# What is the standard deviations of {1,3,5,5,7,9,10,11,13,15,15,17,17,17}?

Nov 12, 2015

Assuming you mean population standard deviation, then $\sigma = 5.3132$

#### Explanation:

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 ${\left({X}_{i} - \mu\right)}^{2}$:

${\left({X}_{1} - \mu\right)}^{2} = {\left(1 - 10.357142857143\right)}^{2} = - 9.35714285714292 = 87.55612244898$

(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

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)

$\sigma = 5.3132$