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Descriptive statistics, variance, correlation & regression.

descriptive statistics · correlation · regression

Descriptive statistics

Quartiles & outliers. \(\mathrm{IQR}=Q_3-Q_1\). Outlier: \(x\lt Q_1-1.5\,\mathrm{IQR}\) or \(x\gt Q_3+1.5\,\mathrm{IQR}\).

Mean. \(\bar x=\frac1n\sum x_i\); grouped estimate \(\bar x=\frac{\sum f_ix_i}{\sum f_i}\) using midpoints.

Variance. Population: \(\sigma^2=\frac1n\sum(x_i-\bar x)^2=\frac{\sum x_i^2}{n}-\bar x^2\); grouped \(\sigma^2=\frac{\sum f_ix_i^2}{\sum f_i}-\bar x^2\). Sample: \(s^2=\frac1{n-1}\sum(x_i-\bar x)^2\). Standard deviation \(=\sqrt{\text{variance}}\).

Linear transforms. \(Y=aX+b\): \(\bar y=a\bar x+b\), \(\sigma_Y=|a|\sigma_X\), \(\sigma_Y^2=a^2\sigma_X^2\).

Correlation

Sums of squares. \(S_{xx}=\sum(x_i-\bar x)^2\), \(S_{yy}=\sum(y_i-\bar y)^2\), \(S_{xy}=\sum(x_i-\bar x)(y_i-\bar y)\).

Pearson. \(r=\dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\), \(-1\le r\le1\).

Regression

Regression y on x. \(y-\bar y=b(x-\bar x)\), \(b=\dfrac{S_{xy}}{S_{xx}}=r\dfrac{s_y}{s_x}\), intercept \(a=\bar y-b\bar x\), so \(y=a+bx\).

Regression x on y. \(x-\bar x=b'(y-\bar y)\), \(b'=S_{xy}/S_{yy}=r(s_x/s_y)\).

Caveats. Prediction only inside data range; correlation \(\ne\) causation.