descriptive statistics: part 2/2 (ch 3) - github pages · descriptive statistics: part 2/2 (ch 3)...

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

Boxplots

NumericalSummaries

Parameters

Outline

Boxplots

Quantile-Quantile (QQ) Plots

Theoretical Quantile-Quantile Plots

Numerical Summaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Generic Boxplot

Will Landau

Boxplots

NumericalSummaries

Parameters

Example: bullet data

Will Landau

Boxplots

NumericalSummaries

Parameters

Example: bullet data (230-grain bullets)

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Outline

Boxplots

Numerical Summaries

Parameters

Will Landau

Boxplots

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

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

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

Boxplots

NumericalSummaries

Parameters

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

Boxplots

NumericalSummaries

Parameters

Outline

Boxplots

Numerical Summaries

Parameters

Will Landau

Boxplots

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

Boxplots

NumericalSummaries

Parameters

n ),Q( 2−.5n ), . . . ,Q( n−.5

Q(p) ≈ 4.9(p0.14 − (1− p)0.14)

Q(p) ≈ −λ−1 log(1− p)

Will Landau

Boxplots

NumericalSummaries

Parameters

n ),Q( 2−.5n ), . . . ,Q( n−.5

Q(p) ≈ 4.9(p0.14 − (1− p)0.14)

Q(p) ≈ −λ−1 log(1− p)

Will Landau

Boxplots

NumericalSummaries

Parameters

n ),Q( 2−.5n ), . . . ,Q( n−.5

Q(p) ≈ 4.9(p0.14 − (1− p)0.14)

Q(p) ≈ −λ−1 log(1− p)

Will Landau

Boxplots

NumericalSummaries

Parameters

n ),Q( 2−.5n ), . . . ,Q( n−.5

Q(p) ≈ 4.9(p0.14 − (1− p)0.14)

Q(p) ≈ −λ−1 log(1− p)

Will Landau

Boxplots

NumericalSummaries

Parameters

n ),Q( 2−.5n ), . . . ,Q( n−.5

Q(p) ≈ 4.9(p0.14 − (1− p)0.14)

Q(p) ≈ −λ−1 log(1− p)

Will Landau

Boxplots

NumericalSummaries

Parameters

n ),Q( 2−.5n ), . . . ,Q( n−.5

Q(p) ≈ 4.9(p0.14 − (1− p)0.14)

Q(p) ≈ −λ−1 log(1− p)

Will Landau

Boxplots

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

Boxplots

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

Boxplots

NumericalSummaries

Parameters

Example: towel breaking strength data

Will Landau

Boxplots

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

Boxplots

NumericalSummaries

Parameters

●●●

●●

58 60 62 64 66 68 70 72

2Normal QQ plot: 200−grain bullet penetration

Sample Quantiles

Will Landau

Boxplots

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

Boxplots

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

Boxplots

NumericalSummaries

Parameters

Outline

Boxplots

Numerical Summaries

Parameters

Will Landau

Boxplots

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

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

I Arithmetic mean:I x = 1

∑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

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

NumericalSummaries

Parameters

Measures of center

x1 x2 x3 x4 x5 x60 1 1 2 3 5

∑ni=1 xi

I Here, x = 16 (0 + 1 + 1 + 2 + 3 + 5) = 2

I Here, Q(0.5) = (1 + 2)/2 = 1.5

Will Landau

Boxplots

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 =

∑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

Boxplots

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

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

Boxplots

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

Boxplots

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

Boxplots

NumericalSummaries

Parameters

Outline

Boxplots

Numerical Summaries

Parameters

Will Landau

Boxplots

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

∑Ni=1(xi − µ)2 if N the finite population size.

I s2 ≈ σ2.

I Population standard deviation (“true” standarddeviation):

I σ =√

∑Ni=1(xi − µ)2 if N is the finite population

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

Will Landau

Boxplots

NumericalSummaries

Parameters

I µ = 1N

I x ≈ µ.

I s2 ≈ σ2.

I σ =√

size.I s ≈ σ.

descriptive statistics: part 2/2 (ch 3) - github pages · descriptive statistics: part 2/2 (ch 3)...

Documents