Univariate Data - 2. Numeric Summaries

Transcription

1 Univariate Data - 2. Numeric Summaries Young W. Lim Mon Young W. Lim Univariate Data - 2. Numeric Summaries Mon 1 / 31

2 Outline 1 Univariate Data Based on Numerical Summaries Young W. Lim Univariate Data - 2. Numeric Summaries Mon 2 / 31

3 Based on "Using R for Introductory Statistics" John Verzani I, the copyright holder of this work, hereby publish it under the following licenses: GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. CC BY SA This file is licensed under the Creative Commons Attribution ShareAlike 3.0 Unported License. In short: you are free to share and make derivative works of the file under the conditions that you appropriately attribute it, and that you distribute it only under a license compatible with this one. Young W. Lim Univariate Data - 2. Numeric Summaries Mon 3 / 31

4 Functions function(x) { sum(x) / length(x) } functions(x) { total <- sum(x) n <- length(x) total / n } my_mean <- function(x) { sum(x) / length(x) } my_mean( c(1,2,3,4) ) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 4 / 31

5 The sample mean x <- c(51, 52, 48, 83, 92) sort(x) mean(x) devs <- x - mean(x) mean(devs) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 5 / 31

6 The trimmed mean mean(x, trim=0.10) # trim 10% of both ends mean(exec.pay) mean(exec.pay, trim=0.10) the Macdonell (HistData) data set w <- Macdonell$frequency / sum(macdonell$frequency) y <- Macdonell$height sum(w*y) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 6 / 31

7 The sample median median(x) n <- length(exec.pay); trim=0.10 lo <- 1 + floor(n * trim) hi <- n lo median(sort(exec.pay)[lo:hi]) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 7 / 31

8 Measures of position x <- 0:5 length(x) mean(sort(x)[3:4]) median(x) quantile(x) quantile(x, 0.25) # 0, 25, 50, pecentiles quantile(x, seq(0, 1, by=0.2)) fivenum(x) y = c("a"=95, "B"=82, "C"=74, "D"=61)y Young W. Lim Univariate Data - 2. Numeric Summaries Mon 8 / 31

9 Other measures of center table(x) table(x) == max(table(x)) which(table(x) == max(table(x))) as.numeric(names( which(table(x) == max(table(x))))) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 9 / 31

10 Spread range(x) diff(range(x)) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 10 / 31

11 Variance and standard deviation var(x) sum( (x - mean(x))^2 ) / (length(x) -1) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 11 / 31

12 Sample standard deviation x <- c(100, 300, 900, NA, 200, 500, 700) range(x, na.rm=true) sd(x, na.rm=true) z_score <- function(x) (x - mean(x)) / sd(x) z_score(x) scale(x)[,1] # to extract the 1st column z <- (x -mean(x)) / sd(x) x[ z >= 1.28] mean(x) * sd(x) z <- (exec.pay- mean(exec.pay)) / sd(exec.pay) out <- abs(z) > 3 sum(out) / length(z) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 12 / 31

13 IQR (InterWuartile Range) median(rivers) IQR(rivers) IQR(rivers) / sd(rivers) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 13 / 31

14 mad (Median Absolute Deviation) mad(rivers) / sd(rivers) ht <- kid.weights$height mad(ht) / sd(ht) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 14 / 31

15 Shape Symmetry and skew Tails Modes Young W. Lim Univariate Data - 2. Numeric Summaries Mon 15 / 31

16 Symmetry and skew skew <- function(x) { n <- length(x) z <- (x - mean(x)) / sd(x) sum(z^3) / n } skew(exec.pay) four_year_hts <- \ kid.weights[kid.weights$age %in% 48:59, "height"] skew(four_year_hts) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 16 / 31

17 Tail kurtosis <- function(x) { n <- length(x) z <- (x -mean(x)) / sd(x) sum(z^4)/n - 3 } kurtosis(galton$parent) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 17 / 31

18 Modes unimodal bimodal multimodal Young W. Lim Univariate Data - 2. Numeric Summaries Mon 18 / 31

19 Viewing the shape of a data set Dot plots Stem and leaf plots Histograms Density plots Box plots Quantile graphs Young W. Lim Univariate Data - 2. Numeric Summaries Mon 19 / 31

20 Dot plots jitter stripchart Young W. Lim Univariate Data - 2. Numeric Summaries Mon 20 / 31

21 Stem-and-leaf plots stem stem(bumpers) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 21 / 31

22 Histogram hist(faithful$waiting) bins <- seq(40, 100, by=5) x <- faithful$waiting out <- cut(x, breaks=bins) head(out) table(out) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 22 / 31

23 Density plots plot( density(bumpers) ) b_hist <- hist(bumpers, plot=false) b_dens <- density(bumpers) hist(bumpers, probability=true, xlim=range(c(b_hist$breaks, b_dens$x)), ylim=range(c(b_hist$density, b_dens$y)) ) lines(b_dens, lwd=2) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 23 / 31

24 Standard plotting arguments xlim ylim xlab ylab main pch cex col lwd lty set x coordinate range set y coordinate range set label for x axis set label for y axis set the main title adjust plot symbols (?pch) adjust size of text and symbols adjust color of objects drawn (?colors) adjust width of lines drawn adjust how line is drawn "blank", "solid", "dashed", dotdash" bty adjust bos type ("o", "l", "7", "c", "u", "]" Young W. Lim Univariate Data - 2. Numeric Summaries Mon 24 / 31

25 Functions to add layers points lines abkube text mtext add points to a graphic add points connected by lines to a graphic add a line of the form a + bx, y=h, or x=v add text to a graphic add text to margins of a graphic Young W. Lim Univariate Data - 2. Numeric Summaries Mon 25 / 31

26 Various plots hist(bumpers, probability=true, xlim = range( c(b_hist$breaks, b_dens$x)), ylim = range( c(b_hist$density, b_dens$y))) lines(b_dens, lwd=2) boxplot(bumpers, horizontal=true, main="bumpers") x <- rep(macdonell$finger, Macdonell$frequency) qqnorm(x) x <- jitter(histdata::galton$child, factor=5) qqnorm(x) Young W. Lim Univariate Data - 2. Numeric Summaries Mon 26 / 31

27 OOO Young W. Lim Univariate Data - 2. Numeric Summaries Mon 27 / 31

28 OOO Young W. Lim Univariate Data - 2. Numeric Summaries Mon 28 / 31

29 OOO Young W. Lim Univariate Data - 2. Numeric Summaries Mon 29 / 31

30 OOO Young W. Lim Univariate Data - 2. Numeric Summaries Mon 30 / 31

31 OOO Young W. Lim Univariate Data - 2. Numeric Summaries Mon 31 / 31