Page 1
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Descriptive Statistics: Part 2/2 (Ch 3)
Will Landau
Iowa State University
January 24, 2013
© Will Landau Iowa State University January 24, 2013 1 / 26
Page 2
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Outline
Boxplots
Quantile-Quantile (QQ) Plots
Theoretical Quantile-Quantile Plots
Numerical Summaries
Parameters
© Will Landau Iowa State University January 24, 2013 2 / 26
Page 3
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Generic Boxplot
© Will Landau Iowa State University January 24, 2013 3 / 26
Page 4
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Example: bullet data
© Will Landau Iowa State University January 24, 2013 4 / 26
Page 5
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Example: bullet data (230-grain bullets)
© Will Landau Iowa State University January 24, 2013 5 / 26
Page 6
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Example: bullet data
© Will Landau Iowa State University January 24, 2013 6 / 26
Page 7
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Outline
Boxplots
Quantile-Quantile (QQ) Plots
Theoretical Quantile-Quantile Plots
Numerical Summaries
Parameters
© Will Landau Iowa State University January 24, 2013 7 / 26
Page 8
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
I Quantile-quantile (QQ) plot: a scatterplot of thesorted values of one dataset on the sorted values ofanother dataset.
I This plot is used to tell if the distributional shapes ofthe datasets are the same or different.
I If the points in the plot lie in a straight line, thedistributional shapes are the same.
I Otherwise, the shapes are different.
I The datasets must be univariate, numerical, and of thesame size.
© Will Landau Iowa State University January 24, 2013 8 / 26
Page 9
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
I Quantile-quantile (QQ) plot: a scatterplot of thesorted values of one dataset on the sorted values ofanother dataset.
I This plot is used to tell if the distributional shapes ofthe datasets are the same or different.
I If the points in the plot lie in a straight line, thedistributional shapes are the same.
I Otherwise, the shapes are different.
I The datasets must be univariate, numerical, and of thesame size.
© Will Landau Iowa State University January 24, 2013 8 / 26
Page 10
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
I Quantile-quantile (QQ) plot: a scatterplot of thesorted values of one dataset on the sorted values ofanother dataset.
I This plot is used to tell if the distributional shapes ofthe datasets are the same or different.
I If the points in the plot lie in a straight line, thedistributional shapes are the same.
I Otherwise, the shapes are different.
I The datasets must be univariate, numerical, and of thesame size.
© Will Landau Iowa State University January 24, 2013 8 / 26
Page 11
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
I Quantile-quantile (QQ) plot: a scatterplot of thesorted values of one dataset on the sorted values ofanother dataset.
I This plot is used to tell if the distributional shapes ofthe datasets are the same or different.
I If the points in the plot lie in a straight line, thedistributional shapes are the same.
I Otherwise, the shapes are different.
I The datasets must be univariate, numerical, and of thesame size.
© Will Landau Iowa State University January 24, 2013 8 / 26
Page 12
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
I Quantile-quantile (QQ) plot: a scatterplot of thesorted values of one dataset on the sorted values ofanother dataset.
I This plot is used to tell if the distributional shapes ofthe datasets are the same or different.
I If the points in the plot lie in a straight line, thedistributional shapes are the same.
I Otherwise, the shapes are different.
I The datasets must be univariate, numerical, and of thesame size.
© Will Landau Iowa State University January 24, 2013 8 / 26
Page 13
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Example: bullet data
© Will Landau Iowa State University January 24, 2013 9 / 26
Page 14
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Example: bullet data
I I can make a QQ plot of the bullet data by plotting thesorted 200-grain depths against the sorted 230-grain depths.
I The points lie in approximately a straight line, so the200-grain depths are similarly shaped in distribution to the230-grain depths.
© Will Landau Iowa State University January 24, 2013 10 / 26
Page 15
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Example: bullet data
I I can make a QQ plot of the bullet data by plotting thesorted 200-grain depths against the sorted 230-grain depths.
I The points lie in approximately a straight line, so the200-grain depths are similarly shaped in distribution to the230-grain depths.
