% Basic Statistical Concepts
% R Bootcamp HTML Slides
% Jared Knowles

Introduction

We will review the following statistical concepts here through the lens of R:

Statistical distributions
Attributes of distributions
Samples
Standard deviations and standard errors
Statistical models
Univariate Regression
Multivariate Regression
Causality

Describing Data

At it's most basic level, statistics is about summarizing and understanding data

- Data are themselves abstractions of real world concepts we care about

Spread of the Data

Data spread describes how scattered a data set is
One type of data, categorical data, describes groups

qplot(factor(cyl), data = mtcars) + labs(x = "cylinder", title = "Car models by Cylinder Count")

plot of chunk barplot

What can we learn here?

Let's try another

What can we learn from this chart type about the data?

data(diamonds)
qplot(factor(cut), data = diamonds) + labs(x = "Cut", title = "Diamonds by Cut Quality")

plot of chunk diamondplot

What else might we want to learn?

How about?

qplot(factor(color), data = diamonds) + labs(x = "Color", title = "Diamonds by Color and Clarity") + 
    facet_wrap(~clarity, nrow = 2)

plot of chunk diamondplot2

Still more

With diamonds we immediately want to look at price

qplot(carat, price, data = diamonds, color = color) + geom_smooth(aes(group = 1))

plot of chunk diamondplot3

What do you see?
Outliers?
Data modes?
Clusters?

Graphical Depictions of Data

These are ways to show data with graphics
Graphical displays are driven by the concept of dimensions
One dimension–a single category
Two dimensions–two categories

Levels of Measurement

Any given dimension may be measured at different levels of measure
Nominal: unordered categories of data
Ordinal: ordered categories of data, relative size and degree of difference between categories is unknown
Interval: ordered categories of data, fixed width, like discrete temperature scales
Continuous (ratio): a measurement scale in a continuous space with a meaningful zero–physical measurements
Derived by Stanley Smith Stevens in the 1940s and 50s

Quiz 1

- Car color is what level of measurement?

Levels of measurement matter

How you depict the data
What you can calculate using the data

Describing Data with Numbers

What types of measures can we use to describe different levels of measurement?

Level of Meas.	Stats
Nominal	mode, Chi-squared
Ordinal	median, percentile, (plus above)
Interval	mean, standard deviation, correlation, ANOVA, plus above
Continuous	geometric mean, harmonic mean, coefficient of variation, logarithms, plus above

Let's talk about these statistics

STATISTIC: a single measure of some attribute of a sample (e.g. its arithmetic mean value). It is calculated by applying a function (statistical algorithm) to the values of the items comprising the sample which are known together as a set of data. (Wikipedia)[http://en.wikipedia.org/wiki/Statistic]
These statistics can measure a number of features of a dataset, but we tend to think of them as measuring either central tendency, spread, or association
We'll focus on these today.

Measures of Central Tendency

These are the three canonical measures of central tendency:
- Mean
- Median
- Mode
How are these different? What properties do they have? Why does this matter?

qplot(hwy, data = mpg, geom = "density") + geom_vline(xintercept = median(mpg$hwy), 
    color = I("blue"), size = I(1.1)) + geom_vline(xintercept = mean(mpg$hwy), 
    color = I("gold"), size = I(1.1)) + geom_vline(xintercept = 26, color = I("orange"), 
    size = I(1.1)) + geom_text(aes(x = median(mpg$hwy) + 1.5, y = 0.08, label = "Median"), 
    size = I(4.5)) + geom_text(aes(x = mean(mpg$hwy) - 1.5, y = 0.06, label = "Mean"), 
    size = I(4.5)) + geom_text(aes(x = 26 + 1.5, y = 0.05, label = "Mode"), 
    size = I(4.5))

plot of chunk centraltend

library(xtable)
print(xtable(table(mpg$hwy)), type = "html")

