Statistics- Landmark Summaries Interpreting Typical Values and Percentiles

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Chapter 4

Landmark Summaries:

Interpreting Typical Values and

Percentiles



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Average or Mean

Add the data, divide by n or N (the number of

elementary units)

Divides total equally. The only such summary

A representative, central number (if data set is

appr oximately normal)

Summation notation

± 7 is capital Greek sigma

n

X X X X

n!

...21

N

X X X N

!Q...21

Sample average

Po pulation average

§!

!

n

i

i X n

X

1

1§!

!Q

N

i

i X N 1

1



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Example: Number of Defects

Def ects measured for each of 10 pr oduction lots

4, 1, 3, 7, 3, 0, 7, 14, 5, 9

0

2

0 5 10 15 20

Def ects per lot

F r e q u e n c y ( l o t s )

Average is 5.1

def ects p

er lot

Fig 4.1.1



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Median

Also summarizes the data

The middle one

± Put data in order

± Pick middleo

ne (o

r average middle two

if n is even)

± Median (9, 4, 5) = Median(4, 5, 9) = 5

± Median (9, 4, 5, 7) = Median (4, 5, 7, 9) = = 6

R ank

of the median is (1+n)/2

± If n=3, rank is (1+3)/2 = 2

± If n=4, rank is (1+4)/2 = 2.5 (so average 2nd and 3rd)

± If n=262, rank is (1+262)/2 = 131.5

5+7

2



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Median (continued)

A representative, central number

± If data set has a center

Less sensitive to outliers than the average

For skewed data, represents the ³typical case´ better than the average does

± e.g., incomes

Average income for a country equally divides the total, which

may include some very high incomes

Median income chooses the middle person (half earn less, half

earn more), giving less inf luence to high incomes (if any)



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Example: Spending

Customers plan to spend ($thousands)

3.8, 1.4, 0.3, 0.6, 2.8, 5.5, 0.9, 1.1

R ank ordered f r om smallest to largest

0.3, 0.6, 0.9, 1.1, 1.4, 2.8, 3.8, 5.51 2 3 4 5 6 7 8

Median is (1.1+1.4)/2 = 1.25 ± Smaller than the average, 2.05

Due to slight skewness?

R ank of median

= (1+8)/2 = 4.5

0 1 2 3 4 5

3 1 8 8 56 4

9

Median Average



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Example: The Crash of 1987

Dow-Jones Industrials, stock-price changes as

each stock began trading that f atef ul morning

Fairly normal

Mean and median are similar

Fig 4.1.2

0

5

-20% -10% 0%

Percent change at o pening

F r e q u e n c y

Average = -8.2%

Median = -8.6%



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Example: Incomes

Personal income of 100 peo ple

Average is higher than median due to skewness

Fig 4.1.3

0

10

20

30

40

50

$0 $100,000 $200,000 Income

Average = $38,710

Median = $27,216

F r e q u e n c y



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Mode

Also summarizes the data

Most common data value

± Middle of tallest histogram bar

Pr o blems:

± Depends on how you draw histogram (bin width)

± Might be more than one mode (two tallest bars)

Good if most data values are ³correct´

Good for nominal data (e.g., elections)

Mode

Mode



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Normal Distribution

Average, median, and mode are identical

± If the data come f r om a normal distribution

Average, median, and modeare identical

in the case of a normal distribution



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Skewed Distribution

Average, median, and mode are different

± The f ew large (or small) values inf luence the mean

more than the median

± The highest point is not in the center

Average

Median

Mode



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Which summary to use?

Average

± Best for normal data

± Preserves totals

Median ± Good for skewed data or data with outliers, pr ovided

you do not need to preserve or estimate total amounts

Mode

± Best for categories (nominal data). ± The mode is the only summary computable for nominal

data!



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Which Summary? (continued)

Average requires quantitative data (numbers)

Median works with quantitative or ordinal

Mode works with quantitative, ordinal, or nominal

Quantitative Ordinal Nominal

Average Yes - -

Median Yes Yes -

Mode Yes Yes Yes



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Weighted Average

Ordinary average gives same weight to allelementary units

Weighted average allows diff erent weights

Weights must add up to 1

± If not, then divide each by their total

n X n

X n

X n

X 1

...11

21!

nn X w X w X w X ! ...2211

1...21!

nwww



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Weighted Average (continued)

Average is per element ary unit

± The average of your course grades is your ³average per

course´

Weighted average is per unit of weig ht

± Your GPA (grade point average) is a weighted average,

using credit hours to def ine the weights. The weighted

average is your ³average per credit hour´



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Example: Portfolio Rate of Return

Portfolio ex pected return (an interest rate,indicating per formance) is the weig ht ed aver a g e

of the ex pected rates of return of assets in the

portfolio, weighted by $dollars invested

Portfolio contains three stocks. One ($1,000

invested) is ex pected to return 20%. Another

($1,800 invested) ex pects 15%. Third is $2,200

and 30%.

Total invested is 1,000+1,800+2,200 = $5,000



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Example (continued)

Weights are

w1 = $1,000/$5,000 = 0.20

w2 = $1,800/$5,000 = 0.36

w3

= $2,200/$5,000 = 0.44

Weighted average is

0.20v(20%) + 0.36v(15%) + 0.44v(30%) = 22.6%

± The ex pected return for the portfolio.

