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CS 147:Computer Systems Performance AnalysisSummarizing Variability and Determining Distributions
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CS 147:Computer Systems Performance AnalysisSummarizing Variability and Determining Distributions
2015
-06-
15
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
2015
-06-
15
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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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
2015
-06-
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CS147Introduction
Summarizing Variability
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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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?
2015
-06-
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CS147Introduction
Why Is Variability Important?
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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Indices of Dispersion
I Measures of how much a data set variesI RangeI Variance and standard deviationI PercentilesI Semi-interquartile rangeI Mean absolute deviation
2015
-06-
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CS147Introduction
Indices of Dispersion
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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CS147Indices of Dispersion
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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CS147Indices of Dispersion
Variance, Standard Deviation, C.V.Variance (and Its Cousins)
Indices of Dispersion Variance, Standard Deviation, C.V.
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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CS147Indices of Dispersion
Variance, Standard Deviation, C.V.Variance Example
Indices of Dispersion Variance, Standard Deviation, C.V.
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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CS147Indices of Dispersion
Variance, Standard Deviation, C.V.Standard Deviation
Indices of Dispersion Variance, Standard Deviation, C.V.
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
2015
-06-
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CS147Indices of Dispersion
Variance, Standard Deviation, C.V.Standard Deviation Example
Indices of Dispersion Variance, Standard Deviation, C.V.
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.
2015
-06-
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CS147Indices of Dispersion
Variance, Standard Deviation, C.V.Coefficient of Variation
Indices of Dispersion Variance, Standard Deviation, C.V.
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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CS147Indices of Dispersion
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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CS147Indices of Dispersion
QuantilesPercentiles
Indices of Dispersion Quantiles
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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CS147Indices of Dispersion
QuantilesRelatives of Percentiles
Indices of Dispersion Quantiles
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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CS147Indices of Dispersion
QuantilesCalculating Quantiles
Indices of Dispersion 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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CS147Indices of Dispersion
QuantilesQuartile Example
Indices of Dispersion Quantiles
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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CS147Indices of Dispersion
QuantilesInterquartile Range
Indices of Dispersion Quantiles
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
2015
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CS147Indices of Dispersion
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)
2015
-06-
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CS147Indices of Dispersion
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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CS147Indices of Dispersion
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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CS147Indices of Dispersion
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
But always remember what you’re looking for2015
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CS147Indices of Dispersion
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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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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CS147Identifying Distributions
Finding a Distribution for Datasets
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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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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CS147Identifying Distributions
Methods of Determining a Distribution
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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CS147Identifying Distributions
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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CS147Identifying Distributions
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
2015
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CS147Identifying Distributions
Kernel Density EstimationKernel Density Estimation
Identifying Distributions Kernel 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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CS147Identifying Distributions
Kernel Density EstimationKDE Equations
Identifying Distributions Kernel Density Estimation
KDE Intuition (Rectangular)
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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Identifying Distributions Kernel Density Estimation
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25
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Rectangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Rectangular)
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KDE Intuition (Rectangular)
0 5 10 15 20
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0 5 10 15 20
0
5
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Rectangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Rectangular)
0 5 10 15 20
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5
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30 / 49
KDE Intuition (Rectangular)
0 5 10 15 20
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5
10
15
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0 5 10 15 20
0
5
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Rectangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
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25
0 5 10 15 20
0
5
10
15
20
25
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
5
10
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20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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5
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
5
10
15
20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
5
10
15
20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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KDE Intuition (Triangular)
0 5 10 15 20
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5
10
15
20
25
0 5 10 15 20
0
5
10
15
20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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5
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
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5
10
15
20
25
0 5 10 15 20
0
5
10
15
20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
5
10
15
20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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5
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31 / 49
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
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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31 / 49
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
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
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0 5 10 15 20
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31 / 49
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
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
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31 / 49
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
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
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25
0 5 10 15 20
0
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31 / 49
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
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
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0 5 10 15 20
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
5
10
15
20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
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31 / 49
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
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
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0 5 10 15 20
0
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20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
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15
20
25
0 5 10 15 20
0
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20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
5
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15
20
25
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
5
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15
20
25
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
10
15
20
25
0 5 10 15 20
0
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15
20
25
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
0
5
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15
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25
0 5 10 15 20
0
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20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
KDE Intuition (Triangular)
0 5 10 15 20
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31 / 49
KDE Intuition (Triangular)
0 5 10 15 20
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0 5 10 15 20
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2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Intuition (Triangular)
Identifying Distributions Kernel Density Estimation
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)
32 / 49
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)
2015
-06-
15
CS147Identifying Distributions
Kernel Density EstimationKDE Example
Identifying Distributions Kernel Density Estimation
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)
2015
-06-
15
CS147Identifying Distributions
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
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsQuantile-Quantile Plots
Identifying Distributions Quantile-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
35 / 49
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
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsObtaining Theoretical Quantiles
Identifying Distributions Quantile-Quantile Plots
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)
36 / 49
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)
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsInverting a Distribution
Identifying Distributions Quantile-Quantile Plots
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
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsIs Our Sample Data Set NormallyDistributed?
