CS 147: Computer Systems Performance Analysisgeoff/classes/hmc.cs147.201201/...INormalizes units of...

CS 147:Computer Systems Performance AnalysisSummarizing Variability and Determining Distributions

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CS 147:Computer Systems Performance AnalysisSummarizing Variability and Determining Distributions

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CS147

Overview

Introduction

Indices of DispersionRangeVariance, Standard Deviation, C.V.QuantilesMiscellaneous MeasuresChoosing a Measure

Identifying DistributionsHistogramsKernel Density EstimationQuantile-Quantile Plots

Statistics of SamplesMeaning of a SampleGuessing the True Value

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Overview

Introduction

Indices of DispersionRangeVariance, Standard Deviation, C.V.QuantilesMiscellaneous MeasuresChoosing a Measure

Identifying DistributionsHistogramsKernel Density EstimationQuantile-Quantile Plots

Statistics of SamplesMeaning of a SampleGuessing the True Value

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CS147

Overview

Introduction

Summarizing Variability

I A single number rarely tells entire story of a data setI Usually, you need to know how much the rest of the data set

varies from that index of central tendency

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I A single number rarely tells entire story of a data setI Usually, you need to know how much the rest of the data set

varies from that index of central tendency

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CS147Introduction


Introduction

Why Is Variability Important?

I Consider two Web servers:I Server A services all requests in 1 secondI Server B services 90% of all requests in .5 seconds

I But 10% in 55 secondsI Both have mean service times of 1 secondI But which would you prefer to use?

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I Consider two Web servers:I Server A services all requests in 1 secondI Server B services 90% of all requests in .5 seconds

I But 10% in 55 secondsI Both have mean service times of 1 secondI But which would you prefer to use?

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CS147Introduction


Introduction

Indices of Dispersion

I Measures of how much a data set variesI RangeI Variance and standard deviationI PercentilesI Semi-interquartile rangeI Mean absolute deviation

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I Measures of how much a data set variesI RangeI Variance and standard deviationI PercentilesI Semi-interquartile rangeI Mean absolute deviation

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CS147Introduction


Indices of Dispersion Range

Range

I Minimum & maximum values in data setI Can be tracked as data values arriveI Variability characterized by difference between minimum and

maximumI Often not useful, due to outliersI Minimum tends to go to zeroI Maximum tends to increase over timeI Not useful for unbounded variables

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Range

I Minimum & maximum values in data setI Can be tracked as data values arriveI Variability characterized by difference between minimum and

maximumI Often not useful, due to outliersI Minimum tends to go to zeroI Maximum tends to increase over timeI Not useful for unbounded variables20

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CS147Indices of Dispersion

RangeRange

Indices of Dispersion Range

Example of Range

I For data set 2, 5.4, -17, 2056, 445, -4.8, 84.3, 92, 27, -10I Maximum is 2056I Minimum is -17I Range is 2073I While arithmetic mean is 268

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Example of Range

I For data set 2, 5.4, -17, 2056, 445, -4.8, 84.3, 92, 27, -10I Maximum is 2056I Minimum is -17I Range is 2073I While arithmetic mean is 268

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RangeExample of Range

Indices of Dispersion Variance, Standard Deviation, C.V.

Variance (and Its Cousins)

I Sample variance is

s2 =1

n − 1

n∑i=1

(xi − x)2

I Expressed in units of the measured quantity, squaredI Which isn’t always easy to understand

I Standard deviation and coefficient of variation are derivedfrom variance

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Variance (and Its Cousins)

I Sample variance is

s2 =1

n − 1

n∑i=1

(xi − x)2

I Expressed in units of the measured quantity, squaredI Which isn’t always easy to understand

I Standard deviation and coefficient of variation are derivedfrom variance20

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Variance, Standard Deviation, C.V.Variance (and Its Cousins)


Variance Example

I For data set 2, 5.4, -17, 2056, 445, -4.8, 84.3, 92, 27, -10I Variance is 413746.6I You can see the problem with variance:

I Given a mean of 268, what does that variance indicate?

