用数值描述统计

Mean

• Population mean: $\mu=\frac{\sum_{i=1}^N x_i}{N}$
• Sample mean: $\bar{x}=\frac{\sum_{i=1}^n x_i}{n}$

Median

Mode 众数

常用于 categorical data．一个众数 = unimodal；两个众数 = bimodal；multimodal

Percentiles and Quartiles

measures that indicate the location, or position, of a value relative to the entire set of data, 适合描述大数据集

$p$-th percentile = approximately $p\%$ of observations $\le$ this value. value located at $\frac{p}{100}(n+1)$-th position.

Quartiles:

• $Q_1: 25\%$ percentiles
• $Q_2: 50\%$ percentiles
• $Q_3: 75\%$ percentiles

Five-Number Summary

minimum, first quartile, median, third quartile, maximum

Range

$\max-\min$

sensitive to outliers

Interquartile Range

$IQR=Q_3-Q_1$

Box-and-Whisker Plot

Inner box 表示 $Q_1\sim Q_3$ 的范围，inner box 中间的分割线表示 median，两条 whiskers 分别从 $Q_1$ 延伸到 minimum 和从 $Q_3$ 延伸到 maximum．

对于实数的 boxplot，upper whisker ends at $\min(V_{\max}, Q_3+1.5\times IQR)$, lower whisker ends at $\max(Q_1-1.5\times IQR, V_{\min})$

Variance

Population variance

$$\sigma^2=\frac{\sum_{i=1}^N (x_i-\mu)^2}{N}=\frac{\sum_{i=1}^N x_i^2}{N}-\mu^2$$

Sample variance

$$s^2=\frac{\sum_{i=1}^n (x_i-\bar{x})^2}{n-1}=\frac{\sum_{i=1}^n x_i^2 -n\bar{x}^2}{n-1}$$

Coefficient of Variance (CV)

measure of relative dispersion that expresses the standard deviation as a percentage of the mean

population CV = $\frac{\sigma}{\mu}\times 100\%$, sample CV = $\frac{s}{\bar{x}}\times 100\%$

Empirical Rule

• 大约 $0.68$ 的数据落在 $[\mu\plusmn \sigma]$ 的区间里
• 大约 $0.95$ 的数据落在 $[\mu\plusmn 2\sigma]$ 的区间里
• 大约 $0.997$ 的数据落在 $[\mu\plusmn 3\sigma]$ 的区间里

z-Score

measures the location or position of a value relative to the mean of the distribution: it is a standardized value that indicates the number of standard deviations a value is from the mean

Population $z_i=\frac{x_i-\mu}{\sigma}$, sample $z_i=\frac{x_i-\bar{x}}{s}$

Skewness of Distribution

$$\text{skewness}=\frac{1}{n}\frac{\sum_{i=1}^n (x_i-\bar{x})^3}{s^3}$$

Covariance

Population covariance

$$Cov(x,y)=\sigma_{x,y}=\frac{\sum_{i=1}^N (x_i-\mu_x)(y_i-\mu_y)}{N}$$

Sample covariance

$$s_{x,y}=\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n}$$

Correlation

free of units and provides both the direction and strength of a relationship

Population correlation

$$\rho_{xy}=\frac{\sigma_{xy}}{\sigma_x\sigma_y}$$

Sample correlation

$$r_{xy}=\frac{s_{xy}}{s_xs_y}$$

Relationship 存在：$|r_{xy}|\ge \frac{2}{\sqrt{n}}$