WZORY

Liczba klas $\mathbf{k} \approx \sqrt{n}$

Różnica R ≈ x_max − x_min

Długość klasy ∝ = …

Średnia arytmetyczna $\overset{\overline{}}{\mathbf{x}} = \ \frac{1}{n}\sum_{i = 1}^{k}{x_{i}n_{i}}$

Mediana PUNKTOWY $\mathbf{m}_{\mathbf{e}} = \left\{ \begin{matrix} x_{\left( \frac{n + 1}{2} \right)} \\ \frac{x_{\left( \frac{n}{2} \right)} + x_{\left( \frac{n}{2} + 1 \right)}}{2} \\ \end{matrix} \right.\ $

KLASY $\mathbf{m}_{\mathbf{e}} = \ x_{i} + \frac{b}{n_{m}}\left( \frac{n}{2} - \sum_{i = 1}^{m - 1}n_{i} \right)$

Moda $\mathbf{m}_{\mathbf{o}} = \ x_{i} + b\frac{n_{m} - n_{m - 1}}{\left( n_{m} - n_{m - 1} \right) + \left( n_{m} - n_{m + 1} \right)}$

Wariacje $\mathbf{S}^{\mathbf{2}} = \ \frac{1}{n}\sum_{i = 1}^{k}{\left( x_{i} - \overset{\overline{}}{x} \right)^{2}n_{i}}$

Odchylenie standardowe $\mathbf{S} = \sqrt{S^{2}}$

Odchylenie przeciętne od średniej $\mathbf{d}_{\mathbf{1}} = \frac{1}{n}\sum_{i = 1}^{k}\left| x_{i} - \overset{\overline{}}{x} \right|n_{i}$

Odchylenie przeciętne od mediany $\mathbf{d}_{\mathbf{2}} = \frac{1}{n}\sum_{i = 1}^{k}\left| x_{i} - m_{e} \right|n_{i}$

Współczynnik asymetrii $\mathbf{g} = \ \frac{M_{3}}{S^{3}}$ $\mathbf{M}_{\mathbf{3}} = \ \frac{1}{n}\sum_{n}^{k}\left( x_{i} - \overset{\overline{}}{x} \right)^{3}n_{i}$

Współczynnik koncentracji $\mathbf{K} = \ \frac{M_{4}}{S^{4}}$ $\mathbf{M}_{\mathbf{4}} = \ \frac{1}{n}\sum_{n}^{k}\left( x_{i} - \overset{\overline{}}{x} \right)^{4}n_{i}$

Współczynnik nierównomierności $\mathbf{H} = \ \frac{d_{1}}{\overset{\overline{}}{x}}*100\%$

Współczynnik zmienności $\mathbf{V} = \frac{S}{\overset{\overline{}}{x}}*100\%$

Moment główny rzędu L $\mathbf{m}_{\mathbf{L}} = \ \frac{1}{n}\sum_{i = 1}^{k}{x_{i}}^{L}n_{i}$

Moment centralny rzędu L $\mathbf{M}_{\mathbf{L}} = \ \frac{1}{n}\sum_{i = 1}^{k}\left( x_{i} - \overset{\overline{}}{x} \right)^{L}n_{i}$

Kwantyl górny $\mathbf{Q}_{\mathbf{1}} = \ \frac{x_{\left( \frac{n}{4} \right)} + x_{\left( \frac{n}{4} + 1 \right)}}{2}$

Kwantyl dolny $\mathbf{Q}_{\mathbf{3}} = \ \frac{x_{\left( \frac{n}{4} \right)} + x_{\left( \frac{n}{4} + 1 \right)}}{2}$

Odchylenie ćwiartkowe $\mathbf{Q} = \ \frac{Q_{1} + Q_{3}}{2}$

Współczynnik korelacji $\mathbf{r}_{\mathbf{\text{xy}}} = \frac{\text{cov\ }\left( x,y \right)}{S_{x}*S_{y}} = \ \frac{\frac{1}{n}\sum_{i = 1}^{n}{\left( x_{i} - \overset{\overline{}}{x} \right)\left( y_{i} - \overset{\overline{}}{y} \right)}}{\sqrt{\frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overset{\overline{}}{x} \right)^{2}}\ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}\left( y_{i} - \overset{\overline{}}{y} \right)^{2}}} = \ \frac{\sum_{i = 1}^{n}{x_{i}y_{i} - n\overset{\overline{}}{x}*\overset{\overline{}}{y}}}{\sqrt{\sum_{i = 1}^{n}{x_{i}}^{2} - n{\overset{\overline{}}{x}}^{2}}\ \sqrt{\sum_{i = 1}^{n}{{y_{i}}^{2} - n{\overset{\overline{}}{y}}^{2}}}}$