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Variance Example

I For data set 2, 5.4, -17, 2056, 445, -4.8, 84.3, 92, 27, -10I Variance is 413746.6I You can see the problem with variance:

I Given a mean of 268, what does that variance indicate?

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Variance, Standard Deviation, C.V.Variance Example


Standard Deviation

I Square root of the varianceI In same units as units of metricI So easier to compare to metric

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Standard Deviation

I Square root of the varianceI In same units as units of metricI So easier to compare to metric

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Variance, Standard Deviation, C.V.Standard Deviation


Standard Deviation Example

I For sample set we’ve been using, standard deviation is 643I Given mean of 268, standard deviation clearly shows lots of

variability from mean

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Standard Deviation Example

I For sample set we’ve been using, standard deviation is 643I Given mean of 268, standard deviation clearly shows lots of

variability from mean

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Variance, Standard Deviation, C.V.Standard Deviation Example


Coefficient of Variation

I Ratio of standard deviation to meanI Normalizes units of these quantities into ratio or percentageI Often abbreviated C.O.V. or C.V.

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Coefficient of Variation

I Ratio of standard deviation to meanI Normalizes units of these quantities into ratio or percentageI Often abbreviated C.O.V. or C.V.

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Variance, Standard Deviation, C.V.Coefficient of Variation


Coefficient of Variation Example

I For sample set we’ve been using, standard deviation is 643I Mean is 268I So C.O.V. is 643/268 ≈ 2.4

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Coefficient of Variation Example

I For sample set we’ve been using, standard deviation is 643I Mean is 268I So C.O.V. is 643/268 ≈ 2.4

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Variance, Standard Deviation, C.V.Coefficient of Variation Example

Indices of Dispersion Quantiles

Percentiles

I Specification of how observations fall into bucketsI E.g., 5-percentile is observation that is at the lower 5% of the

setI While 95-percentile is observation at the 95% boundary

I Useful even for unbounded variables

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Percentiles

I Specification of how observations fall into bucketsI E.g., 5-percentile is observation that is at the lower 5% of the

setI While 95-percentile is observation at the 95% boundary

I Useful even for unbounded variables

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QuantilesPercentiles


Relatives of Percentiles

I Quantiles - fraction between 0 and 1I Instead of percentageI Also called fractiles

I Deciles—percentiles at 10% boundariesI First is 10-percentile, second is 20-percentile, etc.

I Quartiles—divide data set into four partsI 25% of sample below first quartile, etc.I Second quartile is also median

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Relatives of Percentiles

I Quantiles - fraction between 0 and 1I Instead of percentageI Also called fractiles

I Deciles—percentiles at 10% boundariesI First is 10-percentile, second is 20-percentile, etc.

I Quartiles—divide data set into four partsI 25% of sample below first quartile, etc.I Second quartile is also median

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QuantilesRelatives of Percentiles


Calculating Quantiles

To estimate α-quantile:I First sort the setI Then take [(n − 1)α+ 1]th element

I 1-indexedI Round to nearest integer indexI Exception: for small sets, may be better to choose

“intermediate” value as is done for median

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Calculating Quantiles

To estimate α-quantile:I First sort the setI Then take [(n − 1)α+ 1]th element

I 1-indexedI Round to nearest integer indexI Exception: for small sets, may be better to choose

“intermediate” value as is done for median

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QuantilesCalculating Quantiles


Quartile Example

I For data set 2, 5.4, -17, 2056, 445, -4.8, 84.3, 92, 27, -10(10 observations)

I Sort it: -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I First quartile, Q1, is -4.8I Third quartile, Q3, is 92

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Quartile Example

I For data set 2, 5.4, -17, 2056, 445, -4.8, 84.3, 92, 27, -10(10 observations)

I Sort it: -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I First quartile, Q1, is -4.8I Third quartile, Q3, is 92

