Calculate Mean (Average) denoted as μ
mu = frac(Sigma_n)(n)
mu = frac(1 + 3 + 5 + 5 + 7 + 9 + 10 + 11 + 13 + 15 + 15 + 17 + 17 + 17)(14)
mu = frac(145)(14)
mu = 10.357142857143
Let's evaluate the square difference from the mean of each term (X_i - mu)^2:
(X_1 - mu)^2 = (1 - 10.357142857143)^2 = -9.35714285714292 = 87.55612244898
(X_2 - μ)^2 = (3 - 10.357142857143)^2 = -7.3571428571429^2 = 54.127551020408
(X_3 - μ)^2 = (5 - 10.357142857143)^2 = -5.3571428571429^2 = 28.698979591837
(X_4 - μ)^2 = (5 - 10.357142857143)^2 = -5.3571428571429^2 = 28.698979591837
(X_5 - μ)^2 = (7 - 10.357142857143)^2 = -3.3571428571429^2 = 11.270408163265
(X_6 - μ)^2 = (9 - 10.357142857143)^2 = -1.3571428571429^2 = 1.8418367346939
(X_7 - μ)^2 = (10 - 10.357142857143)^2 = -0.35714285714286^2 = 0.12755102040816
(X_8 - μ)^2 = (11 - 10.357142857143)^2 = 0.64285714285714^2 = 0.41326530612245
(X_9 - μ)^2 = (13 - 10.357142857143)^2 = 2.6428571428571^2 = 6.984693877551
(X_10 - μ)^2 = (15 - 10.357142857143)^2 = 4.6428571428571^2 = 21.55612244898
(X_11 - μ)^2 = (15 - 10.357142857143)^2 = 4.6428571428571^2 = 21.55612244898
(X_12 - μ)^2 = (17 - 10.357142857143)^2 = 6.6428571428571^2 = 44.127551020408
(X_13 - μ)^2 = (17 - 10.357142857143)^2 = 6.6428571428571^2 = 44.127551020408
(X_14 - μ)^2 = (17 - 10.357142857143)^2 = 6.6428571428571^2 = 44.127551020408
Add up the terms:
ΣE(X_i - μ)^2 = 87.55612244898 + 54.127551020408 + 28.698979591837 + 28.698979591837 + 11.270408163265 + 1.8418367346939 + 0.12755102040816 + 0.41326530612245 + 6.984693877551 + 21.55612244898 + 21.55612244898 + 44.127551020408 + 44.127551020408 + 44.127551020408
ΣE(X_i - μ)2 = 395.21428571429
Calculate Variance
σ^2 = frac(ΣE(X_i - μ)^2)(n)
σ^2 = frac(395.21428571429)(14)
σ^2 = 28.229591836735
Calculate Standard Deviation:
σ = sqrt(σ^2) = sqrt(28.229591836735)
sigma = 5.3132
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