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Math 1710Class 12

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A SimpleMedian andQuartileExample

Empirical Ruleand Outliers

Proof ofNormalApproximation

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Normal Distributions, Outliers, and Summary StatisticsDr. Back

Sep. 23, 2009

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Per Capita CO2 Emissions

Units are metric tons per person per year.

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8 Most Populous Countries in the World:

Country tons/yr.China 2.3India 1.1

US 19.7Indonesia 1.2

Brazil 1.8Russia 9.8

Pakistan .7? .2

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8 Most Populous Countries in the World:

Country tons/yr.China 2.3India 1.1

US 19.7Indonesia 1.2

Brazil 1.8Russia 9.8

Pakistan .7Bangladesh .2

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In order:

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8 19.7

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In order with positions: (n = 8)

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8 19.7Posn. : 1 2 3 4 5 6 7 8

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In order with positions: (n = 8)

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8 19.7Posn. : 1 2 3 4 5 6 7 8

The median is the middle value.A basic measure of center.When the sample size is even, we average the two middlevalues.

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The median is the middle value.A basic measure of center.When the sample size is even, we average the two middlevalues.Sample size (n = 8)

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8 19.7Posn. : 1 2 3 4 5 6 7 8

median =1.8 + 1.2

2= 1.5.

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tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8 19.7

Histogram of all 8

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tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8 19.7

Histogram of all 8

The US value of 19.7 is not in keeping with the rest if the data.Such a value is called an outlier.

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Histogram of all but US. (n=7)

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Histogram of all but US. (n=7)

Removal of the outlier gives a much more revealing histogram.

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Without the outlier: (n = 7)

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8Posn. : 1 2 3 4 5 6 7

With n odd, the median is just the middle value of 1.2.

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Without the outlier: (n = 7)

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8Posn. : 1 2 3 4 5 6 7

With n odd, the median is just the middle value of 1.2.

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The first quartile Q1 or 25th percentile is defined to be themedian of the bottom half of our data.

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The first quartile Q1 or 25th percentile is defined to be themedian of the bottom half of our data.For a data set of odd sample size, we do not include themedian in the bottom half:

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The first quartile Q1 or 25th percentile is defined to be themedian of the bottom half of our data.For a data set of odd sample size, we do not include themedian in the bottom half:Without the outlier: (n = 7)

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8Posn. : 1 2 3 4 5 6 7

Q1 = .7

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tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8Posn. : 1 2 3 4 5 6 7

Q1 = .7

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tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8Posn. : 1 2 3 4 5 6 7

Q1 = .7

(And Q3 = 2.3.)The convention about not including the middle changed in the3rd edition of our text.

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For the original data set: (n = 8)

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8 19.7Posn. : 1 2 3 4 5 6 7 8

Q1 =.7 + 1.1

2= .9 and Q3 =

2.3 + 9.8

2= 6.05

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The 5-number summary:

min, Q1, median, Q3, maxall 8 .2, .9, 1.5, 6.05, 19.7

w/o US .2, .7, 1.2, 2.3, 9.8

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w/o US .2, .7, 1.2, 2.3, 9.8

Boxplot - Graphical form of the 5 number summary:

all - n=8

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w/o US .2, .7, 1.2, 2.3, 9.8

Boxplot - Graphical form of the 5 number summary:

Without the Outlier - n=7

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Interquartile Range: IQR=Q3 − Q1.A basic measure of spread.

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Interquartile Range: IQR=Q3 − Q1.A basic measure of spread.The median and IQR are usually little affected by outliers.“Resistant to Outliers”

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Interquartile Range: IQR=Q3 − Q1.A basic measure of spread.The median and IQR are usually little affected by outliers.“Resistant to Outliers”

n Q1 med Q3 IQR

with outlier 8 .9 1.5 6.05 5.15w/o outlier 7 .7 1.2 2.3 1.6

It is mostly because of the small sample size that the medianand IQR change as much as they do here.

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Spreadsheets do“wild” things when computing quartiles:

tons/yr : .2 .7 1.1 1.2 1.8 2.3 9.8 19.7

Open Office, an Excel Clone

One way to get such numbers:With 8 numbers there are 7 intervals in between..7 is the 100

7 %ile.1.1 is the 200

7 %ile.25 = 1

4 ·1007 + 3

4 ·2007

So the 25th %ile is 14 · .7 + 3

4 · 1.1 = 1.

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Forbes

790 CEO Salaries 1994

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Forbes

790 CEO Salaries 1994

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Forbes

CEO Salaries 1994 Boxplot

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Forbes

All But Top 9 CEO Salaries 1994

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Forbes

All But Top 9 CEO Salaries 1994

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Forbes

CEO Salaries 1994 Boxplot

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Comparing Measures of Center

n Mean x̄ MedianAll 790 2.82M 1.304MWithout top 9 781 2.24M 1.296M

Means heavily affected by outliers.

Medians resistant to outliers.

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Math 1710Class 12

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Comparing Measures of Spread

n Std. Dev.s IQRAll 790 8.32M 1.731MWithout top 9 781 2.724M 1.662M

Std. Dev heavily affected by outliers.

IQR resistant to outliers.

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Math 1710Class 12

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Math 1710Class 12

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Forbes

Bottom 638 CEO Salaries 1994

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Forbes

Bottom 638 CEO Salaries 1994

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Forbes

Bottom 638 CEO Salaries 1994 Boxplot

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Comparing 84th Percentiles

For a normal distribution N(µ, σ), this would be at µ + σ.