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QuantilesQuartile Example


Interquartile Range

I Yet another measure of dispersionI The difference between Q3 and Q1I Semi-interquartile range is half that:

SIQR =Q3 −Q1

2

I Often interesting measure of what’s going on in middle ofrange

I Basically indicates distance of quartiles from median

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Interquartile Range

I Yet another measure of dispersionI The difference between Q3 and Q1I Semi-interquartile range is half that:

SIQR =Q3 −Q1

2

I Often interesting measure of what’s going on in middle ofrange

I Basically indicates distance of quartiles from median2015

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QuantilesInterquartile Range


Semi-Interquartile Range Example

For data set -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I Q3 is 92I Q1 is -4.8

SIQR =Q3 −Q1

2=

92− (−4.8)2

= 48

I Compare to standard deviation of 643I Suggests that much of variability is caused by outliers

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Semi-Interquartile Range Example

For data set -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I Q3 is 92I Q1 is -4.8

SIQR =Q3 −Q1

2=

92− (−4.8)2

= 48

I Compare to standard deviation of 643I Suggests that much of variability is caused by outliers

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QuantilesSemi-Interquartile Range Example

Indices of Dispersion Miscellaneous Measures

Mean Absolute Deviation

I Yet another measure of variability

I Mean absolute deviation =1n

n∑i=1

|xi − x |

I Good for hand calculation (doesn’t require multiplication orsquare roots)

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Mean Absolute Deviation

I Yet another measure of variability

I Mean absolute deviation =1n

n∑i=1

|xi − x |

I Good for hand calculation (doesn’t require multiplication orsquare roots)

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Miscellaneous MeasuresMean Absolute Deviation

Indices of Dispersion Miscellaneous Measures

Mean Absolute Deviation Example

For data set -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I Mean absolute deviation is

110

10∑i=1

|xi − 268| = 393

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Mean Absolute Deviation Example

For data set -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I Mean absolute deviation is

110

10∑i=1

|xi − 268| = 393

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Miscellaneous MeasuresMean Absolute Deviation Example

Indices of Dispersion Choosing a Measure

Sensitivity To Outliers

I From most to least,I RangeI VarianceI Mean absolute deviationI Semi-interquartile range

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Sensitivity To Outliers

I From most to least,I RangeI VarianceI Mean absolute deviationI Semi-interquartile range

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Choosing a MeasureSensitivity To Outliers

Indices of Dispersion Choosing a Measure

So, Which Index of Dispersion Should I Use?

Bounded?Unimodal

Symmetrical?Percentiles or SIQR

Range C.O.V.

Yes Yes

No No

But always remember what you’re looking for

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So, Which Index of Dispersion Should I Use?

Bounded?Unimodal

Symmetrical?Percentiles or SIQR

Range C.O.V.

Yes Yes

No No

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Choosing a MeasureSo, Which Index of Dispersion Should I Use?

Identifying Distributions

Finding a Distribution for Datasets

I If a data set has a common distribution, that’s the best way tosummarize it

I Saying a data set is uniformly distributed is more informativethan just giving mean and standard deviation

I So how do you determine if your data set fits a distribution?

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I If a data set has a common distribution, that’s the best way tosummarize it

I Saying a data set is uniformly distributed is more informativethan just giving mean and standard deviation

I So how do you determine if your data set fits a distribution?

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CS147Identifying Distributions


Identifying Distributions

Methods of Determining a Distribution

I Plot a histogramI Kernel density estimationI Quantile-quantile plotI Statistical methods (not covered in this class)

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I Plot a histogramI Kernel density estimationI Quantile-quantile plotI Statistical methods (not covered in this class)

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Identifying Distributions Histograms

Plotting a Histogram

Suitable if you have relatively large number of data points

Procedure:1. Determine range of observations2. Divide range into buckets3. Count number of observations in each bucket4. Divide by total number of observations and plot as column

chart

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Plotting a Histogram

Suitable if you have relatively large number of data points

Procedure:1. Determine range of observations2. Divide range into buckets3. Count number of observations in each bucket4. Divide by total number of observations and plot as column

chart

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HistogramsPlotting a Histogram