© Will Landau Iowa State University January 24, 2013 10 / 26
Page 16
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Outline
Boxplots
Quantile-Quantile (QQ) Plots
Theoretical Quantile-Quantile Plots
Numerical Summaries
Parameters
© Will Landau Iowa State University January 24, 2013 11 / 26
Page 17
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Theoretical quantile-quantile (QQ) plotsI Theoretical quantile-quantile (QQ) plot: a
scatterplot with:I The sorted values x1, x2, . . . xn of some real data set on
the x axis.I Q( 1−.5
n ),Q( 2−.5n ), . . . ,Q( n−.5
n ) on the y axis.I Q is some theoretical quantile function: the quantile
function we would expect from a dataset if thatdataset had a certain shape.
I Example theoretical quantile functions:I “Standard” bell-shaped data should have:
Q(p) ≈ 4.9(p0.14 − (1− p)0.14)
I “Exponentially distributed” data (a kind of highlyright-skewed data) should have:
Q(p) ≈ −λ−1 log(1− p)
where λ is some constant.
© Will Landau Iowa State University January 24, 2013 12 / 26
Page 18
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Theoretical quantile-quantile (QQ) plotsI Theoretical quantile-quantile (QQ) plot: a
scatterplot with:I The sorted values x1, x2, . . . xn of some real data set on
the x axis.I Q( 1−.5
n ),Q( 2−.5n ), . . . ,Q( n−.5
n ) on the y axis.I Q is some theoretical quantile function: the quantile
function we would expect from a dataset if thatdataset had a certain shape.
I Example theoretical quantile functions:I “Standard” bell-shaped data should have:
Q(p) ≈ 4.9(p0.14 − (1− p)0.14)
I “Exponentially distributed” data (a kind of highlyright-skewed data) should have:
Q(p) ≈ −λ−1 log(1− p)
where λ is some constant.
© Will Landau Iowa State University January 24, 2013 12 / 26
Page 19
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Theoretical quantile-quantile (QQ) plotsI Theoretical quantile-quantile (QQ) plot: a
scatterplot with:I The sorted values x1, x2, . . . xn of some real data set on
the x axis.I Q( 1−.5
n ),Q( 2−.5n ), . . . ,Q( n−.5
n ) on the y axis.I Q is some theoretical quantile function: the quantile
function we would expect from a dataset if thatdataset had a certain shape.
I Example theoretical quantile functions:I “Standard” bell-shaped data should have:
Q(p) ≈ 4.9(p0.14 − (1− p)0.14)
I “Exponentially distributed” data (a kind of highlyright-skewed data) should have:
Q(p) ≈ −λ−1 log(1− p)
where λ is some constant.
© Will Landau Iowa State University January 24, 2013 12 / 26
Page 20
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Theoretical quantile-quantile (QQ) plotsI Theoretical quantile-quantile (QQ) plot: a
scatterplot with:I The sorted values x1, x2, . . . xn of some real data set on
the x axis.I Q( 1−.5
n ),Q( 2−.5n ), . . . ,Q( n−.5
n ) on the y axis.I Q is some theoretical quantile function: the quantile
function we would expect from a dataset if thatdataset had a certain shape.
I Example theoretical quantile functions:I “Standard” bell-shaped data should have:
Q(p) ≈ 4.9(p0.14 − (1− p)0.14)
I “Exponentially distributed” data (a kind of highlyright-skewed data) should have:
Q(p) ≈ −λ−1 log(1− p)
where λ is some constant.
© Will Landau Iowa State University January 24, 2013 12 / 26
Page 21
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Theoretical quantile-quantile (QQ) plotsI Theoretical quantile-quantile (QQ) plot: a
scatterplot with:I The sorted values x1, x2, . . . xn of some real data set on
the x axis.I Q( 1−.5
n ),Q( 2−.5n ), . . . ,Q( n−.5
n ) on the y axis.I Q is some theoretical quantile function: the quantile
function we would expect from a dataset if thatdataset had a certain shape.
I Example theoretical quantile functions:I “Standard” bell-shaped data should have:
Q(p) ≈ 4.9(p0.14 − (1− p)0.14)
I “Exponentially distributed” data (a kind of highlyright-skewed data) should have:
Q(p) ≈ −λ−1 log(1− p)
where λ is some constant.
© Will Landau Iowa State University January 24, 2013 12 / 26
Page 22
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Theoretical quantile-quantile (QQ) plotsI Theoretical quantile-quantile (QQ) plot: a
scatterplot with:I The sorted values x1, x2, . . . xn of some real data set on
the x axis.I Q( 1−.5
n ),Q( 2−.5n ), . . . ,Q( n−.5
n ) on the y axis.I Q is some theoretical quantile function: the quantile
function we would expect from a dataset if thatdataset had a certain shape.
