STATISTICAL METHODS IN DATA SCIENCE AND LABORATORY I
Professor Perone
Expectation and Variance[edit | edit source]
The variance Var(x) of a random variable X is the number: <math>Var(x)=E[(X - E[X])^2]</math>. So the variance is the expectation of the Mean Squared of the expectation. Another useful parameter is the Standard Deviation: <math>\sqrt{(Var(X))}</math>
Exploratory Data Analysis[edit | edit source]
It's important to make a distinction between probability theory and statistics. In the former the distribution function is given and we want to analyze the outcomes, in statistics we have the outcomes and we want to infer knowledge by modeling those outcomes
Graphical Summaries[edit | edit source]
Sample Statistic | Formula | Distribution Feature | Formula |
Empirical Distribution Function | <math>F_{n}</math> | Distribution Function | <math>F</math> |
Kernel Density Estimate (Histogram) | <math>f_{n,h}</math> | Probability Density | <math>f</math> |
(Number of | Probability Mass Function | <math>p(a)</math> |
Numerical Summaries[edit | edit source]
The numerical summaries provided in probability theory have their analogous in data analysis as reported in the table
Sample Statistic | Formula | Distribution Feature | Formula |
Sample Mean | <math>X_{n}</math> | Expectation | <math>\mu</math> |
Sample Median | <math>Med(X_{1},X_{2},...,X_{n})</math> | Median | <math>q_{0.5} = F^{inv}(0.5)</math> |
pth Empirical Quantile | <math>q_{n}(p)</math> | 100pth percentile | <math>q_{p} = F^{inv}(p)</math> |
Sample Variance | <math>S^{2}_{n}</math> | Variance | <math>Var(x)=E[(X - E[X])^2]</math> |
Sample Standard Deviation | <math>S_{n}</math> | Standard Deviation | <math>\sqrt{(Var(X))}</math> |
Median of absolute deviation (MAD) | <math>MAD(X_{1},X_{2},...X_{n})</math> | Median of absolute deviation (MAD) | <math>F^{inv}(0.75)-F^{inv}(0.5) for symmetric F</math> |
- Sample Mean
- Sample Variance
- Sample Standard Deviation
- Median of absolute deviation (MAD)
In this chapter
Statistical Models and their Features' Estimation[edit | edit source]
Basic Statistical Models[edit | edit source]
Bootstrap[edit | edit source]
Estimators[edit | edit source]
Biased and Unbiased Estimators[edit | edit source]
Estimators Efficiency Comparison: Variance and Mean Squared Error[edit | edit source]
When we have two unbiased estimators and we would like to know which one is the best to estimate parameters we use variance, instead for unbiased estimators we use Mean Squared Errors