Identifying Distributions Histograms

Problems With Histogram Approach

I Determining cell sizeI If too small, too few observations per cellI If too large, no useful details in plot

I If fewer than five observations in a cell, cell size is too small

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Problems With Histogram Approach

I Determining cell sizeI If too small, too few observations per cellI If too large, no useful details in plot

I If fewer than five observations in a cell, cell size is too small

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HistogramsProblems With Histogram Approach

Identifying Distributions Kernel Density Estimation

Kernel Density Estimation

I Basic idea: any observation represents probability of highnear near that observation

I Example:I Seeing 7 means pdf is high all around 7I Seeing 6.5 also means pdf is high near 7

I “Average out” observations to get smooth histogram

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Kernel Density Estimation

I Basic idea: any observation represents probability of highnear near that observation

I Example:I Seeing 7 means pdf is high all around 7I Seeing 6.5 also means pdf is high near 7

I “Average out” observations to get smooth histogram

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Kernel Density EstimationKernel Density Estimation


KDE Equations

I Want to estimate continuous p(x):

p̂(x) =1

nh

n∑i=1

K(

x − xi

h

)I Where K (x) is kernel functionI Must integrate to unity:

∫∞−∞ K (x)dx = 1

I Purpose is to select nearby samplesI h is bandwidth parameterI Controls how many nearby samples selectedI Large bandwidth⇒ more smoothing, less detail

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KDE Equations

I Want to estimate continuous p(x):

p̂(x) =1

nh

n∑i=1

K(

x − xi

h

)I Where K (x) is kernel functionI Must integrate to unity:

∫∞−∞ K (x)dx = 1

I Purpose is to select nearby samplesI h is bandwidth parameterI Controls how many nearby samples selectedI Large bandwidth⇒ more smoothing, less detail20

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Kernel Density EstimationKDE Equations


KDE Intuition (Rectangular)

0 5 10 15 20

0

5

10

15

20

25

0 5 10 15 20

0

5

10

15

20

25

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KDE Intuition (Rectangular)

0 5 10 15 20

0

5

10

15

20

25

0 5 10 15 20

0

5

10

15

20

25

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Kernel Density EstimationKDE Intuition (Rectangular)


KDE Intuition (Triangular)

0 5 10 15 20

0

5

10

15

20

25

0 5 10 15 20

0

5

10

15

20

25

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KDE Intuition (Triangular)

0 5 10 15 20

0

5

10

15

20

25

0 5 10 15 20

0

5

10

15

20

25

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Kernel Density EstimationKDE Intuition (Triangular)


KDE Example

I Sample data set: -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I One observation per sampleI KDE with Gaussian window (RHS dropped):

-50 0 50 100 150

0.00

0.01

0.02

0.03

p(x)

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KDE Example

I Sample data set: -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I One observation per sampleI KDE with Gaussian window (RHS dropped):

-50 0 50 100 150

0.00

0.01

0.02

0.03

p(x)

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Kernel Density EstimationKDE Example


KDE Example #2

I Same data setI Narrower Gaussian windowI (Again, RHS dropped):

-50 0 50 100 150

0.00

0.01

0.02

0.03

p(x)

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KDE Example #2

I Same data setI Narrower Gaussian windowI (Again, RHS dropped):

-50 0 50 100 150

0.00

0.01

0.02

0.03

p(x)

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Kernel Density EstimationKDE Example #2

Identifying Distributions Quantile-Quantile Plots

Quantile-Quantile Plots

I More suitable than KDE for small data setsI Basically, guess a distributionI Plot where quantiles of data should fall in that distribution

I Against where they actually fallI If plot is close to linear, data closely matches guessed

distribution

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Quantile-Quantile Plots

I More suitable than KDE for small data setsI Basically, guess a distributionI Plot where quantiles of data should fall in that distribution

