altre misure di centro e - unipr.itbrunic22/mysite/tda5.pdf · chiorri journal of statistics...
TRANSCRIPT
![Page 1: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/1.jpg)
altre misure di centro e
dispersione
![Page 2: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/2.jpg)
media geometricala radice n-esima del prodotto degli n dati
2, 8 ---->dati G
4, 1, 1/32 ->
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applicazioni
poco utilizzata nelle scienze sociali
è utile per rappresentare la tendenza centrale in
distribuzioni non simmetriche
![Page 4: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/4.jpg)
relazione con la media di log(x)
se logM è la media aritmetica di log(x)
allora G = antilog(logM)
![Page 5: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/5.jpg)
il logaritmo di x è il numero a cui va elevata la base per ottenere x: log10(100) = 2
l’antilogaritmo di log è il numero che si ottiene elevando la base alla potenza log: antilog10(2) =10^2 = 100
![Page 6: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/6.jpg)
media armonicail reciproco della media
dei reciproci dei dati
1, 2, 4 dati
H
![Page 7: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/7.jpg)
applicazioni
poco utilizzata nelle scienze sociali
usata in fisica in situazioni in cui occorre mediare rapporti o tassi
di crescita
![Page 8: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/8.jpg)
S V 100 km 200 km/h 100 km 100 km/h
200 km
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S V T 100 km 200 km/h 0.5 h 100 km 100 km/h 1 h
V media sui 200 km = S/T = 200/1.5
= 133 km/h
media aritmetica = (200 + 100)/2 = 150 km/h!
media armonica = 1/[(1/200 + 1/100)/2] = 133 km/h
![Page 10: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/10.jpg)
> d <- read.table("IQ.txt", header = TRUE) > ma <- mean(d$Height) > mg <- prod(d$Height)^(1/length(d$Height)) > mh <- 1/mean(1/d$Height)
> ma [1] 68.5375
> mg [1] 68.42843
> mh [1] 68.32095
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> mg [1] 68.42843
> 10^mean(log10(d$Height)) [1] 68.42843
> exp(mean(log(d$Height))) [1] 68.42843
![Page 12: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/12.jpg)
deviazione mediana assoluta
mediana degli scarti non segnati dalla media
Median Absolute Deviation
mad()
![Page 13: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/13.jpg)
> mad(d$Height) [1] 3.33585
> sd(d$Height) [1] 3.943816
![Page 14: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/14.jpg)
statistiche per la forma di una distribuzione
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asimmetria (skewness)
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skewness positiva
skewness negativa
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Student (1927) Biometrika, 19, 160
curtosi
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LEPtokurtic
MESOkurtic
PLATYkurtic
![Page 19: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/19.jpg)
Journal of Statistics Education, Volume 19, Number 2(2011)
3
Skewed Left
Long tail points left Symmetric Normal Tails are balanced
Skewed Right Long tail points right
Figure 1. Sketches showing general position of mean, median, and mode in a population.
Next, a textbook might present stylized sample histograms, as in Figure 2. Figures like these ������� �������������������������� ���������������������������������������-� ������������������(b) extreme data values in one tail are not unusual in real data; and (c) real samples may not resemble any simple histogram prototype. The instructor can discuss causes of asymmetry (e.g., why waiting times are exponential, why earthquake magnitudes follow a power law, why home prices are skewed to the right) or the effects of outliers and extreme data values (e.g., how a ����������� ������������������������� ���������������������� ������� ����������������affects the health insurance premiums for a pool of employees).
Skewed Left Symmetric Skewed Right
One Mode Bell-Shaped One Mode
Two Modes Bimodal Bimodal
Left Tail Extremes Uniform (no mode) Right Tail Extremes
Figure 2. Illustrative prototype histograms.
