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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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