I Against where they actually fallI If plot is close to linear, data closely matches guessed

distribution

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Quantile-Quantile PlotsQuantile-Quantile Plots


Obtaining Theoretical Quantiles

I Need to determine where quantiles should fall for a particulardistribution

I Requires inverting CDF for that distributionI Then determining quantiles for observed pointsI Then plugging quantiles into inverted CDF

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Obtaining Theoretical Quantiles

I Need to determine where quantiles should fall for a particulardistribution

I Requires inverting CDF for that distributionI Then determining quantiles for observed pointsI Then plugging quantiles into inverted CDF

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Quantile-Quantile PlotsObtaining Theoretical Quantiles


Inverting a Distribution

I Many common distributions have already been inverted (howconvenient...)

I For others that are hard to invert, tables and approximationsoften available (nearly as convenient)

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Inverting a Distribution

I Many common distributions have already been inverted (howconvenient...)

I For others that are hard to invert, tables and approximationsoften available (nearly as convenient)

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Quantile-Quantile PlotsInverting a Distribution


Is Our Sample Data Set Normally Distributed?

I Our data set was -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I Does this match normal distribution?I Normal distribution doesn’t invert nicely

I But there is an approximation:

xi = 4.91(

q 0.14i − (1− qi)

0.14)

I Or invert numerically

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Is Our Sample Data Set Normally Distributed?

I Our data set was -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056I Does this match normal distribution?I Normal distribution doesn’t invert nicely

I But there is an approximation:

xi = 4.91(

q 0.14i − (1− qi)

0.14)

I Or invert numerically

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Quantile-Quantile PlotsIs Our Sample Data Set NormallyDistributed?


Data For Example Normal Quantile-Quantile Plot

i qi yi xi1 0.05 -17.0 -1.646842 0.15 -10.0 -1.034813 0.25 -4.8 -0.672344 0.35 2.0 -0.383755 0.45 5.4 -0.125106 0.55 27.0 0.125107 0.65 84.3 0.383758 0.75 92.0 0.672349 0.85 445.0 1.03481

10 0.95 2056.0 1.64684

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Data For Example Normal Quantile-Quantile Plot

i qi yi xi1 0.05 -17.0 -1.646842 0.15 -10.0 -1.034813 0.25 -4.8 -0.672344 0.35 2.0 -0.383755 0.45 5.4 -0.125106 0.55 27.0 0.125107 0.65 84.3 0.383758 0.75 92.0 0.672349 0.85 445.0 1.03481

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Quantile-Quantile PlotsData For Example Normal Quantile-QuantilePlot


Example Normal Quantile-Quantile Plot

-1 0 1

-500

0

500

1000

1500

2000

2500

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Example Normal Quantile-Quantile Plot

-1 0 1

-500

0

500

1000

1500

2000

2500

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Quantile-Quantile PlotsExample Normal Quantile-Quantile Plot


Analysis

I Definitely not normalI Because it isn’t linearI Tail at high end is too long for normal

I But perhaps the lower part of graph is normal?

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Analysis

I Definitely not normalI Because it isn’t linearI Tail at high end is too long for normal

I But perhaps the lower part of graph is normal?

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Quantile-Quantile PlotsAnalysis


Quantile-Quantile Plot of Partial Data

-1 0

0

50

100

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Quantile-Quantile Plot of Partial Data

-1 0

0

50

100

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Quantile-Quantile PlotsQuantile-Quantile Plot of Partial Data


Analysis of Partial Data Plot

I Again, at highest points it doesn’t fit normal distributionI But at lower points it fits somewhat wellI So, again, this distribution looks like normal with longer tail to

right

I (Really need more data points)I You can keep this up for a good, long time

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right

I (Really need more data points)I You can keep this up for a good, long time

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Quantile-Quantile PlotsAnalysis of Partial Data Plot




rightI (Really need more data points)

I You can keep this up for a good, long time

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rightI (Really need more data points)

I You can keep this up for a good, long time

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rightI (Really need more data points)I You can keep this up for a good, long time

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rightI (Really need more data points)I You can keep this up for a good, long time

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Interpreting Quantile-Quantile Plots

Mnemonic: Q-Q plot shaped like “S” has Short tails;opposite has long ones.

