CIS 2033 Based on Textbook A Modern Introduction

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CIS 2033 Based on Textbook: A Modern Introduction to Probability and Statistics. 2007 Instructor:

CIS 2033 Based on Textbook: A Modern Introduction to Probability and Statistics. 2007 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:

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, 41, 41, 42, 43, 58, 41

Outliers an outlier is an observation that is numerically distant from the rest of

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

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. Median of Absolute Deviation (MAD): The Median of the Absolute Deviations of

Variability cont. 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

Empirical quantiles The order statistics consist of the same elements as the original dataset x 1, x 2 x 3, …, 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

Quick exercise 16. 5. Compute the 55 th empirical percentile for the Wick temperature

Quick exercise 16. 5. Compute the 55 th empirical percentile for the Wick temperature data. The order statistics of the Wick temperature data is

Quartiles • Lower quartile: qn(. 25) • Upper quartile: qn(. 75) • Interquartile Range

Quartiles • Lower quartile: qn(. 25) • Upper quartile: qn(. 75) • Interquartile Range (IQR) • IQR = qn(0. 75) − qn(0. 25) • Median (Middle Quartile): qn(. 50)

Quick exercise 16. 6. Compute the five-number summary for the (uncorrected) Wick temperature data.

Quick exercise 16. 6. Compute the five-number summary for the (uncorrected) Wick temperature data.

The box-and-whisker plot • Advantages: • Good representation of statistical data • Shows quartiles,

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 Up from the upper quartile we measure out a distance of 1. 5 times the IQR and draw a so-called whisker up to the largest observation that lies within this distance, where we put a horizontal line. Similarly, down from the lower quartile we measure out a distance of 1. 5 times the IQR and draw a whisker to the smallest observation that lies within this distance, where we also put a horizontal line. All other observations beyond the whiskers are marked by ◦. Such an observation is called an outlier.

Using boxplots to compare several datasets Boxplots become useful if we want to compare

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