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Effect Size Formula

Effect Size Formula is used to compare two given observations. It can be used to generate various results from the comparison of two set of information. It is also used to forecast and predict many possibilities by comparing them. First we calculate the mean of both the observations and then subtract second from the first. We calculate the standard deviations for both the observations and find their squares. By plugging all the value in the following formula, we get Cohen's index or Cohen's value:
Effect Size Formula
From this Cohen's index, we are able to find effect-size coefficient with the help of following formula:
Formula for Effect Size

Where,
d = Cohen's index
M1 = Mean of first observation.
M2 = Mean of second observation.
S1 = Standard deviation of first observation.
S2 = Standard deviation of second observation.
r = Effect-size coefficient.

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Effect Size Problems

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The few problems based on effect size formula are as follows:

Solved Examples

Question 1: There are two set of observations. 10, 20, 15, 30 and 2, 4, 3, 8, 1, 5. Calculate the effect size coefficient ?

Solution:
 
Calculate mean of both observations.
Mean is given by the following formula:
$\bar{x}$ = $\frac{\sum x}{n}$

Mean for first observation:
$M_{1}$ = $\frac{10+20+15+30}{4}$

               = 18.75
Mean for second observation:
$M_{2}$ = $\frac{2+4+3+8+1+5}{6}$

                = 3.83
Calculate standard deviation for both observations
Standard deviation is given by the following formula:

S = $\sqrt{\frac{\sum (x-\bar{x})^{2}}{n}}$

Standard deviation for first observation:

          x           
        $x-\bar{x}$        $(x-\bar{x})^{2}$
10 -8.75
76.56
20
 1.25 1.56
15
 -3.75 14.06
30
 11.25 126.56
         $\sum (x-\bar{x})^{2}$ = 218.74    

S1 = $\sqrt{\frac{218.74}{4}}$

      = 7.39

Standard deviation for second observation:

               x                                    $x-\bar{x}$                     $(x-\bar{x})^{2}$
2
-1.83
3.35
4
0.17
0.03
3
-0.83
0.69
8
4.17
17.39
1
-2.83
8.01
5
1.17
1.37
        $\sum (x-\bar{x})^{2}$ = 30.84       

S2 = $\sqrt{\frac{30.84}{6}}$

= 2.27

d = $\frac{M_{1}-M_{2}}{\sqrt{\frac{S_{1}^{2}+S_{2}^{2}}{2}}}$

d = $\frac{18.75-3.83}{\sqrt{\frac{7.39^{2}+2.27^{2}}{2}}}$

= 2.73

r = $\frac{d}{\sqrt{d^{2}+4}}$

r = $\frac{2.73}{\sqrt{2.73^{2}+4}}$

r = 0.8066 = 0.81 (approx)

 

Question 2: Calculate the effect-size coefficient for two sets of data given below:
100, 200, 300, 400 and 82, 78, 60 ?

Solution:
 
Calculate mean for both sets of data.
Formula of mean is given by:
$\bar{x}$ = $\frac{\sum x}{n}$

Mean for first set of data:
M1 = $\frac{100+200+300+400}{4}$

= 250

Mean for second set of data:
M= $\frac{82+78+60}{3}$

Calculate standard deviation for both sets of data.
Formula of standard deviation is given by:

S = $\sqrt{\frac{\sum (x-\bar{x})^{2}}{n}}$

Standard deviation for first set of data:

          x                  $x-\bar{x}$       
$(x-\bar{x})^{2}$
  100   
-150
22500
 200 
-50
2500
300
50
2500
400
150
22500
             $\sum (x-\bar{x})^{2}$ = 50000     

S1 = $\sqrt{\frac{50000}{4}}$

S1 = 111.80

Standard deviation of second set of data:

        x      
        $x-\bar{x}$                         $(x-\bar{x})^{2}$                
      82
             8.67
                        75.17
      78
             4.67                         21.81
      60
          -13.33
                      177.69
         $\sum (x-\bar{x})^{2}$ = 274.67       

S2 = $\sqrt{\frac{274.67}{3}}$

S2 = 9.57

d = $\frac{M_{1}-M_{2}}{\sqrt{\frac{S_{1}^{2}+S_{2}^{2}}{2}}}$

d = $\frac{250-73.33}{\sqrt{\frac{111.80^{2}+9.57^{2}}{2}}}$

d = 2.23

r = $\frac{d}{\sqrt{d^{2}+4}}$

r = $\frac{2.23}{\sqrt{2.23^{2}+4}}$

r = 0.74

 

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