## 计算标准偏差

#### 目录

 ◦这个计算器怎么用？ ◦什么是标准差公式？ ◦人口标准差 ◦样本标准差 ◦未校正样本标准偏差 ◦修正样本标准偏差 ◦无偏样本标准差 ◦标准差的应用 ◦参考

“SD”代表标准差，是使用最广泛的缩写。

## 未校正样本标准偏差

{x₁, x₂, x₃, ..., xₙ} = values of the sample items
x̄ = mean value of values
N = size of the sample (the square root of the variance)

## 无偏样本标准差

unbiased estimator for the normal distribution = s/c₄

Refined approximation = N - 1.5 + 1 / (8(N - 1))

Y₂ = excess kurtosis

kurtosis: a₄ = m₄ / m₂²
excess kurtosis: g₂ = a₄ - 3
m = m₄ = ∑(x−x̅)⁴
m₂ = ∑(x−x̅)² / N

## 参考

David, H. A., et al. “The Distribution of the Ratio, in a Single Normal Sample, of Range to Standard Deviation.” Biometrika, vol. 41, no. 3/4, [Oxford University Press, Biometrika Trust], 1954, pp. 482–93, https://doi.org/10.2307/2332728.
Delmas, R. and Liu, Y., 2005. Exploring students’ conceptions of the standard deviation. Statistics Education Research Journal, 4(1), pp.55-82.
Premaratne, G. and Bera, A.K., 2000. Modeling asymmetry and excess kurtosis in stock return data. Illinois Research & Reference Working Paper No. 00-123.
Weisstein, Eric W. "Chi Distribution." From MathWorld--A Wolfram Web Resource, https://mathworld.wolfram.com/ChiDistribution.html
Measures of Shape: Skewness and Kurtosis, Stan Brown, https://brownmath.com/stat/shape.htm

John Cruz

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