$\mathbf{\text{cov}}\left( \mathbf{x,y} \right) = \ \frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overset{\overline{}}{x} \right)\left( y_{i} - \overset{\overline{}}{y} \right) = \frac{1}{n}\sum_{i = 1}^{n}{x_{i}y_{i} - \overset{\overline{}}{x}\overset{\overline{}}{y}}$

$\mathbf{S}_{\mathbf{x}} = \ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overset{\overline{}}{x} \right)^{2}} = \ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{x_{i}}^{2} - {\overset{\overline{}}{x}}^{2}} = \ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}\left( x - 2x_{i}*\overset{\overline{}}{x} + {\overset{\overline{}}{x}}^{2} \right)} = \ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{x_{i}}^{2} - 2\overset{\overline{}}{x}*\frac{1}{n}\sum_{i = 1}^{n}{x_{i} + \frac{{\overset{\overline{}}{x}}^{2}}{n}\sum_{i = 1}^{n}}} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{{x_{i}}^{2} - 2{\overset{\overline{}}{x}}^{2} + {\overset{\overline{}}{x}}^{2}}}$

$\mathbf{S}_{\mathbf{y}} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}\left( y_{i} - \overset{\overline{}}{y} \right)^{2}} = \ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}y^{2} - {\overset{\overline{}}{y}}^{2}}$

Współczynnik korelacji

cechy X,Y

$\mathbf{r}_{\mathbf{\text{xy}}} = \ \frac{\frac{1}{n}\sum_{i = 1}^{m}{\sum_{j = 1}^{l}{{\overset{\overline{}}{x}}_{i}{\overset{\overline{}}{y}}_{i}}}{n_{i}}_{j} - \overset{\overline{}}{x}\overset{\overline{}}{y}}{\sqrt{\frac{1}{n}\sum_{i = 1}^{m}{{\overset{\overline{}}{x}}_{i}}^{2}*n_{i} - {\overset{\overline{}}{x}}^{2}}\text{\ \ }\sqrt{\frac{1}{n}\sum_{j = 1}^{l}{{\overset{\overline{}}{y}}_{i}}^{2}*y_{i} - {\overset{\overline{}}{y}}^{2}}\ } = \ \frac{\frac{1}{n}\sum_{i = 1}^{m}{x_{i}\sum_{j = 1}^{l}{\overset{\overline{}}{y_{i}}n_{i} - \overset{\overline{}}{X}\overset{\overline{}}{Y}}}}{\sqrt{\frac{1}{n}\sum_{i = 1}^{m}{{{\overset{\overline{}}{x}}_{i}}^{2}*n_{i} - {\overset{\overline{}}{x}}^{2}}}\ \sqrt{\frac{1}{n}\sum_{j = 1}^{l}{{{\overset{\overline{}}{y}}_{i}}^{2}*n_{\text{.j}} - {\overset{\overline{}}{y}}^{2}}}} = \ \frac{\frac{1}{n}\sum_{j = 1}^{l}{\overset{\overline{}}{y_{i}}\sum_{i = 1}^{m}{\overset{\overline{}}{x_{i}}n_{\text{.j}} - \overset{\overline{}}{X}\overset{\overline{}}{Y}}}}{\sqrt{\frac{1}{n}\sum_{i = 1}^{m}{{\overset{\overline{}}{x}}_{i}}^{2}n_{i} - {\overset{\overline{}}{x}}^{2}}\ \sqrt{\frac{1}{n}\sum_{j = 1}^{l}{{\overset{\overline{}}{y}}_{i}}^{2}n_{\text{.j}} - {\overset{\overline{}}{y}}^{2}}}$

$\overset{\overline{}}{\mathbf{x}} = \ \frac{1}{n}\sum_{i = 1}^{m}{\overset{\overline{}}{x_{i}}n_{i}}$

$\overset{\overline{}}{\mathbf{y}} = \ \frac{1}{n}\sum_{j = 1}^{l}{\overset{\overline{}}{y_{i}}n_{\text{.j}}}$