![Page 20: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/20.jpg)
coefficiente di skewness di Fisher
Journal of Statistics Education, Volume 19, Number 2(2011)
6
appearance varies if we alter the bin limits, so any single histogram may not give a definitive view of population shape. But students like clear-cut answers. The next question is likely to be: ,�����%$�� '������������"������%#$�'��#�����$'����$�������������������$ �#�)�$��$�$���
! !%��$� ���#�#��'�����#�.$�$��"��# �������� ��$�#$�� "�#��'��##�- �$%���$#�'� �� $����$���#��'��##�#$�$�#$�������(���.#�Descriptive Statistics may ask more specific questions. For example: ,�)�#��!���#��'��##�#$�$�#$����" ���(�����#�+0.308. So can I say that my sample of 12
items came from a left-#��'���! !%��$� ��- ,����)�#��!��� ������$��#��$���#��!�������������(����#�$���#��!����������������)�$�
my skewness statistic is negative +������ '�����$��$����- 3. Skewness Statistics Since Karl Pearson (1895), statisticians have studied the properties of various statistics of skewness, and have discussed their utility and limitations. This research stream covers more than a century. For an overview, see Arnold and Groenveld (1995), Groenveld and Meeden (1984), and Rayner, Best and Matthews (1995). Empirical studies have examined bias, mean squared error, Type I error, and power for samples of various sizes drawn from various populations. A recent study by Tabor (2010) ranked 11 different statistics in terms of their power for detecting skewness in samples from populations with varying degrees of skewness. MacGillivray (1986) � ���%��#�$��$�,*$���"���$�&����! "$����� ��$��������"��$� rderings and measures depends on ��"�%�#$����#�������$��#�%������)�$��$���)� ���� %��������#�"������#�� #$���! "$��$�*-��� ��notes that describing skewness is really a special case of comparing distributions. This key point is perhaps a bit subtle for students. Students (and instructors) merely need to bear in mind that we are not testing for symmetry in general. Rather, the (often implicit) null hypothesis must refer to a specific symmetric population. Because the most common reference point is the normal distribution (especially in an introductory statistics class) we will limit our discussion accordingly. Mathematicians discuss skewness in terms of the second and third moments around the mean,
i.e., 22
1
1( )
n
ii
m x xn �
� �� and 33
1
1( )
n
ii
m x xn �
� �� . Mathematical statistics textbooks and a few
software packages (e.g., Stata, Visual Statistics, early versions of Minitab) report the traditional Fisher-Pearson coefficient of skewness:
[1a]
3
3 11 3/23/2
2 2
1
1 ( )
1 ( )
n
ii
n
ii
x xm ngm
x xn
�
�
�� �
� ��� �
�
�
�.
![Page 21: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/21.jpg)
Chiorri
Journal of Statistics Education, Volume 19, Number 2(2011)
7
������ ����������������� ����this statistic is sometimes referred to as 1� which is awkward because g1 can be negative. Pearson and Hartley (1970) provide tables for g1as a test for departure from normality (i.e., testing the sample against one particular symmetric distribution). Although well documented and widely referenced in the literature, this formula does not correspond to what students will see in most software packages nowadays. Major software packages available to educators (e.g., Minitab, Excel, SPSS, SAS) include an adjustment for sample size, and provide the adjusted Fisher-Pearson standardized moment coefficient1:
[1b] 3
11( 1)( 2)
ni
i
x xnGn n s�
�� �� � � � � ��
. In large samples, g1 and G1 will be similar. Few students will be aware of this formula because it is buried within the help files for the software. The formula for G1 is probably not even in the textbook unless the student is studying mathematical statistics. However, this statistic is included in E������Data Analysis > Descriptive Statistics and is calculated by the Excel function =SKEW(Array), so it will be seen by millions of ����������������������������������� ��Wikipedia, they will find a different-looking but equivalent formula:
[1c]
3
11 3
22
1
1 ( )( 1)
2 1 ( )
n
ii
n
ii
x xn n nGn
x xn
�
�
� ��
� �� �� � � ��� �� �� �
�
� .
This alternate formulation (1c) has the attraction of showing that the adjustment for sample size approaches unity as n increases. Joanes and Gill (1998) compare bias and mean squared error (MSE) of different measures of skewness in samples of various sizes from normal and skewed populations. G1 is shown to perform well, for example, having small MSE in samples from skewed populations. Unfortunately, none of these formulas is likely to convey very much to a student. An ambitious instructor can dissect such formulas to impart grains of understanding to the best students, while the rest of the class groans as the discussion turns to second and third moments around the mean. The resulting insights are, at best, likely to be short-lived. Yet once students realize that there is a formula for skewness and see it in Excel, they will want to know how to interpret it. The instructor must decide what to say about a statistic such as G1 without spending more time than the topic is worth. A minimalist might say that
� Its sign reflects the direction of skewness. � It compares the sample with a normal (symmetric) distribution.
1 Not many years ago, computer packages reported g1without an adjustment for sample size. Although the adjustment is now incorporated in software packages, textbooks (with a few exceptions) do not report an adapted version of the Pearson-Hartley tables. For that matter, many textbooks show no table of critical values at all. Without a table, why even mention the sample skewness statistic? !