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n = 790 n = 781 n = 638

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For a normal distribution N(µ, σ), this would be at µ + σ.n = 790 n = 781 n = 638

Both n = 790 and n = 781 are strongly skewed to the rightwith lots of outliers.

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n s x̄ 84%ile x̄ + s

All 790 8.32M 2.82M 3.392M 11.14MWithout top 9 781 2.724M 2.24M 3.347M 4.97MBottom 638 638 .689M 1.25M 2.076M 1.94M

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Agreement of x̄ + s with 84%ile is poor for the first two.

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Agreement of x̄ + s with 84%ile is poor for the first two.

Agreement of x̄ + s with 84%ile is good when n = 638.

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Comparing IQR’s

For a normal distribution N(µ, σ),the quartiles are at µ± .675−σ,so the IQR would be at 1.35σ.

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Comparing IQR’s

For a normal distribution N(µ, σ),the quartiles are at µ± .675−σ,so the IQR would be at 1.35σ.n = 790 n = 781 n = 638

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Comparing IQR’s

For a normal distribution N(µ, σ),the quartiles are at µ± .675−σ,so the IQR would be at 1.35σ.n = 790 n = 781 n = 638

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Comparing IQR’s

n Std. Dev. s x̄ IQR 1.35s

All 790 8.32M 2.82M 1.73M 11.23MWithout top 9 781 2.724M 2.24M 1.66M 3.68MBottom 638 638 .689M 1.25M .979M .930M

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Comparing IQR’s

Agreement of 1.35s with IQR is poor for the first two.

Math 1710Class 12

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Comparing IQR’s

Agreement of 1.35s with IQR is poor for the first two.

Agreement of 1.35s with IQR is good when n = 638.

Math 1710Class 12

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Comparing 97.7%iles

For a normal distribution N(µ, σ), this would be at µ + 2σ.

Math 1710Class 12

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Comparing 97.7%iles

n = 790 n = 781 n = 638

Math 1710Class 12

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Comparing 97.7%iles

For a normal distribution N(µ, σ), this would be at µ + 2σ.n = 790 n = 781 n = 638

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Comparing 97.7%iles

n s x̄ 97.7%ile x̄ + 2s

All 790 8.32M 2.82M 14.85M 19.46MWithout top 9 781 2.724M 2.24 12.33M 7.68MBottom 638 638 .689M 1.25M 2.79M 2.63M

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Comparing 97.7%iles

n s x̄ 97.7%ile x̄ + 2s

Agreement of x̄ + 2s with 97.7%ile is poor for the first two.

Math 1710Class 12

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Comparing 97.7%iles

n s x̄ 97.7%ile x̄ + 2s

Agreement of x̄ + 2s with 97.7%ile is poor for the first two.

Agreement of x̄ + 2s with 97.7%ile is good when n = 638.

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Normal Distribution Formula

N(0,1)

f (x) =1√2π

e−x2

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N(µ, σ)

f (x) =1√2πσ

e−(x−µ)2

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N(µ, σ)

f (x) =1√2πσ

e−(x−µ)2

These formulas and the following argument are far above thebasic level of our course.

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Why Normal Out of Binomial?

Suppose X ∼ Binom(2m, .5)

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Suppose X ∼ Binom(2m, .5)µ = m and σ =

√.5m.

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Suppose X ∼ Binom(2m, .5)We want to understand why X ∼ N(m,

√.5m) approximately.

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ak = P(X = m + k) =

)(.5)2m

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ak = P(X = m + k) =

)(.5)2m

The z-score of m + k is k√

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ak = P(X = m + k) =

)(.5)2m

The z-score of m + k is k√

So we want to show ak ∼ ce−k2

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ak = P(X = m + k) =

)(.5)2m

So we want to show ak ∼ ce−k2

m .i.e. ln ak ∼ ln c − k2

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ak = P(X = m + k) =

)(.5)2m

i.e. ln ak ∼ ln c − k2

m .Strategy: Compare ak to a0 using the approximationln (1 + x) ∼ x for x small.

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ak = P(X = m + k) =

)(.5)2m

ak =(2m)!(.5)2m

(m + k)!(m − k)!= a0

(m)(m − 1) . . . (m − k + 1)

(m + k)(m + k − 1) . . . (m + 1)

= a0(1)(1− 1

m ) . . . (1− k−1m )

(1 + km )(1 + k−1

m ) . . . (1 + 1m )

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ak = P(X = m + k) =

)(.5)2m

ak =(2m)!(.5)2m

(m + k)!(m − k)!= a0

(m)(m − 1) . . . (m − k + 1)

(m + k)(m + k − 1) . . . (m + 1)

= a0(1)(1− 1

m ) . . . (1− k−1m )

(1 + km )(1 + k−1

m ) . . . (1 + 1m )

So using ln (1 + x) ∼ x ,

ln ak ∼ ln a0 − 2

m+ . . . +

k − 1

)− k

Math 1710Class 12

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ak = P(X = m + k) =

)(.5)2m

m .So using ln (1 + x) ∼ x ,

m+ . . . +

k − 1

)− k

But 1 + 2 + . . . k − 1 = k(k−1)2 , so

Math 1710Class 12

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ak = P(X = m + k) =

)(.5)2m

m .So using ln (1 + x) ∼ x ,

m+ . . . +

k − 1

)− k

But 1 + 2 + . . . k − 1 = k(k−1)2 , so

ln ak ∼ ln a0 −k2

as desired.

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Math 1710Class 12

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