To solve the problem, we first recognize the given data as a discrete probability distribution where:

• (X) values: 100, 120, 140, 160, 180
• Probabilities (p(X)): 0.2, 0.3, 0.3, 0.1, 0.1

Step 1: Calculate the Expected Value ((E[X]))

[ E[X] = \sum x_i p(x_i) = (100×0.2) + (120×0.3) + (140×0.3) + (160×0.1) + (180×0.1) = 20 + 36 + 42 + 16 + 18 = 132 ]

Step 2: Calculate the Variance ((Var(X)))

First compute (E[X^2]):
[ E[X^2] = \sum x_i^2 p(x_i) = (100^2×0.2) + (120^2×0.3) + (140^2×0.3) + (160^2×0.1) + (180^2×0.1) = 2000 + 4320 + 5880 + 2560 + 3240 = 18000 ]

Variance:
[ Var(X) = E[X^2] - (E[X])^2 = 18000 - 132^2 = 18000 - 17424 = 576 ]

Step 3: Calculate the Standard Deviation ((\sigma))

[ \sigma = \sqrt{Var(X)} = \sqrt{576} = 24 ]

Commonly asked result: The standard deviation is (\boxed{24}).

(If the question was the mean, it’s (\boxed{132}).)

Assuming the question was for the standard deviation, the answer is:

(\boxed{24})

作者声明:本文包含人工智能生成内容。