Journal of Statistics Education, Volume 19, Number 2(2011)
7
������ ����������������� ����this statistic is sometimes referred to as 1� which is awkward because g1 can be negative. Pearson and Hartley (1970) provide tables for g1as a test for departure from normality (i.e., testing the sample against one particular symmetric distribution). Although well documented and widely referenced in the literature, this formula does not correspond to what students will see in most software packages nowadays. Major software packages available to educators (e.g., Minitab, Excel, SPSS, SAS) include an adjustment for sample size, and provide the adjusted Fisher-Pearson standardized moment coefficient1:
[1b] 3
11( 1)( 2)
ni
i
x xnGn n s�
�� �� � � � � ��
. In large samples, g1 and G1 will be similar. Few students will be aware of this formula because it is buried within the help files for the software. The formula for G1 is probably not even in the textbook unless the student is studying mathematical statistics. However, this statistic is included in E������Data Analysis > Descriptive Statistics and is calculated by the Excel function =SKEW(Array), so it will be seen by millions of ����������������������������������� ��Wikipedia, they will find a different-looking but equivalent formula:
[1c]
3
11 3
22
1
1 ( )( 1)
2 1 ( )
n
ii
n
ii
x xn n nGn
x xn
�
�
� ��
� �� �� � � ��� �� �� �
�
� .
This alternate formulation (1c) has the attraction of showing that the adjustment for sample size approaches unity as n increases. Joanes and Gill (1998) compare bias and mean squared error (MSE) of different measures of skewness in samples of various sizes from normal and skewed populations. G1 is shown to perform well, for example, having small MSE in samples from skewed populations. Unfortunately, none of these formulas is likely to convey very much to a student. An ambitious instructor can dissect such formulas to impart grains of understanding to the best students, while the rest of the class groans as the discussion turns to second and third moments around the mean. The resulting insights are, at best, likely to be short-lived. Yet once students realize that there is a formula for skewness and see it in Excel, they will want to know how to interpret it. The instructor must decide what to say about a statistic such as G1 without spending more time than the topic is worth. A minimalist might say that
� Its sign reflects the direction of skewness. � It compares the sample with a normal (symmetric) distribution.
1 Not many years ago, computer packages reported g1without an adjustment for sample size. Although the adjustment is now incorporated in software packages, textbooks (with a few exceptions) do not report an adapted version of the Pearson-Hartley tables. For that matter, many textbooks show no table of critical values at all. Without a table, why even mention the sample skewness statistic? !
Journal of Statistics Education, Volume 19, Number 2(2011)
6
appearance varies if we alter the bin limits, so any single histogram may not give a definitive view of population shape. But students like clear-cut answers. The next question is likely to be: ,�����%$�� '������������"������%#$�'��#�����$'����$�������������������$ �#�)�$��$�$���
! !%��$� ���#�#��'�����#�.$�$��"��# �������� ��$�#$�� "�#��'��##�- �$%���$#�'� �� $����$���#��'��##�#$�$�#$�������(���.#�Descriptive Statistics may ask more specific questions. For example: ,�)�#��!���#��'��##�#$�$�#$����" ���(�����#�+0.308. So can I say that my sample of 12
items came from a left-#��'���! !%��$� ��- ,����)�#��!��� ������$��#��$���#��!�������������(����#�$���#��!����������������)�$�
my skewness statistic is negative +������ '�����$��$����- 3. Skewness Statistics Since Karl Pearson (1895), statisticians have studied the properties of various statistics of skewness, and have discussed their utility and limitations. This research stream covers more than a century. For an overview, see Arnold and Groenveld (1995), Groenveld and Meeden (1984), and Rayner, Best and Matthews (1995). Empirical studies have examined bias, mean squared error, Type I error, and power for samples of various sizes drawn from various populations. A recent study by Tabor (2010) ranked 11 different statistics in terms of their power for detecting skewness in samples from populations with varying degrees of skewness. MacGillivray (1986) � ���%��#�$��$�,*$���"���$�&����! "$����� ��$��������"��$� rderings and measures depends on ��"�%�#$����#�������$��#�%������)�$��$���)� ���� %��������#�"������#�� #$���! "$��$�*-��� ��notes that describing skewness is really a special case of comparing distributions. This key point is perhaps a bit subtle for students. Students (and instructors) merely need to bear in mind that we are not testing for symmetry in general. Rather, the (often implicit) null hypothesis must refer to a specific symmetric population. Because the most common reference point is the normal distribution (especially in an introductory statistics class) we will limit our discussion accordingly. Mathematicians discuss skewness in terms of the second and third moments around the mean,
i.e., 22
1
1( )
n
ii
m x xn �
� �� and 33
1
1( )
n
ii
m x xn �
� �� . Mathematical statistics textbooks and a few
software packages (e.g., Stata, Visual Statistics, early versions of Minitab) report the traditional Fisher-Pearson coefficient of skewness:
[1a]
3
3 11 3/23/2
2 2
1
1 ( )
1 ( )
n
ii
n
ii
x xm ngm
x xn
�
�
�� �
� ��� �
�
�
�.
