Mean, Spread and Outliers

Center

Sample mean

\bar{x} = \frac{\sum x_{i}}{n}

Population mean

μ = \frac{\sum x_{i}}{N}

Weighted mean

{\bar{x}}_{w} = \frac{\sum w_{i} x_{i}}{\sum w_{i}}

Mean absolute deviation (MAD)

MAD = \frac{\sum | x_{i} - \bar{x} |}{n}

Sample variance

s^{2} = \frac{\sum (x_{i} - \bar{x})^{2}}{n - 1}

Sample standard deviation

s = \sqrt{\frac{\sum (x_{i} - \bar{x})^{2}}{n - 1}}

Population variance

σ^{2} = \frac{\sum (x_{i} - μ)^{2}}{N}

Population standard deviation

σ = \sqrt{\frac{\sum (x_{i} - μ)^{2}}{N}}

Interquartile range

IQR = Q_{3} - Q_{1}

Outlier fences

Q_{1} - 1.5 IQR and Q_{3} + 1.5 IQR

z-score

z = \frac{x - μ}{σ} = \frac{x - \bar{x}}{s}

Empirical rule (normal data)

μ \pm 1 σ \approx 68 % μ \pm 2 σ \approx 95 % μ \pm 3 σ \approx 99.7 %

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