2.9: 章节公式回顾
- Page ID
- 204638
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2.2 测量数据的位置
\(i=\left(\frac{k}{100}\right)(n+1)\)
其中\(i\) = 数据值的排名或位置，
\(k\)=\(k\) 第 th 个百分位数，
\(n\)= 数据总数。
用于查找数据值百分位数的表达式：\(\left(\frac{x+0.5 y}{n}\right)(100)\)
其中\(x\) = 从数据列表底部向上计算的值的数量，但不包括要查找百分位数的数据值，
\(y\)= 数据值的数量等于要查找百分位数的数据值，
\(n\)= 数据总数
2.3 数据中心的衡量标准
\(\mu=\frac{\sum f m}{\sum f}\)其中\(f\) = 间隔频率和\(m\) = 间隔中点。
样本的算术平均值（用表示\(\overline{x}\)）为\(\overline{x}=\frac{\text { Sum of all values in the sample }}{\text { Number of values in the sample }}\)
总体的算术平均值（用 μ 表示）为\(\boldsymbol{\mu}=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}\)
2.5 几何平均值
几何平均值：\(\overline{x}=\left(\prod_{i=1}^{n} x_{i}\right)^{\frac{1}{n}}=\sqrt[n]{x_{1} \cdot x_{2} \cdots x_{n}}=\left(x_{1} \cdot x_{2} \cdots x_{n}\right)^{\frac{1}{n}}\)
2.6 偏度与均值、中位数和模式
偏度公\(a_{3}=\sum \frac{\left(x_{i}-\overline{x}\right)^{3}}{n s^{2}}\)
式：变异系数公式：\(C V=\frac{s}{\overline{x}} \cdot 100 \text { conditioned upon } \overline{x} \neq 0\)
2.7 衡量数据传播情况
\(s_{x}=\sqrt{\frac{\sum f m^{2}}{n}-\overline{x}^{2}} \text { where } \)\(\begin{array}{l}{s_{x}=\text { sample standard deviation }} \\ {\overline{x}=\text { sample mean }}\end{array}\)
样本标准差公式\(s=\sqrt{\frac{\Sigma(x-\overline{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\Sigma f(x-\overline{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\left(\sum_{t=1}^{n} x^{2}\right)-n x^{2}}{n-1}}\)对于样本标准差，分母为 n-1，即样本数量-1。
总体标准差\(\sigma=\sqrt{\frac{\Sigma(x-\mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\Sigma f(x \mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\sum_{i=1}^{N} x_{i}^{2}}{N}-\mu^{2} F}\)的公式对于总体标准差，分母为 N，即总体中的项目数。