Journal of Statistics Education, Volume 19, Number 2(2011)
7
������ ����������������� ����this statistic is sometimes referred to as 1� which is awkward because g1 can be negative. Pearson and Hartley (1970) provide tables for g1as a test for departure from normality (i.e., testing the sample against one particular symmetric distribution). Although well documented and widely referenced in the literature, this formula does not correspond to what students will see in most software packages nowadays. Major software packages available to educators (e.g., Minitab, Excel, SPSS, SAS) include an adjustment for sample size, and provide the adjusted Fisher-Pearson standardized moment coefficient1:
[1b] 3
11( 1)( 2)
ni
i
x xnGn n s�
�� �� � � � � ��
. In large samples, g1 and G1 will be similar. Few students will be aware of this formula because it is buried within the help files for the software. The formula for G1 is probably not even in the textbook unless the student is studying mathematical statistics. However, this statistic is included in E������Data Analysis > Descriptive Statistics and is calculated by the Excel function =SKEW(Array), so it will be seen by millions of ����������������������������������� ��Wikipedia, they will find a different-looking but equivalent formula:
[1c]
3
11 3
22
1
1 ( )( 1)
2 1 ( )
n
ii
n
ii
x xn n nGn
x xn
�
�
� ��
� �� �� � � ��� �� �� �
�
� .
This alternate formulation (1c) has the attraction of showing that the adjustment for sample size approaches unity as n increases. Joanes and Gill (1998) compare bias and mean squared error (MSE) of different measures of skewness in samples of various sizes from normal and skewed populations. G1 is shown to perform well, for example, having small MSE in samples from skewed populations. Unfortunately, none of these formulas is likely to convey very much to a student. An ambitious instructor can dissect such formulas to impart grains of understanding to the best students, while the rest of the class groans as the discussion turns to second and third moments around the mean. The resulting insights are, at best, likely to be short-lived. Yet once students realize that there is a formula for skewness and see it in Excel, they will want to know how to interpret it. The instructor must decide what to say about a statistic such as G1 without spending more time than the topic is worth. A minimalist might say that
� Its sign reflects the direction of skewness. � It compares the sample with a normal (symmetric) distribution.
1 Not many years ago, computer packages reported g1without an adjustment for sample size. Although the adjustment is now incorporated in software packages, textbooks (with a few exceptions) do not report an adapted version of the Pearson-Hartley tables. For that matter, many textbooks show no table of critical values at all. Without a table, why even mention the sample skewness statistic? !
Journal of Statistics Education, Volume 19, Number 2(2011)
7
������ ����������������� ����this statistic is sometimes referred to as 1� which is awkward because g1 can be negative. Pearson and Hartley (1970) provide tables for g1as a test for departure from normality (i.e., testing the sample against one particular symmetric distribution). Although well documented and widely referenced in the literature, this formula does not correspond to what students will see in most software packages nowadays. Major software packages available to educators (e.g., Minitab, Excel, SPSS, SAS) include an adjustment for sample size, and provide the adjusted Fisher-Pearson standardized moment coefficient1:
[1b] 3
11( 1)( 2)
ni
i
x xnGn n s�
�� �� � � � � ��
. In large samples, g1 and G1 will be similar. Few students will be aware of this formula because it is buried within the help files for the software. The formula for G1 is probably not even in the textbook unless the student is studying mathematical statistics. However, this statistic is included in E������Data Analysis > Descriptive Statistics and is calculated by the Excel function =SKEW(Array), so it will be seen by millions of ����������������������������������� ��Wikipedia, they will find a different-looking but equivalent formula:
[1c]
3
11 3
22
1
1 ( )( 1)
2 1 ( )
n
ii
n
ii
x xn n nGn
x xn
�
�
� ��
� �� �� � � ��� �� �� �
�
� .
This alternate formulation (1c) has the attraction of showing that the adjustment for sample size approaches unity as n increases. Joanes and Gill (1998) compare bias and mean squared error (MSE) of different measures of skewness in samples of various sizes from normal and skewed populations. G1 is shown to perform well, for example, having small MSE in samples from skewed populations. Unfortunately, none of these formulas is likely to convey very much to a student. An ambitious instructor can dissect such formulas to impart grains of understanding to the best students, while the rest of the class groans as the discussion turns to second and third moments around the mean. The resulting insights are, at best, likely to be short-lived. Yet once students realize that there is a formula for skewness and see it in Excel, they will want to know how to interpret it. The instructor must decide what to say about a statistic such as G1 without spending more time than the topic is worth. A minimalist might say that
� Its sign reflects the direction of skewness. � It compares the sample with a normal (symmetric) distribution.
