# MEASURES OF CENTRALITY. Last lecture summary Mode Distribution.

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MEASURES OF CENTRALITY

Last lecture summary Mode Distribution

Life expectancy data

Minimum Sierra Leone minimum = 47.8

Maximum Japan maximum = 84.3

Life expectancy data all countries

Life expectancy data 1 197 Egypt 99 73.2 half larger half smaller

Life expectancy data Minimum = 47.8 Maximum = 83.4 Median = 73.2

Q1 1 197 Sao Tomé & Príncipe 50 (¼ way) 1 st quartile = 64.7

Q1 ¾ larger¼ smaller 1 st quartile = 64.7

Q3 1 197 Netherland Antilles 148 (¾ way) 3 rd quartile = 76.7

Q3 3 rd quartile = 76.7 ¾ smaller¼ larger

Life expectancy data Minimum = 47.8 Maximum = 83.4 Median = 73.2 1 st quartile = 64.7 3 rd quartile = 76.7

Box Plot

Box plot 1 st quartile 3 rd quartile median minimum maximum

Quartiles, median – how to do it? 79, 68, 88, 69, 90, 74, 87, 93, 76 Find min, max, median, Q1, Q3 in these data. Then, draw the box plot.

Another example Min. 1st Qu. Median 3rd Qu. Max. 68.00 75.00 81.00 88.50 93.00 78, 93, 68, 84, 90, 74

Percentiles věk [roky] http://www.rustovyhormon.cz/on-line-rustove-grafy

Skeleton data Estimate age at death from skeletal remains Common problem in forensic anthropology Based on wear and deterioration of certain bones Measurements on 400 skeletons Two estimation methods Di Gangi et al., aspects of the first rib Suchey-Brooks, most common, pubic bone http://www.bestcoloringpagesforkids.com/wp-content/uploads/2013/07/Skeleton-Coloring-Page.gif

400 skeletons, the estimated and the actual age of death

DiGangi

Modified boxplot Min. Q1 Median Q3 Max. -60.00 -23.00 -13.00 -5.00 32.00

Mean

Median = -13 Mean = -14.2 Mean is not a robust statistic. Median is a robust statistic. Robust statistic

Median = -13 Mean = -14.2 10% trimmed mean … eliminate upper and lower 10% of data (i.e. 40 points). 10% trimmed mean = mean of 320 middle data values = -13.8 Trimmed mean is more robust. Trimmed mean

Salary o 25 players of the American football (NY red Bulls) in 2012. 33 750 44 000 45 566 65 000 95 000 103 500 112 495 138 188 141 666 181 500 185 000 190 000 194 375 195 000 205 000 292 500 301 999 4 600 000 5 600 000 median = 112 495 mean = 518 311 8% trimmed mean = 128 109

MEASURES OF VARIABILITY

Navození atmosféry

QUESTION Mean1 Mean2 Mode1 Mode2 Median1 Median2

range (variační rozpětí) MAX - min

Range Range changes when we add new data into dataset Always Sometimes Never

Cut off data IQR, mezikvartilové rozpětí

Interquartile range, IQR Let’ take this quiz, answer yes ot not. 1. About 50% of the data fall within the IQR. 2. The IQR is affected by every value in the data set. 3. The IQR is not affected by outliers. 4. The mean is always between Q1 and Q3. 0 1 1 1 2 2 2 2 2 3 3 3 90 Q2Q1=1 Q3=3

Define outlier Sample \$38,946 \$43,420 \$49,191 \$50,430 \$50,557 \$52,580 \$53,595 \$54,135 \$60,181 \$10,000,000 What values are outliers for this data set? 1.\$60,000 2.\$80,000 3.\$100,000 4.\$200,000

Problem with IQR normal bimodal uniform

Options for measuring variability Find the average distance between all pairs of data values. Find the average distance between each data value and either the max or the min. Find the average distance between each data value and the mean.

Average distance from mean Sample 10 5 3 2 19 1 7 11 1 1

Average distance from mean Sample 10 5 3 2 19 1 7 11 1 1

Average distance from mean Sample 104 5 3-3 2-4 1913 1-5 71 115 1-5 1 Find the average distance between each data value and the mean.

Preventing cancellation How can we prevent the negative and positive deviations from cancelling each out? 1. Ignore (i.e. delete) the negative sign. 2. Multiply each deviation by two. 3. Square each deviation. 4. Take absolute value of each deviation.