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P x x∈X Mean E(X) = n Standard Deviation: 1 n . P x∈X (x − x)2 = E(X 2 ) − E(X)2 . S1 = {70, 65, 68, 69, 70}, S2 = {66, 69, 65, 66, 64}. Sample Mean 1 n Pn i=1 xi 70 + 65 + 68 + 69 + 70 = 68.4. 5 66 + 69 + 65 + 66 + 64 x2 = = 66.0. 5 x1 = Sample Standard Deviation n s21 = 1 X = (xi − x)2 . n − 1 i=1 (1.6)2 + (−3.4)2 + (−0.4)2 + (0.6)2 + (1.6)2 2.56 + 11.56 + 0.16 + 0.36 + 2.56 s21 = 17.20 Sample Variance s21 = 17.2, sample standard deviation s1 = √ 17.2 ≈ 4.15. s22 = 02 + 32 + (−1)2 + 02 + (−2)2 = 14. √ Sample Variance s22 = 14, √ s2√= 14 ≈ 3.74. Mean s1 / n, 3.44 ≈ 1.85 √ √ variance s2 / n = 2.8 ≈ 1.67. Are Sample means different? x − x2 2.4 2.4 2.4 q 12 ≈ 0.96. ≈ =√ =√ 2 s1 s2 2.498 3.44 + 2.8 6.24 + n1 n2 No, the sample means are less than two standard deviations apart. Therefore the means do not differ in a statistically significant way. 1