1 Not many years ago, computer packages reported g1without an adjustment for sample size. Although the adjustment is now incorporated in software packages, textbooks (with a few exceptions) do not report an adapted version of the Pearson-Hartley tables. For that matter, many textbooks show no table of critical values at all. Without a table, why even mention the sample skewness statistic? !
correzione per n piccoli
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Abstract
This paper discusses common approaches to presenting the topic of skewness in the classroom, and explains why students need to know how to measure it. Two skewness statistics are examined: the Fisher-Pearson standardized third moment coefficient, and the Pearson 3 coefficient that compares the mean and median.
Journal of Statistics Education, Volume 19, Number 2(2011)
1
Measuring Skewness: A Forgotten Statistic? David P. Doane Oakland University Lori E. Seward University of Colorado Journal of Statistics Education Volume 19, Number 2(2011), www.amstat.org/publications/jse/v19n2/doane.pdf Copyright © 2011 by David P. Doane and Lori E. Seward all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor. Key Words: Moment Coefficient; Pearson 2; Monte Carlo; Type I Error; Power; Normality Test. Abstract This paper discusses common approaches to presenting the topic of skewness in the classroom, and explains why students need to know how to measure it. Two skewness statistics are examined: the Fisher-Pearson standardized third moment coefficient, and the Pearson 2 coefficient that compares the mean and median. The former is reported in statistical software packages, while the latter is all but forgotten in textbooks. Given its intuitive appeal, why did Pearson 2 disappear? Is it ever useful? Using Monte Carlo simulation, tables of percentiles are created for Pearson 2. It is shown that while Pearson 2 has lower power, it matches classroom explanations of skewness and can be calculated when summarized data are available. This paper suggests reviving the Pearson 2 skewness statistic for the introductory statistics course because it compares the mean to the median in a precise way that students can understand. The paper reiterates warnings about what any skewness statistic can actually tell us. 1. Introduction
In an introductory level statistics course, instructors spend the first part of the course teaching students three important characteristics used when summarizing a data set: center, variability, and shape. The instructor typically begins by introducing visu���������������������������������data. The concept of center (also location or central tendency) is familiar to most students and ����������������������� �������������������������������������������������������� �����
This paper suggests reviving the Pearson 3 skewness statistic for the introductory statistics course because it compares the mean to the median in a precise way that students can understand. The paper reiterates warnings about what any skewness statistic can actually tell us.
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Journal of Statistics Education, Volume 19, Number 2(2011)
3
Skewed Left
Long tail points left Symmetric Normal Tails are balanced
Skewed Right Long tail points right
Figure 1. Sketches showing general position of mean, median, and mode in a population.
Next, a textbook might present stylized sample histograms, as in Figure 2. Figures like these ������� �������������������������� ���������������������������������������-� ������������������(b) extreme data values in one tail are not unusual in real data; and (c) real samples may not resemble any simple histogram prototype. The instructor can discuss causes of asymmetry (e.g., why waiting times are exponential, why earthquake magnitudes follow a power law, why home prices are skewed to the right) or the effects of outliers and extreme data values (e.g., how a ����������� ������������������������� ���������������������� ������� ����������������affects the health insurance premiums for a pool of employees).
Skewed Left Symmetric Skewed Right
One Mode Bell-Shaped One Mode
Two Modes Bimodal Bimodal
Left Tail Extremes Uniform (no mode) Right Tail Extremes
Figure 2. Illustrative prototype histograms.