Współczynnik korelacji Spearmana $\mathbf{r}_{\mathbf{\text{xy}}} = \frac{\text{cov\ }\left( RX,RY \right)}{\sqrt{\text{Var}\left( \text{RX} \right)}\ \sqrt{\text{Var\ }\left( \text{RY} \right)}} = \ \frac{\frac{1}{n}\sum_{i = 1}^{m}\left( \text{RX}_{i} - \frac{n + 1}{2} \right)\left( \text{RY}_{i} - \frac{n + 1}{2} \right)}{\sqrt{\frac{n^{2} - 1}{12}}\ *\ \sqrt{\frac{n^{2} - 1}{12}}}$

Regresja liniowa Y= aX + b

$\left\{ \begin{matrix} \mathbf{a} = \frac{\frac{1}{n}\sum_{i = 1}^{n}{x_{i}y_{i} - \overset{\overline{}}{X}\overset{\overline{}}{Y}}}{\frac{1}{n}\sum_{i = 1}^{n}{{x_{i}}^{2} - \left( \overset{\overline{}}{x} \right)^{2}}} = \ r_{\text{xy}}\frac{S_{y}}{S_{x}} \\ \mathbf{b} = \ \overset{\overline{}}{Y} - a\overset{\overline{}}{X}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ $

X= a’Y + b’

$\left\{ \begin{matrix} \mathbf{a'} = \frac{\frac{1}{n}\sum_{i = 1}^{n}{x_{i}y_{i} - \overset{\overline{}}{X}\overset{\overline{}}{Y}}}{\frac{1}{n}\sum_{i = 1}^{n}{{y_{i}}^{2} - \left( \overset{\overline{}}{y} \right)^{2}}} = \ r_{\text{xy}}\frac{S_{x}}{S_{y}} \\ \mathbf{b'} = \ \overset{\overline{}}{X} - a'\overset{\overline{}}{Y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ $

$\mathbf{S}_{\mathbf{x}} = \ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}\left( x_{i} - \overset{\overline{}}{x} \right)^{2}} = \ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{x_{i}}^{2} - {\overset{\overline{}}{x}}^{2}}$

$\mathbf{S}_{\mathbf{y}} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}\left( y_{i} - \overset{\overline{}}{y} \right)^{2}} = \ \sqrt{\frac{1}{n}\sum_{i = 1}^{n}y^{2} - {\overset{\overline{}}{y}}^{2}}$

RANGI:

• Własności rang regularnych $\mathbf{\text{ERX}}_{\mathbf{i}} = ER\overset{\overline{}}{X} = \frac{1}{n}\sum_{i = 1}^{n}{\text{RX}_{i} = \frac{1}{n}\sum_{i = 1}^{n}i = \frac{1}{n}\left( 1 + 2 + \ldots + n \right) = \frac{1}{n}\frac{n + 1}{2}*n = \frac{n + 1}{2}}$

• Wariacja $\mathbf{\text{Var}}\left( \mathbf{\text{RX}} \right) = \frac{1}{n}\sum_{i = 1}^{n}{\left( \text{RX}_{i} - R\overset{\overline{}}{X} \right)^{2} = \frac{1}{n}\sum_{i = 1}^{n}{\left( i - \frac{n + 1}{2} \right)^{2} = \frac{1}{n}\left( \sum_{i = 1}^{n}{i^{2} - 2\frac{n + 1}{2}*\sum_{i = 1}^{n}{i + n\left( \frac{n + 1}{2} \right)^{2}}} \right)}} = \ \frac{1}{n}\left( \frac{1}{6}n\left( n + 1 \right)\left( 2n + 1 \right) - \left( n + 1 \right)\frac{n + 1}{2}*n + n\frac{\left( n + 1 \right)^{2}}{4} \right) = \frac{1}{6}\left( n + 1 \right)\left( 2n + 1 \right) - \frac{\left( n + 1 \right)^{2}}{2} + \frac{\left( n + 1 \right)^{2}}{2} = \ \frac{1}{6}\left( n + 1 \right)\left( 2n + 1 \right) - \frac{\left( n + 1 \right)^{2}}{2} = \ \left( n + 1 \right)\left( \frac{2n + 1}{6} - \frac{n + 1}{4} \right) = \ \left( n + 1 \right)\frac{4n + 2 - 3n - 3}{12} = \frac{\left( n + 1 \right)\left( n - 1 \right)}{12} = \frac{n^{2} - 1}{12}$