## <!-- html table generated in R 2.15.1 by xtable 1.7-0 package -->
## <!-- Wed Sep 26 16:57:47 2012 -->
## <TABLE border=1>
## <TR> <TH>  </TH> <TH> V1 </TH>  </TR>
##   <TR> <TD align="right"> 12 </TD> <TD align="right">   5 </TD> </TR>
##   <TR> <TD align="right"> 14 </TD> <TD align="right">   2 </TD> </TR>
##   <TR> <TD align="right"> 15 </TD> <TD align="right">  10 </TD> </TR>
##   <TR> <TD align="right"> 16 </TD> <TD align="right">   7 </TD> </TR>
##   <TR> <TD align="right"> 17 </TD> <TD align="right">  31 </TD> </TR>
##   <TR> <TD align="right"> 18 </TD> <TD align="right">  10 </TD> </TR>
##   <TR> <TD align="right"> 19 </TD> <TD align="right">  13 </TD> </TR>
##   <TR> <TD align="right"> 20 </TD> <TD align="right">  11 </TD> </TR>
##   <TR> <TD align="right"> 21 </TD> <TD align="right">   2 </TD> </TR>
##   <TR> <TD align="right"> 22 </TD> <TD align="right">   7 </TD> </TR>
##   <TR> <TD align="right"> 23 </TD> <TD align="right">   7 </TD> </TR>
##   <TR> <TD align="right"> 24 </TD> <TD align="right">  13 </TD> </TR>
##   <TR> <TD align="right"> 25 </TD> <TD align="right">  15 </TD> </TR>
##   <TR> <TD align="right"> 26 </TD> <TD align="right">  32 </TD> </TR>
##   <TR> <TD align="right"> 27 </TD> <TD align="right">  14 </TD> </TR>
##   <TR> <TD align="right"> 28 </TD> <TD align="right">   7 </TD> </TR>
##   <TR> <TD align="right"> 29 </TD> <TD align="right">  22 </TD> </TR>
##   <TR> <TD align="right"> 30 </TD> <TD align="right">   4 </TD> </TR>
##   <TR> <TD align="right"> 31 </TD> <TD align="right">   7 </TD> </TR>
##   <TR> <TD align="right"> 32 </TD> <TD align="right">   4 </TD> </TR>
##   <TR> <TD align="right"> 33 </TD> <TD align="right">   2 </TD> </TR>
##   <TR> <TD align="right"> 34 </TD> <TD align="right">   1 </TD> </TR>
##   <TR> <TD align="right"> 35 </TD> <TD align="right">   2 </TD> </TR>
##   <TR> <TD align="right"> 36 </TD> <TD align="right">   2 </TD> </TR>
##   <TR> <TD align="right"> 37 </TD> <TD align="right">   1 </TD> </TR>
##   <TR> <TD align="right"> 41 </TD> <TD align="right">   1 </TD> </TR>
##   <TR> <TD align="right"> 44 </TD> <TD align="right">   2 </TD> </TR>
##    </TABLE>

Session Info

It is good to include the session info, e.g. this document is produced with knitr version 0.8. Here is my session info:

print(sessionInfo(), locale = FALSE)

## R version 2.15.1 (2012-06-22)
## Platform: x86_64-pc-mingw32/x64 (64-bit)
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] xtable_1.7-0    mgcv_1.7-21     ggplot2_0.9.2.1 knitr_0.8      
## 
## loaded via a namespace (and not attached):
##  [1] colorspace_1.1-1   dichromat_1.2-4    digest_0.5.2      
##  [4] evaluate_0.4.2     formatR_0.6        grid_2.15.1       
##  [7] gtable_0.1.1       labeling_0.1       lattice_0.20-10   
## [10] MASS_7.3-21        Matrix_1.0-9       memoise_0.1       
## [13] munsell_0.4        nlme_3.1-104       plyr_1.7.1        
## [16] proto_0.3-9.2      RColorBrewer_1.0-5 reshape2_1.2.1    
## [19] scales_0.2.2       stringr_0.6.1      tools_2.15.1

Attribution and License

This work (R Tutorial for Education, by Jared E. Knowles), in service of the Wisconsin Department of Public Instruction, is free of known copyright restrictions.