skewness media, mediana
e moda
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SK3 di Pearson
SK3 = 3(media - mediana) / DS
SK1 = (media - moda) / DS SK2 = 3(media - moda) / DS
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coefficiente di curtosi di Fisher
Journal of Statistics Education, Volume 19, Number 2(2011)
6
appearance varies if we alter the bin limits, so any single histogram may not give a definitive view of population shape. But students like clear-cut answers. The next question is likely to be: ,�����%$�� '������������"������%#$�'��#�����$'����$�������������������$ �#�)�$��$�$���
! !%��$� ���#�#��'�����#�.$�$��"��# �������� ��$�#$�� "�#��'��##�- �$%���$#�'� �� $����$���#��'��##�#$�$�#$�������(���.#�Descriptive Statistics may ask more specific questions. For example: ,�)�#��!���#��'��##�#$�$�#$����" ���(�����#�+0.308. So can I say that my sample of 12
items came from a left-#��'���! !%��$� ��- ,����)�#��!��� ������$��#��$���#��!�������������(����#�$���#��!����������������)�$�
my skewness statistic is negative +������ '�����$��$����- 3. Skewness Statistics Since Karl Pearson (1895), statisticians have studied the properties of various statistics of skewness, and have discussed their utility and limitations. This research stream covers more than a century. For an overview, see Arnold and Groenveld (1995), Groenveld and Meeden (1984), and Rayner, Best and Matthews (1995). Empirical studies have examined bias, mean squared error, Type I error, and power for samples of various sizes drawn from various populations. A recent study by Tabor (2010) ranked 11 different statistics in terms of their power for detecting skewness in samples from populations with varying degrees of skewness. MacGillivray (1986) � ���%��#�$��$�,*$���"���$�&����! "$����� ��$��������"��$� rderings and measures depends on ��"�%�#$����#�������$��#�%������)�$��$���)� ���� %��������#�"������#�� #$���! "$��$�*-��� ��notes that describing skewness is really a special case of comparing distributions. This key point is perhaps a bit subtle for students. Students (and instructors) merely need to bear in mind that we are not testing for symmetry in general. Rather, the (often implicit) null hypothesis must refer to a specific symmetric population. Because the most common reference point is the normal distribution (especially in an introductory statistics class) we will limit our discussion accordingly. Mathematicians discuss skewness in terms of the second and third moments around the mean,
i.e., 22
1
1( )
n
ii
m x xn �
� �� and 33
1
1( )
n
ii
m x xn �
� �� . Mathematical statistics textbooks and a few
software packages (e.g., Stata, Visual Statistics, early versions of Minitab) report the traditional Fisher-Pearson coefficient of skewness:
[1a]
3
3 11 3/23/2
2 2
1
1 ( )
1 ( )
n
ii
n
ii
x xm ngm
x xn
�
�
�� �
� ��� �
�
�
�.
4
2
4
2
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> df <- read.table("~/Desktop/dati completi.txt", header = TRUE)
> head(df)
OvsR Sex HAND RVF LIKF Ts SPAF Eng compito 1 O f dx -3.9873143 1.500000 190 1.0 1 c 2 O f dx -0.4353741 4.166667 413 5.0 2 cd 3 R f dx 3.6857530 3.500000 491 2.5 1 cd 4 O m dx -1.4842264 5.666667 277 6.0 1 c 5 R m dx -1.3211927 6.000000 480 7.0 2 cd 6 R m dx -0.2262290 4.166667 265 3.0 2 c
> hist(df$Ts)
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Histogram of df$Ts
df$Ts
Frequency
200 400 600 800 1000
05
1015
2025
30
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df <- read.table("~/Desktop/dati completi.txt", header = TRUE)
m <- mean(df$Ts)n <- length(df$Ts)s <- sqrt(sum((df$Ts - m)^2)/n) num <- sum((df$Ts - m)^3)/nden <- (sum((df$Ts - m)^2)/n)^1.5sk <- num/den
num <- sum((df$Ts - m)^4)/nden <- (sum((df$Ts - m)^2)/n)^2ku <- num/den
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> sk[1] 0.911130
> ku[1] 2.884776
> library(moments)
> skewness(df$Ts)[1] 0.9111309
> kurtosis(df$Ts)[1] 2.884776
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riassumendola skewness è 0 se la distribuzione è simmetrica, <0 se ha coda sinistra e >0 se ha coda a destra
la curtosi è 3 se la distribuzione è normale, >3 se è leptocurtica e <3 se è platicurtica
alcuni programmi calcolano il coefficiente di eccesso ku - 3, in tal caso la distribuzione normale ha curtosi 0 (R non lo fa)
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facciamo un po’ di esercizio
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x
-3 -2 -1 0 1 2 3
0.0
0.1
0.2
0.3