I Example theoretical quantile functions:I “Standard” bell-shaped data should have:
Q(p) ≈ 4.9(p0.14 − (1− p)0.14)
I “Exponentially distributed” data (a kind of highlyright-skewed data) should have:
Q(p) ≈ −λ−1 log(1− p)
where λ is some constant.
© Will Landau Iowa State University January 24, 2013 12 / 26
Page 23
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Theoretical quantile-quantile (QQ) plotsI Theoretical quantile-quantile (QQ) plot: a
scatterplot with:I The sorted values x1, x2, . . . xn of some real data set on
the x axis.I Q( 1−.5
n ),Q( 2−.5n ), . . . ,Q( n−.5
n ) on the y axis.I Q is some theoretical quantile function: the quantile
function we would expect from a dataset if thatdataset had a certain shape.
I Example theoretical quantile functions:I “Standard” bell-shaped data should have:
Q(p) ≈ 4.9(p0.14 − (1− p)0.14)
I “Exponentially distributed” data (a kind of highlyright-skewed data) should have:
Q(p) ≈ −λ−1 log(1− p)
where λ is some constant.
© Will Landau Iowa State University January 24, 2013 12 / 26
Page 24
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Normal quantile-quantile (QQ) Plots
I Normal quantile-quantile (QQ) plot: a theoreticalQQ plot where the quantile function, Q, is the quantilefunction for “standard” bell-shaped(normally-distributed) data.
I If the points in a normal QQ plot are in a straight line,the dataset in question is bell-shaped. Otherwise, thedata is not bell-shaped.
© Will Landau Iowa State University January 24, 2013 13 / 26
Page 25
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Normal quantile-quantile (QQ) Plots
I Normal quantile-quantile (QQ) plot: a theoreticalQQ plot where the quantile function, Q, is the quantilefunction for “standard” bell-shaped(normally-distributed) data.
I If the points in a normal QQ plot are in a straight line,the dataset in question is bell-shaped. Otherwise, thedata is not bell-shaped.
© Will Landau Iowa State University January 24, 2013 13 / 26
Page 26
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Example: towel breaking strength data
© Will Landau Iowa State University January 24, 2013 14 / 26
Page 27
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Example: towel breaking strength data
I The points are roughly straight-line-shaped, so thebreaking strength data is roughly bell-shaped.
© Will Landau Iowa State University January 24, 2013 15 / 26
Page 28
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
●
●
●
●●●
●
●
●●
●
●
●●
●
●
●
●
●
●
58 60 62 64 66 68 70 72
−2
−1
01
2Normal QQ plot: 200−grain bullet penetration
Sample Quantiles
The
oret
ical
Qua
ntile
s
© Will Landau Iowa State University January 24, 2013 16 / 26
Page 29
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Observations
I Since the points in the normal QQ plot are not quitearranged in a straight line, the 200-grain penetrationdepths are not quite bell-shaped. However, thedeparture from normality is not severe.
I The QQ plot of the bullet data from before revealedthat the 200-grain depths had the same distributionalshape as the 200-grain bullet depths. Thus, the230-grain bullet data is not quite bell-shaped either.
© Will Landau Iowa State University January 24, 2013 17 / 26
Page 30
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Observations
I Since the points in the normal QQ plot are not quitearranged in a straight line, the 200-grain penetrationdepths are not quite bell-shaped. However, thedeparture from normality is not severe.
I The QQ plot of the bullet data from before revealedthat the 200-grain depths had the same distributionalshape as the 200-grain bullet depths. Thus, the230-grain bullet data is not quite bell-shaped either.
© Will Landau Iowa State University January 24, 2013 17 / 26
Page 31
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Outline
Boxplots
Quantile-Quantile (QQ) Plots
Theoretical Quantile-Quantile Plots
Numerical Summaries
Parameters
© Will Landau Iowa State University January 24, 2013 18 / 26
Page 32
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 33
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 34
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 35
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 36
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 37
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 38
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 39
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 40
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 41
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 42
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 43
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Numerical summariesI Numerical summary (statistic)
I A number or list of numbers calculated using the data(and only the data).
I Numerical summaries highlight important features ofthe data (shape, center, spread, outliers).