Identifying Distributions Quantile-Quantile Plots
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
38 / 49
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 1.646842015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsData For Example Normal Quantile-QuantilePlot
Identifying Distributions Quantile-Quantile Plots
Example Normal Quantile-Quantile Plot
-1 0 1
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Example Normal Quantile-Quantile Plot
-1 0 1
-500
0
500
1000
1500
2000
2500
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsExample Normal Quantile-Quantile Plot
Identifying Distributions Quantile-Quantile Plots
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?
40 / 49
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?
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsAnalysis
Identifying Distributions Quantile-Quantile Plots
Quantile-Quantile Plot of Partial Data
-1 0
0
50
100
41 / 49
Quantile-Quantile Plot of Partial Data
-1 0
0
50
100
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsQuantile-Quantile Plot of Partial Data
Identifying Distributions Quantile-Quantile Plots
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
42 / 49
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
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsAnalysis of Partial Data Plot
Identifying Distributions Quantile-Quantile Plots
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
rightI (Really need more data points)
I You can keep this up for a good, long time
42 / 49
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
rightI (Really need more data points)
I You can keep this up for a good, long time
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsAnalysis of Partial Data Plot
Identifying Distributions Quantile-Quantile Plots
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
rightI (Really need more data points)I You can keep this up for a good, long time
42 / 49
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
rightI (Really need more data points)I You can keep this up for a good, long time
2015
-06-
15
CS147Identifying Distributions
Quantile-Quantile PlotsAnalysis of Partial Data Plot
Identifying Distributions Quantile-Quantile Plots
Interpreting Quantile-Quantile Plots
Mnemonic: Q-Q plot shaped like “S” has Short tails;opposite has long ones.
43 / 49
Interpreting Quantile-Quantile Plots
Mnemonic: Q-Q plot shaped like “S” has Short tails;opposite has long ones.
2015
-06-
15
CS147Identifying Distributions
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
2015
-06-
15
CS147Statistics of Samples
Meaning of a SampleWhat is a Sample?
Statistics of Samples Meaning of 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
45 / 49
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
2015
-06-
15
CS147Statistics of Samples
Meaning of a SampleSample Statistics
Statistics of Samples Meaning of a Sample
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?
2015
-06-
15
CS147Statistics of Samples
Meaning of a SampleEstimating Population from Samples
Statistics of Samples Meaning of a Sample
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
47 / 49
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
2015
-06-
15
CS147Statistics of Samples
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
48 / 49
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
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueEstimating the Value of a Random Variable
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
48 / 49
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
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueEstimating the Value of a Random Variable
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 tall
I Yeah, rightI Safer to assume a range
48 / 49
Estimating the Value of a Random Variable
I How tall is Fred?I Suppose average human height is 170 cm
∴ Fred is 170 cm tall
I Yeah, rightI Safer to assume a range
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueEstimating the Value of a Random Variable
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
48 / 49
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
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueEstimating the Value of a Random Variable
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
48 / 49
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
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueEstimating the Value of a Random Variable
Statistics of Samples Guessing the True Value
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
49 / 49
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
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueConfidence Intervals
Statistics of Samples Guessing the True Value
Confidence 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
49 / 49
Confidence 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
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueConfidence Intervals
Statistics of Samples Guessing the True Value
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
49 / 49
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
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueConfidence Intervals
Statistics of Samples Guessing the True Value
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
49 / 49
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
2015
-06-
15
CS147Statistics of Samples
Guessing the True ValueConfidence Intervals