moda, media e mediana sono circa:
a) 0, 0, 0 b) 0, 0.5, -0.5 c) 0, -0.5, 0.5
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x
-3 -2 -1 0 1 2 3
0.0
0.1
0.2
0.3
DS e GIQ sono circa:
a) 1, 1.5 b) 2, 3 c) 3, 6
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x
-3 -2 -1 0 1 2 3
0.0
0.1
0.2
0.3
skewness e curtosi sono circa:
a) -1, 0 b) 1, 1 c) 0, 3
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x
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8 moda, media e
mediana sono circa:
a) 0, 1, 1.5 b) 2, 4, 3 c) 0, 1.5, 1
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x
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8 DS e GIQ sono circa:
a) 1, 1.5 b) 3, 3 c) 4, 5
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x
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8 skewness e curtosi
sono circa:
a) 1.5, 2 b) -1.5, 0 c) 1.5, difficile dirlo
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xx
0 10 20 30 40 50
0.00
0.02
0.04
0.06
0.08 moda, media e
mediana sono circa:
a) 4, 20, 30 b) 46, 30, 20 c) ce ne sono due, 25, 25
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xx
0 10 20 30 40 50
0.00
0.02
0.04
0.06
0.08 moda, media e
mediana sono circa:
a) 4, 20, 30 b) 46, 30, 20 c) ce ne sono due, 25, 25
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xx
0 10 20 30 40 50
0.00
0.02
0.04
0.06
0.08 la skewness e la
curtosi sono circa:
a) 0, 1 b) 3, 0 c) -3, 0
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standardizzazione
![Page 42: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/42.jpg)
> d <- read.table("~/Desktop/VL.txt", header = TRUE)
> head(d) UN VL PR 1 parma 92 1 2 parma 94 1 3 parma 93 1 4 parma 90 1 5 roma 109 0 6 parma 103 1
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57 studenti ammessi
voto di laurea90 95 100 105 110
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
![Page 44: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/44.jpg)
bologna
catania
chieti
firenzemessina
milano
napoli
parma
paviapisa
roma
sienatriesteverona
ateneo di provenienza
![Page 45: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/45.jpg)
altri parma
9095
100
105
110
ateneo di provenienza
voto
di l
aure
a
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standardizzazione
una traslazione
+
un cambio di scala
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x
0 50 100 150 200
0.0000.0100.020
x + 50
0 50 100 150 200
0.0000.0100.020
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x
0 50 100 150 200
0.0000.0100.020
x/100
0.0 0.5 1.0 1.5 2.0
0.0
1.0
2.0
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esempio
Fahrenheit --> Celsius
C = (F - 32) / 1.8
traslazionecambio di scala
![Page 50: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/50.jpg)
> (10 - 32)/1.8 [1] -12.22222
> (32 - 32)/1.8 [1] 0
> (70 - 32)/1.8 [1] 21.11111
> (90 - 32)/1.8 [1] 32.22222
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standardizzazione
traslazione: M --> 0
+
cambio di scala: DS = 1
![Page 52: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/52.jpg)
> a <- d$VL[d$PR == 0] > p <- d$VL[d$PR == 1]
> zp <- (p - mean(p))/sd(p) > za <- (a - mean(a))/sd(a)
> boxplot(za, zp, names = c("altri", "parma"), xlab = "ateneo di provenienza", cex.lab = 2, col = "grey", ylab = "z(voto di laurea)")
![Page 53: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/53.jpg)
altri parma
-10
12
ateneo di provenienza
z(vo
to d
i lau
rea)
![Page 54: altre misure di centro e - unipr.itbrunic22/mysite/TdA5.pdf · Chiorri Journal of Statistics Education, Volume 19, Number 2(2011) 7 this statistic is sometimes referred to as 1 which](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8d0c70278c7171f65c4cc9/html5/thumbnails/54.jpg)
> tapply(d$VL, d$PR, scale)
scorciatoia
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$`0` [,1] [1,] 0.95244689 [2,] -1.29819815 [3,] 0.05218887 [4,] 0.65236088 [5,] 1.10248989 [6,] 0.65236088 [7,] 1.10248989 [8,] -1.29819815 [9,] -0.84806915 [10,] 0.05218887 [11,] 1.10248989 [12,] -1.59828416 [13,] -0.69802614 [14,] 0.35227488 [15,] 1.10248989 [16,] 0.65236088 [17,] -0.39794014 [18,] 0.05218887 [19,] -1.29819815 [20,] 1.10248989 [21,] -1.44824116 [22,] -1.14815515 [23,] 1.10248989 attr(,"scaled:center") [1] 102.6522 attr(,"scaled:scale") [1] 6.664756
$`1` [,1] [1,] -0.95196357 [2,] -0.62169049 [3,] -0.78682703 [4,] -1.28223664 [5,] 0.86453834 [6,] -1.28223664 [7,] 2.02049410 [8,] 1.19481141 [9,] 0.03885566 [10,] -0.29141742 [11,] 0.69940180 [12,] -0.78682703 [13,] 0.53426527 [14,] 0.53426527 [15,] -1.11710010 [16,] 0.36912873 [17,] -0.95196357 [18,] 1.02967488 [19,] -0.29141742 [20,] -1.28223664 [21,] 0.69940180 [22,] -1.11710010 [23,] -1.11710010
