TESTING THE TEST 2
| X Scores out of 28 |
x = individual deviation from mean | X squared | x squared |
|---|---|---|---|
| 3 | x = 10.667 - 3 x = 7.67 | 9 | 58.78 |
| 5 | x = 10.667 - 5 x = 5.67 | 25 | 32.11 |
| 5 | x = 10.667 - 5 x = 5.67 | 25 | 32.11 |
| 5 | x = 10.667 - 5 x = 5.67 | 25 | 32.11 |
| 6 | x = 10.667 - 6 x = 4.67 | 36 | 21.78 |
| 6 | x = 10.667 - 6 x = 4.67 | 36 | 21.78 |
| 6 | x = 10.667 - 6 x = 4.67 | 36 | 21.78 |
| 7 | x = 10.667 - 7 x = 3.67 | 49 | 13.45 |
| 7 | x = 10.667 - 7 x = 3.67 | 49 | 13.45 |
| 8 | x = 10.667 - 8 x = 2.67 | 64 | 7.11 |
| 9 | x = 10.667 - 9 x = 1.67 | 81 | 2.78 |
| 10 | x = 10.667 - 10 x = 0.67 | 100 | 0.44 |
| 11 | x = 10.667 - 11 x = -0.33 | 121 | 0.11 |
| 11 | x = 10.667 - 11 x = -0.33 | 121 | 0.11 |
| 12 | x = 10.667 - 12 x = -1.33 | 144 | 1.78 |
| 12 | x = 10.667 - 12 x = -1.33 | 144 | 1.78 |
| 13 | x = 10.667 - 13 x = -2.33 | 169 | 5.44 |
| 14 | x = 10.667 - 14 x = -3.33 | 196 | 11.11 |
| 15 | x = 10.667 - 15 x = -4.33 | 225 | 18.77 |
| 16 | x = 10.667 - 16 x = -5.33 | 256 | 28.44 |
| 16 | x = 10.667 - 16 x = -5.33 | 256 | 28.44 |
| 17 | x = 10.667 - 17 x = -6.33 | 289 | 40.11 |
| 20 | x = 10.667 - 20 x = -9.33 | 400 | 87.10 |
| 22 | x = 10.667 - 22 x = -11.33 | 484 | 128.44 |
| Σ X = 256 | 609.31 | ||
|
The Mean is the sum total of the students' scores divided by N
(the number of students or scripts) Σ X / 24 = 10.667 |
Standard Deviation: (s) = the square root of [Σ x², the sum of the individual deviation from the mean squared, divided by (number of scripts − 1)] |
Σ x² = 609.31 / 23 = 26.49. The square root of 26.49 is 5.147. s = 5.147 |
|
Data needed to check the MEAN, RELIABILITY & STANDARD DEVIATION
N = 24 Swedish Pensioners tested together or the number of scripts sampled.
n = the number of items in the test = 28
X = raw scores 3, 5, 5, 5, 6, 6, 6, 7, 7, 8, 9, 10, 11, 11, 12, 12, 13, 14, 15, 16, 16, 17, 20, 22.
X² is the student's raw score multiplied by itself, e.g. 3 × 3 = 9.
Σ X = the sum of all the raw scores = 256
M = the MEAN (a measure of central tendency): M = Σ X / N, i.e. 256/24 = 10.667.
x = the individual deviation from the MEAN = (Mean minus each student's raw score).
x² = individual deviation from the mean multiplied by itself in each case.
Σ x² = the sum of the above (individual deviations from the mean squared) = 609.31.
| S (standard deviation) | = | The square root of [ Σ x² divided by (N-1)] |
|---|---|---|
| The square root of [609.31 / 24-1] = the square root of 26.49 = 5.147 |
s (standard deviation) = 5.147
To calculate the reliability of a test, use the Kuder-Richardson formula:
| R (reliability) | = | 1 − [10.667 × (28 − 10.667) / (28 × 26.49)] |
|---|---|---|
| 1 − 0.249 = 0.751 |
r (reliability) = 0.751
Results from a larger sample
I administered the same test on a younger sample of 52 students of mixed nationality with the following results:
N = 52 (number of students or scripts)
n = 28 (number of items in the test)
M = Σ X / N (606/52) = 11.654 (sum all the scores and divide by the number of scripts)
s (standard deviation) = 5.5338
r (reliability) = 0.778
Note: The reliability based on my larger sample of younger students of mixed nationality was somewhat higher than the reliability based on my smaller sample of elderly Swedish learners.
A scientific pocket calculator or computer statistics software will allow you to calculate standard deviations and reliability a lot more quickly.
Before celebrating the fact that the reliability of the test is 78% (i.e. greater than 75%), the test's validity also needs to be investigated.