± Each stock is represented in pr o portion to $ invested



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Percentiles

Landmark summaries in the same measurementunits as the data

± e.g., dollars, peo ple, miles per gallon, «

Some f amiliar percentiles

± Smallest data value is 0th percentile

± Median is 50th percentile

± Largest data value is 100th percentile

± 90th

percentile is larger than 90%of

elementary units Finding percentiles

± Diff icult to see f r om histogram

± Easy using CDF (Cumulative Distribution Function)



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Cumulative Distribution Function

Data axis horizontally (as in histogram)

Cumulative percent vertically

Equal vertical jump at each data value

0.3, 0.6, 0.9, 1.1, 1.4, 2.8, 3.8, 5.5

0%

50%

100%

$0 $2 $4 $6

Spending

C u m

u l a t i v e

P e r c e n t

80th percentile

is $3.80

80%



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Five-Number Summary

Selected landmarks to represent entire data set

± Median = 50th percentile

± Quartiles

LQ = Lower Quartile = 25th percentile

± R ank =

UQ = U pper Quartile = 75th percentile

± R ank is n+1±[rank of lower quartile]

± Extremes

Smallest = 0th percentile

Largest = 100th percentile

2

2

1int1 ¼½

»¬-

«

n

R ank of median

Discard decimal,

if any.

int(10.5)=10int(35)=35



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Five-Number Summary (continued)

Pr ovides information about

± Central summary

Median

± R ange of the data

Largest ± smallest

± ³Middle half ́ of the data

Fr om LQ to UQ

± Skewness

If median is not appr oximately half way between quartiles



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Box Plot

Displays f ive-number summary

Less detail than histogram

± Easier to compare many gr oups

0 2 4 6 8

Smallest Largest

Lower

QuartileU pper

Quartile

Median

{

Middle half of the data



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Spending rank ordered f r om smallest to largest

0.3, 0.6, 0.9, 1.1, 1.4, 2.8, 3.8, 5.5

1 2 3 4 5 6 7 8

LQ is (0.6+0.9)/2 = 0.75

UQ is (2.8+3.8)/2 = 3.3

Example: Spending

R ank of median= (1+8)/2 = 4.5

R ank of UQ= 8+1-2.5=6.5

R ank of LQ= (1+4)/2 = 2.5

4 = int(4.5)



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Example: Spending (continued)

Five-number summary

0.3, 0.75, 1.25, 3.3, 5.5

Smallest, LQ, Median, UQ, Largest

Box plot

± Shows some skewness (lack of symmetry)

0 5

Spending ($thousands)



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Identifying Outliers

Outliers are def ined as o bservations, if any, either:

± More than UQ + 1.5 (UQ LQ), or

± Less than LQ 1.5 (UQ LQ)

Outliers aref

ar f

r o

m the center of

the distributio

n ± and may be interesting as special cases

UQ LQ

LQ UQ

1.5(UQ LQ)1.5(UQ LQ) U pper

outliers

Lower

outliers



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Example: Technology CEO Pay

CEO compensation in technology companies

± Detailed box plot identif ies outliers

and identif ies the most extreme non-outliers,

gives more detail than the (ordinary) box plot

Fig 4.2.3

$0 $5,000,000 $10,000,000

Detailed Box Plot

$0 $5,000,000 $10,000,000

IBMAMD

Sun

Micr osystems

A pple

Computer

Box Plot



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Example: CEO Compensation

Box plots to compare f irms within industry gr oups

± Utilities gr oup generally shows lower compensation

± Highest-paid are in Financial Services gr oup

Fig 4.2.3

$0 $10,000,000 $20,000,000 $30,000,000

Energy

Financial

Technology

Utilities



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CEO Compensation (continued)

Detailed box plots (with outliers and most extremenon-outliers named)

Fig 4.2.3

IBMAMD

Enr on

Citigr oupGoldmanSachs

Bear Stearns

MerrillLynch

Morgan StanleyDean Witter

LehmanBr others

Phillips Petr oleum

SunMicr osystems

DukeEnergy

GPU

A ppleComputer

Baker Hughes

BerkshireHathaway

$0 $10,000,000 $20,000,000 $30,000,000

Energy

Financial

Technology

Utilities



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Mining the Donations Database

More f requent donors (to p) tend to give smaller current donation amounts (shif t to lef t)

Fig 4.2.4

$0 $50 $100

Size of current donation

N u m b

e r o f

p r e v i o u s

g i f t s

p a s t 2 y e a r s

1

2

3

4+



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Example: Business Failures

Per million peo ple, by state90th percentile is 432.4

50th percentile is 260.2

0%

50%

100%

0 100 200 300 400 500 600 700

Failures

C u m u

l a t i v e P e r c e n t

Fig 4.2.9



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Example: Business Failures

Compare histogram, box plot, and CDF

Histogram

Box plot

CDF

0

10

0 500Failures

0 500Failures

0%

100%

0 500Failures

Fig 4.2.10

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Statistics- Landmark Summaries Interpreting Typical Values and Percentiles