[24,] 1.52508449 [25,] 0.36912873 [26,] -0.45655396 [27,] -0.12628088 [28,] -0.78682703 [29,] -0.12628088 [30,] -0.45655396 [31,] 2.02049410 [32,] -0.12628088 [33,] 0.03885566 [34,] 2.02049410 attr(,"scaled:center") [1] 97.76471 attr(,"scaled:scale") [1] 6.055595
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trasformazione logaritmica
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> df <- read.table("~/Desktop/dati completi.txt", header = TRUE)
> head(df)
OvsR Sex HAND RVF LIKF Ts SPAF Eng compito 1 O f dx -3.9873143 1.500000 190 1.0 1 c 2 O f dx -0.4353741 4.166667 413 5.0 2 cd 3 R f dx 3.6857530 3.500000 491 2.5 1 cd 4 O m dx -1.4842264 5.666667 277 6.0 1 c 5 R m dx -1.3211927 6.000000 480 7.0 2 cd 6 R m dx -0.2262290 4.166667 265 3.0 2 c
> hist(df$Ts)
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Histogram of df$Ts
df$Ts
Frequency
200 400 600 800 1000
05
1015
2025
30
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> hist(log10(df$Ts))
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Histogram of log10(df$Ts)
log10(df$Ts)
Frequency
2.2 2.4 2.6 2.8 3.0
05
1015
20
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> library(moments)
> skewness(df$Ts) [1] 0.9111309
> skewness(log10(df$Ts)) [1] 0.1454733
> kurtosis(df$Ts) [1] 2.884776
> kurtosis(log10(df$Ts)) [1] 2.129199
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log10(t) e normale
log10(t)
Density
1.5 2.0 2.5 3.0 3.5
0.0
0.5
1.0
1.5
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> qqnorm(log10(df$Ts)) > qqline(log10(df$Ts))
plot quantile-quantile
normale
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-2 -1 0 1 2
2.2
2.4
2.6
2.8
3.0
Normal Q-Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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Histogram of df$Ts
df$Ts
Frequency
200 400 600 800 1000
05
1015
2025
30
Histogram of log10(df$Ts)
log10(df$Ts)
Frequency
2.2 2.4 2.6 2.8 3.0
05
1015
20
> mean(df$Ts) [1] 418.3663
> mean(log10(df$Ts)) [1] 2.567262
> mg <- 10^(mean(log10(df$Ts))) > mg [1] 369.1999
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grafici in R (elementi di)
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menu
vedi:
Packages & Data Package Manager Package Installer Data Manager
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graphics
funzioni di basso livello per gli elementi grafici
funzioni di alto livello per grafici preconfezionati
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tipica maniera di procedere
generare i grafici che mi servono con funzioni di alto livello
“annotare” usando ulteriori funzioni di basso livello
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0 20 40 60 80 100
020
4060
80100
Index
x
esempio> x <- runif(100, 0, 100)
> plot(x)
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0 20 40 60 80 100
020
4060
80100
Index
x
esempio> x <- runif(100, 0, 100)
> abline(h = 50)
> plot(x)
0 20 40 60 80 100
020
4060
80100
Index
x
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0 20 40 60 80 100
020
4060
80100
Index
xciao mondo
esempio
> plot(x) > abline(h = 50) > text(60, 60, "ciao mondo", col = "red", cex = 2)
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farsi un’idea
> demo(graphics)
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d <- read.table("~/Desktop/LT.txt", header = TRUE)op <- par(mfrow = c(2, 2))hist(d$LT, prob = TRUE, main = "hist()", xlab = "LT", col = "orange")lines(density(d$LT, col = "blue"))barplot(table(d$LT)/length(d$LT), main = "barplot()", xlab = "LT", ylab= "Density")boxplot(d$LT, ylab = "LT", main = "boxplot()", col = "green")pie(table(d$LT)/length(d$LT), main = "pie()")par(op)
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hist()
LT
Density
90 95 100 105 110
0.00
0.02
0.04
0.06
0.08
90 92 94 96 98 102 107
barplot()
LT
Density
0.00
0.04
0.08
0.12
9095
100
105
110
boxplot()
LT
90
919293
949596
97
98
100101102103105 107
109
110
pie()
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diagramma a barre
dependent variable
one group
standard error of mean
mean of group
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box-plot
dependent variable
one group
Q1
Q3
median
MAX
MIN
outlier
5-number summary
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dietad lib restricted
age
at d
eath
(da
ys)