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Stat 700 Handout
Computing the Mean and Variance of Sample Data
Data Set for Illustration: Scores in an examination after a training method. The scores for training
method A are as follows:
71, 75, 65, 69, 73, 66, 68, 71, 74, 68
Xi
71
75
65
69
73
66
68
71
74
68
Sum of X's = 700
Xi - Mean
1
5
-5
-1
3
-4
-2
1
4
-2
Sum of Deviations = 0
(Xi - Mean)2
1
25
25
1
9
16
4
1
16
4
Sum of Squared
Deviations = 102
(Xi)2
5041
5625
4225
4761
5329
4356
4624
5041
5476
4624
Sum of the Squared
X's = 49102
Sample Size = n = 10
Sample Mean = Sum/n
= 700/10
= 70
Using Definitional Formula for Variance
Sample Variance = S2
= (Sum of Squared Deviations)/(n - 1)
= 102/(10 - 1) = 11.33333333
Using Machine or Computational Formula for Variance
Sample Variance = S2
= [(Sum of Squared X's) - (Sum of X's)2/n]/(n - 1)
= [49102 - (700)2/10]/(10 - 1)
= [49102 - 490000/10]/9
= [49102 - 49000]/9
= 102/9
= 11.333333
Sample Standard Deviation = S
= +SquareRoot of Variance
= (11.333333)1/2
= 3.366501