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Interpreting Quantile-Quantile Plots

Mnemonic: Q-Q plot shaped like “S” has Short tails;opposite has long ones.

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Quantile-Quantile PlotsInterpreting Quantile-Quantile Plots

Statistics of Samples Meaning of a Sample

What is a Sample?

I How tall is a human?I Could measure every person in the worldI Or could measure everyone in this room

I Population has parametersI Real and meaningful

I Sample has statisticsI Drawn from populationI Inherently erroneous

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What is a Sample?

I How tall is a human?I Could measure every person in the worldI Or could measure everyone in this room

I Population has parametersI Real and meaningful

I Sample has statisticsI Drawn from populationI Inherently erroneous

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CS147Statistics of Samples

Meaning of a SampleWhat is a Sample?


Sample Statistics

I How tall is a human?I People in B126 have a mean heightI People in Edwards have a different mean

I Sample mean is itself a random variableI Has own distribution

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Sample Statistics

I How tall is a human?I People in B126 have a mean heightI People in Edwards have a different mean

I Sample mean is itself a random variableI Has own distribution

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Meaning of a SampleSample Statistics


Estimating Population from Samples

I How tall is a human?I Measure everybody in this roomI Calculate sample mean xI Assume population mean µ equals x

I What is the error in our estimate?

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Estimating Population from Samples

I How tall is a human?I Measure everybody in this roomI Calculate sample mean xI Assume population mean µ equals x

I What is the error in our estimate?

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Meaning of a SampleEstimating Population from Samples


Estimating Error

I Sample mean is a random variable⇒ Mean has some distribution∴ Multiple sample means have “mean of means”

I Knowing distribution of means, we can estimate error

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Estimating Error

I Sample mean is a random variable⇒ Mean has some distribution∴ Multiple sample means have “mean of means”

I Knowing distribution of means, we can estimate error

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Meaning of a SampleEstimating Error

Statistics of Samples Guessing the True Value

Estimating the Value of a Random Variable

I How tall is Fred?

I Suppose average human height is 170 cm∴ Fred is 170 cm tallI Yeah, right

I Safer to assume a range

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I How tall is Fred?

I Suppose average human height is 170 cm∴ Fred is 170 cm tallI Yeah, right


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Guessing the True ValueEstimating the Value of a Random Variable



I How tall is Fred?I Suppose average human height is 170 cm

∴ Fred is 170 cm tallI Yeah, right


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∴ Fred is 170 cm tall

I Yeah, rightI Safer to assume a range

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∴ Fred is 170 cm tall

I Yeah, rightI Safer to assume a range

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Confidence Intervals

I How tall is Fred?

I Suppose 90% of humans are between 155 and 190 cm∴ Fred is between 155 and 190 cm

I We are 90% confident that Fred is between 155 and 190 cm

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I How tall is Fred?

I Suppose 90% of humans are between 155 and 190 cm∴ Fred is between 155 and 190 cm


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Guessing the True ValueConfidence Intervals



I How tall is Fred?I Suppose 90% of humans are between 155 and 190 cm

∴ Fred is between 155 and 190 cmI We are 90% confident that Fred is between 155 and 190 cm

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I How tall is Fred?I Suppose 90% of humans are between 155 and 190 cm

∴ Fred is between 155 and 190 cmI We are 90% confident that Fred is between 155 and 190 cm

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I How tall is Fred?I Suppose 90% of humans are between 155 and 190 cm∴ Fred is between 155 and 190 cm


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I How tall is Fred?I Suppose 90% of humans are between 155 and 190 cm∴ Fred is between 155 and 190 cm


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CS 147: Computer Systems Performance Analysisgeoff/classes/hmc.cs147.201201/...INormalizes units of...

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