STAT7055

Last updated Aug 3, 2023

• Established a research question, or hypothesis, that can be tested.
• Determined some relevant variables.
• Identified our population of interest.
• Gathered some data by taking a sample from the population.
• Analysed our data and the relevant variables.
• Formed an inference or conclusion regarding the original hypothesis

population mean $\mu$ population variation $\sigma^2$

sample mean $\bar{X}$ sample variation $s^2$

• Categorical Data
  • Nomial, 各种无关联形容词
  • Ordinal, poor/fair/good
• Numeric
  • 离散
  • 连续

mode, frequency, bar chart, pie chart

ordinal 只是多一个 order

• Mean, median, mode.
• Quantiles.
• Range, interquartile range, variance, coefficient of variation.
• Covariance, correlation.
• Histograms.
• Boxplots.
• central trendency

measure 某个数据在总体的位置, 比如 quatiles, 31st percentile

$L_p = (n+1)\frac{p}{100}$

数据有8个(n=8), 那么 31st percentil($L_{31}$) 在

$L_{31} = (8+1)\frac{31}{100} = 2.79$

$IQR = Q_3 - Q_1$

population variance $\sigma^2$

$\sigma^2 = \frac{1}{N} \sum^N_{i=1}(X_i - \mu)^2$

sample variance $s^2$

$s^2 = \frac{1}{n-1} \sum^N_{i=1}(X_i - \bar{X})^2$

shortcut

为什么样本方差（sample variance）的分母是 n-1？_样本方差的分母为什么是n-1_羊老羊的博客-CSDN博客

= Variance 开方

population sd $\sigma$

sample sd $s$

用来比较两组(很可能是规格不同的)数据的分散情况的标化指标, cv越大, 表示越分散

population $CV=\frac{\sigma}{\mu}$

sample $cv=\frac{s}{\bar{X}}$

协方差是两个变量之间线性关系的度量，描述它们如何相互关联

population covariance $\sigma_{XY} = \frac{1}{N}\sum^N_{i=1}(X_i-\mu_X)(Y_i-\mu_Y)$

sample covariance $s{XY} = \frac{1}{n-1}\sum^N_{i=1}(X_i-\bar{X})(Y_i-\bar{Y})$

shortcut

数据相关性的标化指标, 比如有 A,B 两组数据, 想知道 A与C 更相关, 还是 B与C更相关, 就可以用这个指标

值的范围是-1.0至1.0. 越接近1.0表示越强的正相关, 越接近 -1.0表示越强的负相关

population cor $\rho_{XY} = \frac{\sigma_{XY}}{\sigma_{X}\sigma_Y}$

sample cor $r_{XY} = \frac{s_{XY}}{s_{X}s_Y}$

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> # 把数据存入变量 X
> X <- c(8.3, -6.2, 20.9, -2.7, 33.6, 42.9, 24.4, 5.2, 3.1, 30.5)

> # 把另一组数据存入变量 Y
> Y <- c(12.1, -2.8, 6.4, 12.2, 27.8, 25.3, 18.2, 10.7, -1.3, 11.4)

> # average/mean
> mean_X = mean(X) # Xbar = 16

> # Standard Deviation
> sd_X = sd(X) # s = 16.74336

> # Variance
> var_X = var(X) # s^2 = 280.34

> # Coefficient of Variation
> cv_X = sd(X)/mean(X)

> # Covariation
> cov_XY = cov(X, Y)

> # Correlation Coefficient
> cor_XY = cor(X, Y)