Exploratory data analysis: numerical summaries CIS 2033 Based on Textbook: A Modern Introduction to Probability and Statistics. 2007 Slides: QUINCY R WALKER Modified by the instructor: Dr. Longin Jan Latecki Chapter 16 Exploratory data analysis: numerical summaries

16.1 The Center of the Data Set Center of the Data= sample mean: n = the sample size Example: Sample mean of the following data is 44.7 43, 43, 41, 41, 41, 42, 43, 58, 58, 41, 41

Outliers an outlier is an observation that is numerically distant from the rest of the data Sample median is more robust in the presence of outliers.

Variability in A Data Set Variance: Standard Deviation: where n is the number samples Why we choose the factor 1/(n−1) instead of 1/n will be explained later (in Chapter 19).

Variability cont. Medn= median of sample Median of Absolute Deviation (MAD): The Median of the Absolute Deviations of a Sample. Medn= median of sample Absolute Deviation: The absolute value of the distance Of a point xi in a data set from the median

Empirical quantiles The order statistics consist of the same elements as the original dataset x1, x2 x3,…, xk , but in ascending order. Denote by the kth element in the ordered list. Then: The pth quartile corresponds to pth quartile of a cdf: Finv(p) where F(p) is the cumulative distribution function of the data

Quartiles Lower quartile: qn(.25) Upper quartile: qn(.75) Interquartile Range (IQR) IQR = qn(0.75) − qn(0.25) Median(Middle Quartile): qn(.50)

The box-and-whisker plot Advantages: Good representation of statistical data Shows quartiles, median and outliers Disadvantages poor graphical display of the dataset histogram and kernel density estimate are more informative displays of a single dataset

Using boxplots to compare several datasets Boxplots become useful if we want to compare several sets of data in a simple graphical display: