Reconceptualizing language teaching: an in-service teacher education course in uzbekistan

Measures of Dispersion. Apart from the Measures of Central Tenden-
cy indicators (i.e., mean, median, and mode), we are also interested in how 
spread out the scores are from the mean. These mathematical procedures 
are called Measures of Dispersion (i.e., standard deviation).
Standard Deviation is the average distance of scores from the mean. 
The lower number you receive for standard deviation to 0, the more the 
students in the class are similar. The larger number you obtain for standard 
deviation, the less similar (i.e., more different) the students are in the class. 
The standard deviation formula is mathematically represented as follows:
In other words, there are five steps we need to take to complete the 
standard deviation (if we calculate the standard deviation by hand). Let’s 
refer to our data set from the reading quiz:

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CHAPTER THREE: LANGUAGE ASSESSMENT/TESTING
Student Number
1
2
3
4
5
6
7
8
9
10
Score
14 18
19
20
21
21
21
26
26
27
1) Find the mean. The mean is represented by X bar in the formula. We 
found the mean to be 21.3.
2) For each data point, find the square of its distance to the mean: 
Here is an example for the first data point: 14, from Student 1:
a. (14 - 21.3) = -7.3
b. (-7.3)2 = 53.29
Student 
Number
1
2
3
4
5
6
7
8
9
10
Score
14
18
19
20
21
21
21
26
26
27
Square of 
the dis-
tance from 
the mean 
(
x–x)
2
53.29 10.89 5.29 1.69 0.09 0.09 0.09 22.09 22.09 32.49
3) Sum the values:
a. 53.29 + 10.89 + 5.29 + 1.69 + 0.09 + 0.09 + 0.09 + 22.09 + 22.09 + 32.49
b. Sum = 148.10
4) Divide by the number of data points minus one.
a. 10 students took the class; 10-1 = 9
b. 148.10 divided by 9 equals 16.45
5) Take the square root. 
a. √16.45
b. 4.06
Interpreting the standard deviation: The closer the number is to 0, the 
more similar the class is; the farther away from 0 the number is, the more 
different the students in the class are. Usually, for language teachers, you 
would like your class standard deviation to be between 0.00 and 1.00. How-
ever, the standard deviation for the groups of students here is 4.06, which 
means the students are very spread out and you have various ranges of 
levels of students in your class.