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 (N1)] 


The square root of [ 609. 31 / 241] = The square root of 26.49 = 5.147 
s (standard deviation) = 5.147
To calculate the reliability of a test, use the KuderRichardson 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 s² 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.