I Examples:I Measures of center:
I Arithmetic meanI MedianI Mode
I Measures of spread:I Sample varianceI Sample standard deviationI RangeI IQR
I Measures of shape:I All the quantiles togetherI Skew (beyond the scope of the class)I Kurtosis (beyond the scope of the class)
© Will Landau Iowa State University January 24, 2013 19 / 26
Page 44
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 45
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 46
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 47
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 48
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 49
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 50
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 51
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 52
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 53
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 54
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 55
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of center
x1 x2 x3 x4 x5 x60 1 1 2 3 5
I Arithmetic mean:I x = 1
n
∑ni=1 xi
I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2
I Median: Q(0.5).I A shortcut to calculating Q(0.5) is:
I Q(0.5) = xdn/2e if n is oddI Q(0.5) = (xn/2 + xn/2+1)/2 if n is even.
I Here, Q(0.5) = (1 + 2)/2 = 1.5
I Mode (of a discrete or categorical dataset)I the most frequently-occurring valueI Here, mode = 1.
© Will Landau Iowa State University January 24, 2013 20 / 26
Page 56
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Measures of spread
x1 x2 x3 x4 x5 x6xi 0 1 1 2 3 5
i−.5n
.083 0.25 0.417 0.583 0.75 0.917
I Sample variance
I s2 = 1n−1
∑ni=1(xi − x)2
I Here, s2 = 16−1 [(0− 2)2 + (1− 2)2 + (1− 2)2 + (2−
2)2 + (3− 2)2 + (5− 2)2] = 3.2
I Sample standard deviation
I s =√s2 =
√1
n−1
∑ni=1(xi − x)2
I Here, s =√
3.2 = 1.7889
I Range
I Range = Maximum - MinimumI Here, Range = 5 - 0 = 5
I Interquartile range
I IQR = Q(0.75)− Q(0.25)I Here, IQR = 3− 1 = 2.
© Will Landau Iowa State University January 24, 2013 21 / 26
Page 57
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Your turn: sensitivity to outliers
Compare:
x1 x2 x3 x4 x5 x6xi 0 1 1 2 3 5i−.5n .083 0.25 0.417 0.583 0.75 0.917
to:
y1 y2 y3 y4 y5 y6xi 0 1 1 2 3 817263489i−.5n .083 0.25 0.417 0.583 0.75 0.917
which measures of center and spread differ drasticallybetween the xi ’s and the yi ’s? Which ones are about thesame?
© Will Landau Iowa State University January 24, 2013 22 / 26
Page 58
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Answers: sensitivity to outliers
Data xi yiMean 2 1.3621× 108
Median 1.5 1.5Mode 1 1Sample Variance 3.2 1.1132× 1017
Sample Std. Dev. 1.7889 3.3365× 108
Range 5 8.1726× 108
IQR 2 2
© Will Landau Iowa State University January 24, 2013 23 / 26
Page 59
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Sensitivity of numerical summaries
I Numerical summaries sensitive to outliers and skewness:
I MeanI Sample varianceI Sample standard deviationI Range
I Less sensitive numerical summaries:I MedianI ModeI IQR
© Will Landau Iowa State University January 24, 2013 24 / 26
Page 60
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Outline
Boxplots
Quantile-Quantile (QQ) Plots
Theoretical Quantile-Quantile Plots
Numerical Summaries
Parameters
© Will Landau Iowa State University January 24, 2013 25 / 26
Page 61
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 62
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 63
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 64
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 65
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 66
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 67
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 68
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 69
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 70
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26
Page 71
DescriptiveStatistics: Part2/2 (Ch 3)
Will Landau
Boxplots
Quantile-Quantile(QQ) Plots
TheoreticalQuantile-QuantilePlots
NumericalSummaries
Parameters
Statistics and parameters
I Statistic: numerical summary of data on the sampleI Parameter: numerical summary of a theoretical
distribution or data on an entire population.I Population mean (“true” mean):
I µ = 1N
∑Ni=1 xi if N the finite population size.
I x ≈ µ.
I Population variance (“true” variance):I σ2 = 1
N
∑Ni=1(xi − µ)2 if N the finite population size.
I s2 ≈ σ2.
I Population standard deviation (“true” standarddeviation):
I σ =√
1N
∑Ni=1(xi − µ)2 if N is the finite population
size.I s ≈ σ.
© Will Landau Iowa State University January 24, 2013 26 / 26