkk = reliability of the test k times as long as the original test
r11 = reliability of original test
k = factor by which the length of the test is changed
Lengthen test
Example using Spearman-Brown Formula:
A test is made up of 10 items and has a reliability of
.67. Will reliability improve if the number of items is
doubled, assuming new items are just like the existing
ones?
k = 20/10 = 2
rkk = 2(.67)/[1 + (2 - 1).67] = 1.34/1.67 = .80
Hoping this will be helpful,
Rafik
Dear James,
Please read the following text. For more details, see attached file.
KR20
Dichotomously scored items with a range of difficulty:
Multiple choice
Short answer
Fill in the blank
Formula:
KR20 = [n/(n - 1)] x [1 - (Σpq)/Var]
KR20 = estimated reliability of the full-length test
n = number of items
Var = variance of the whole test (standard deviation squared)
Σpq = sum the product of pq for all n items
p = proportion of people passing the item
q = proportion of people failing the item (or 1-p)
Coefficient Alpha
Items that have more than dichotomous, right-wrong
scores:
Likert scale (e.g rate 1 to 5)
Short answer
Partial credit
Formula:
Alpha = [n/(n - 1)] x [(Vart - ΣVari
)/Vart
]
Alpha = estimated reliability of the full-length test
n = number of items
Vart = variance of the whole test (standard deviation squared)
ΣVari = sum the variance for all n items
KR21
Used for dichotomously scored items that are all
about the same difficulty
Formula:
KR21 = [n/(n - 1)] x [1 - (M x (n - M) / (n x Var))]
KR21 = estimated reliability of the full-length test
n = number of items
Var = variance of the whole test (standard deviation squared)
M = mean score on the test
Limitations of KR20 and KR21
1. Single moment in time
2. Generalization across domains
3. Speededness
Lengthen test
Spearman-Brown Formula
rkk
= k(r11) / [1 + (k - 1)r11]
rkk = reliability of the test k times as long as the original test
r11 = reliability of original test
k = factor by which the length of the test is changed
Lengthen test
Example using Spearman-Brown Formula:
A test is made up of 10 items and has a reliability of
.67. Will reliability improve if the number of items is
doubled, assuming new items are just like the existing
ones?
k = 20/10 = 2
rkk = 2(.67)/[1 + (2 - 1).67] = 1.34/1.67 = .80
Hoping this will be helpful,
Rafik
Hi Rafik, the literature assisted me a lot. I managed to compile my paper. Thanks.
Dear Sifuna,
You can use the KR-20, KR-21 and Cronbach Alfa reliability coefficients when all of the following conditions are met: Data should be parallel, equivalent or uniform measurements, not congeneric; The test must be one dimensional, The test should not be speed test.
There are small distinctions as to how each one is used, but I will not go into details as previous respondents have already explained them.
Good Luck
Usually for a test you use KR20 in a dichotomously scored test items, KR21 is also applied when estimating reliability of a test. However, Alpha is more appropriate for a likert type survey. So for test reliability please use KR20
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