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

73.2half larger

half smaller

Life expectancy data

Minimum = 47.8

Maximum = 83.4

Median = 73.2

Q1

1 197

Sao Tomé & Príncipe

50 (¼ way)

1st quartile = 64.7

Q1

¾ larger¼ smaller

1st quartile = 64.7

Q3

1 197

NetherlandAntilles

148 (¾ way)

3rd quartile = 76.7

Q3

3rd quartile = 76.7

¾ smaller ¼ larger

Life expectancy data

Minimum = 47.8

Maximum = 83.4

Median = 73.2

1st quartile = 64.7

3rd quartile = 76.7

Box Plot

Box plot

1st quartile

3rd quartilemedian

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• Mathematical notation:

• … Greek letter capital sigma• means SUM in mathematics

• Another measure of the center of the data: mean (average)

• Data values:

Median = -13Mean = -14.2

Mean is not a robust statistic.

Median is a robust statistic.

Robust statistic

Median = -13Mean = -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

33 750

33 750

33 750

44 000

44 000

44 000

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 495mean = 518 3118% trimmed mean = 128 109

MEASURES OF VARIABILITY

Navození atmosféry

QUESTION

Mean1 Mean2Mode1 Mode2Median1 Median2

range(variační rozpětí)

MAX - min

RangeRange changes when we add new data into dataset

• Always• Sometimes• Never

Adding Mark Zuckerberg

Cut off data

IQR, mezikvartilové rozpětí

Interquartile range, IQRLet’ 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 outlierOR

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. $60,0002. $80,0003. $100,0004. $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 meanSample

10

5

3

2

19

1

7

11

1

1

Average distance from meanSample Deviation from mean

10

5

3

2

19

1

7

11

1

1

Average distance from meanSample Deviation from mean

10 4

5 -1

3 -3

2 -4

19 13

1 -5

7 1

11 5

1 -5

1 -5

∑ (𝑥 𝑖−𝑥 )=0Find 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.