Semester 3MathematicsApplied StatisticsFundamental FormulasExpected Value (Mean) xˉ=∑i=1nxin(Sample),μ=∑i=1NxiN(Population)\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \quad (\text{Sample}), \qquad \mu = \frac{\sum_{i=1}^{N} x_i}{N} \quad (\text{Population})xˉ=n∑i=1nxi(Sample),μ=N∑i=1Nxi(Population) Variance σ2=∑i=1N(xi−μ)2N(Population)\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} \quad (\text{Population})σ2=N∑i=1N(xi−μ)2(Population) s2=∑i=1n(xi−xˉ)2n−1(Sample — uses n−1 to correct bias)s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1} \quad (\text{Sample — uses } n-1 \text{ to correct bias})s2=n−1∑i=1n(xi−xˉ)2(Sample — uses n−1 to correct bias) Z Score z=x−μσ(Population),z=x−xˉs(Sample)z = \frac{x - \mu}{\sigma} \quad (\text{Population}), \qquad z = \frac{x - \bar{x}}{s} \quad (\text{Sample})z=σx−μ(Population),z=sx−xˉ(Sample) Covariance sxy=∑i=1n(xi−xˉ)(yi−yˉ)n−1s_{xy} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{n - 1}sxy=n−1∑i=1n(xi−xˉ)(yi−yˉ) Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y)Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y) Correlation r=sxysxsy=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2∑(yi−yˉ)2r = \frac{s_{xy}}{s_x s_y} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}r=sxsysxy=∑(xi−xˉ)2∑(yi−yˉ)2∑(xi−xˉ)(yi−yˉ) Corr(X,Y)=Cov(X,Y)Var(X)⋅Var(Y)\text{Corr}(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}Corr(X,Y)=Var(X)⋅Var(Y)Cov(X,Y)Confidence IntervalPrevious PageFundementals of StatisticsNext Page