Types of (random) variables Categorical (Nominal) Variables favorite color: red,blue, sky blue, orange ) No natural order, no clear distance measure Ordinal Variables: A natural order, but the measure of distance not defined / imprecise / arbitrary (e.g. rate on a scale of 1-10 the quality of this your teacher) Interval Variables: Usually random variables with real number values Natural order, and clear-defined measure of distance Ratio Variables: Similar to Interval variables, but with a lower value limit at 0.0 NOTE: We will not make a formal distinction between the last two, rather one should be aware of the physical meaning of the values: e.g. temperatures expressed in F or C would be Interval variables; expressed in Kelvin, they qualify for Ratio Variables But still for our statistical analysis it wouldn’t make a difference (far away from 0K)

Description of random data samples (real-valued variables) (Random data that can be sorted by size) Sample Size Center or location of the sample Measure of the range of the sample Symmetry of the sample distribution (sometimes the range of possible outcomes of the random process is well-known by physical constraints)

Measure for the center of the sample Arithmetic mean: x x Summed up i i

Measure for the range of the sample Standard deviation: x “Bessel Correction:” gives an unbiased estimate. i

Measure for the range of the sample Variance: x Again: You often find the denominator (n-1) instead for an unbiased estimate i

R-Commands mean(x) var(x) (R uses the Bessel Correction (n-1)) sd(x) summary(x) is a more general function that gives a statistical summary of the data Example: summary(c(1,2,42,3,24,52)) returns Min. 1st Qu. Median Mean 3rd Qu. Max. 1.00 2.25 13.50 20.67 37.50 52.00

R Commands Median(x) is another measure of the data’s center point When you sort the data sample in ascending order, you find the mid-point of the sample: Example: x<-c(1,2,3,2,5,62) median(x) returns 2.5 mean(x) returns 12.5 We see the mean is not robust against outliers in the sample, but the median give a robust result.

R Commands Imagine, we did a small typo in the last example, and the real data sample had seven elements x<-c(1,2,3,2,5,6,2) median(x) returns 2 mean(x) returns 3

Quartiles and Quantiles Similar to finding the median of the sample data, one can define the lower and upper quartiles of the data sample That is, you sort the data x in ascending order {x1, x2, x3, … , xn} are ordered data with xi indicating the i-th smallest data if you have for example 100 data then the lower quartile value is closest to (or interpolated between the closest ranks) the position i such that 25% of the values are lower than the quartile value xi=26 The upper quantile would be at xi=76

Quartiles and Quantiles In small sample sizes and the even and odd samples sizes require modification to the estimation of median and quartiles In general: with large sample size one can sort the data and use the probability estimate for the chances of exceeding a certain value in the sorted sample: Let n be your sample size (say n=1000) then the p-th quantile is the value (in the same units of your sample data {xi, i=1,2,…,n}) that exceeds the values of your sample with a probability of p

Quantiles x n samples p-th quantile value: qp k Rank i p=k/n Sample data sorted in ascending order 0<= p <= 1

R Commands Visualization of data samples: hist(x) Albany Airport January 2014 daily mean temperatures [F] histogram. Sample size with n=31 is small. We count the number of days with temperatures in a certain range (bins). Instructions for R: Run script albany2.R after running script execute: hist(tday) Name is tday not tavg

R Commands Visualization of data samples: boxplot(x) Albany Airport January 2014 daily mean, min. and max. temperatures [F] We count the number of days with temperatures in a certain range (bins). R instructions: (make sure you have run script albany2.R) Execute: boxplot(list(tmin=jan$MIN,tavg=tday,tmax=jan$MAX)) Note: boxplot is best used if you want to compare two or more sample distributions visually. To achieve that, boxplot is given a list of data, each data sample get’s it’s own name in the list of data, and boxplot creates for each named data set of the list it’s own boxplot diagram)

Boxplot: max(tmin) Upper quartile median Lower quartile min(tmin) Note: Different flavors of boxplots circulate around. Oftentimes ‘outliers’ are plotted as extra dots, and the boxplot symbols are caculated without the outliers.