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## Designing a placement test

### Using statistics to calculate the MEAN, STANDARD DEVIATION & RELIABILITY

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 S X = 256 609.31 The Mean is the sum total of the students' scores divides by N (the number of students or scripts) S X / 24 = 10.667 Standard Deviation: (s) = the square root of [ S x ² the sum of the individual deviation from the Mean squared divided by (Number of scripts - 1)] S x squared = 609.31 / 23 = 26.49. The square Root of 26.49 is 5.147 S x ² = 609.31 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 x 3 = 9

S X the sum of all the raw scores = 256

M = the MEAN (a measure of central tendency): M = S X / N i.e. 256/24=10.667.

x = the individual deviation from the MEAN = (Mean minus each student's raw score)

x² =individual differentiation from the mean multiplied by itself in each case

S x² = the sum of the above (individual differentiations from the mean²) = 609.31

 S (standard deviation) = The square root of [ S 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 minus [M the Mean multiplied by (n the number of items minus the Mean) divided by (n the number of items multiplied by the standard deviation squared) ].

 R (reliability) = 1 minus [10.667 multiplied by (28 - 10.667) divided by (28 x 26.49)] = 1 minus 0.249 = 0.751

r (reliability) = 0.751

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 = S X / N (606/52) = 11.654 ( sum all the scores and divide by the no. 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.

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