skewnesskurtosis峰度,偏度介绍

Skewness, Kurtosis, and the Normal Curve

?

Skewness

In everyday language, the terms “skewed” and “askew” are used to refer to

something that is out of line or distorted on one side. When referring to the shape of

frequency or probability distributions,

“skewness” refers to asymmetry of the distribution.

A distribution with an asymmetric tail extending out to the right is referred to as

“positively skewed” or “skewed to the right,” while a distribution with an asymmetric tail

extending out to the left is

referred to as “negatively skewed” or “skewed to the left.”

Skewness can range from minus infinity to positive infinity.

Karl Pearson (1895) first suggested measuring skewness by standardizing the

?

?

mode

difference between the mean and the mode, that is,

sk

?

. Population modes

?

are not well estimated from sample modes, but one can estimate the difference

between the mean and the mode as being three times the difference between the mean

and the median (Stuart & Ord, 1994), leading to the following estimate of skewness:

3

(

M

?

median)

. Many statisticians use this measure but with the ‘3’ eliminated,

sk

est

?

s

(

M

?

median)

that is,

sk

?

. This statistic ranges from -1 to +1. Absolute values above

s

0.2 indicate great skewness (Hildebrand, 1986).

Skewness has also been defined with respect to the third moment about the

?

(

X

?

?

)

3

mean:

?

1

?

, which is simply the expected value of the distribution of cubed

z

n

?

3

scores. Skewness measured in this way is sometimes referred to as “Fisher’s

skew

ness.” When the deviations from the mean are greater in one direction than in the

other direction, this statistic will deviate from zero in the direction of the larger

deviations. From sample data, Fisher’s skewness is most often estimated by:

n

?

z

3

g

1

?

. For large sample sizes (

n

> 150),

g

1

may be distributed

(

n

?

1

)(

n

?

2

)

approximately normally, with a standard error of approximately

6

/

n

. While one could

use this sampling distribution to construct confidence intervals for or tests of

hypotheses about

?

1

, there is rarely any value in doing so.

The most commonly used measures of skewness (those discussed here) may

produce surprising results, such as a negative value when the shape of the distribution

appears skewed to the right. There may be superior alternative measures not

commonly used (Groeneveld & Meeden, 1984).

It is important for behavioral researchers to notice skewness when it appears in

their data. Great skewness may motivate the researcher to investigate outliers. When

making decisions about which measure of location to report (means being drawn in the

direction of the skew) and which inferential statistic to employ (one which assumes

normality or one which does not), one should take into consideration the estimated

skewness of the population. Normal distributions have zero skewness. Of course, a

distribution can be perfectly symmetric but far from normal. Transformations commonly

employed to reduce (positive) skewness include square root, log, and reciprocal

transformations.

Also see

Skewness and the Relative Positions of Mean, Median, and Mode

Kurtosis

Karl Pearson (1905) defined a distribution’s degree of kurtosis as

?

?

?

2

?

3

,

?

(

X

?

)

4

where

?

2

?

, the expected value of the distribution of

Z

scores which have

< br>n

?

4

been raised to the 4

th

power.

?

2

is often referred to as “Pearson’s kurtosis,” and

?

2

- 3

(often symbolized with

?

2

) as “kurtosis excess” or “Fisher’s kurtosis,” even though it was

Pearson who defined kurtosis as

?

2

- 3. An unbiased estimator for

?

2

is

n

(

n

?

1< /p>

)

?

Z

4

3

(

n

?

)

2

g

2

?

?

. For large sample sizes (

n

> 1000),

g

2

may be

(

n

?

1

)(

n

?

2

)(

n

?

3

)

(

n

?

2

)(

n

?

3

)

distributed approximately normally, with a standard error of approximately

24

/

n

(Snedecor, & Cochran, 1967). While one could use this sampling distribution to

construct confidence intervals for or tests of hypotheses about

?

2

, there is rarely any

value in doing so.

Pearson (1905) introduced kurtosis as a measure of how flat the top of a

symmetric distribution is when compared to a normal distribution of the same variance.

He referred to more flat-topped distributions (

?

2

< 0) as “platykurtic,” less flat

-topped

distributions (

?

2

> 0) as “leptokurtic,” and equally flat

-topped distributions as

“mesokurtic” (

?

2

?

0). Kurtosis is actually more influenced by scores in the tails of the

distribution than scores in the center of a distribution (DeCarlo, 1967). Accordingly, it is

often appropriate to describe a leptokurtic distribution as “fat in the tails” and a

platykurtic distribution as “thin in the tails.”

Student (1927,

Biometrika

,

19

, 160) published a cute description of kurtosis,

which I quote here: “Platykurtic curves have shorter ‘tails’ than the normal curve of

error and leptokurtic longer ‘tails.’ I

myself bear in mind the meaning of the words by

the above

memoria technica

, where the first figure represents platypus and the second

kangaroos, noted for lepping.” Please point your browser to

./jeff570/

, scroll down to “kurtosis,” and look at Student’s drawings.

Moors (1986) demonstrated that

?

2

?

Var

(

Z

2

)

?

1

. Accordingly, it may be best to

treat kurtosis as the extent to which scores are dispersed away from the shoulders of a

distribution, where the shoulders are the points where

Z

2

= 1, that is,

Z

=

?

1. Balanda

and MacGillivray (1988) wrote “it is best to define kurtosis vaguely as the location

- and

scale-free movement of probability mass from the shoulders of a distribution into its

centre and tails.” If one starts with a normal distribution and mo

ves scores from the

shoulders into the center and the tails, keeping variance constant, kurtosis is increased.

The distribution will likely appear more peaked in the center and fatter in the tails, like a

6

Laplace distribution

(

?

2< /p>

?

3

) or

Student’s

t

with few degrees of freedom (

?

2

?

).

df

?

4

St arting again with a normal distribution, moving scores from the tails and the

center to the shoulders will decrease kurtosis. A

uniform distribution

certainly has a flat

top, with

< br>?

2

?

?

1

.

2

, but

?

2

can reach a minimum value of

?

2 when two score values are

equally probably and all other score values have probability zero (a

rectangular U

distribution

, that is, a binomial distribution with

n

=1,

p

= .5). One might object that the

rectangular U distribution has all of its scores in the tails, but closer inspection will

reveal that it has no tails, and that all of its scores are in its shoulders, exactly one

standard deviation from its mean. Values of

g

2

less than that expected for an uniform

distribution (

?

1.2) may suggest that the distribution is bimodal (Darlington, 1970), but

bimodal distributions can have high kurtosis if the modes are distant from the shoulders.

One leptokurtic distribution we shall deal with is Student’s

t

distribution. The

kurtosis of

t

is infinite when

df

< 5, 6 when

df

= 5, 3 when

df

= 6. Kurtosis decreases

further (towards zero) as

df

increase and

t

approaches the normal distribution.

Kurtosis is usually of interest only when dealing with approximately symmetric

distributions. Skewed distributions are always leptokurtic (Hopkins & Weeks, 1990).

Among the several alternative measures of kurtosis that have been proposed (none of

which has often been employed), is one which adjusts the measurement of kurtosis to

remove the effect of skewness (Blest, 2003).

There is much confusion about how kurtosis is related to the shape of

distributions. Many authors of textbooks have asserted that kurtosis is a measure of

the peakedness of distributions, which is not strictly true.

It is easy to confuse low kurtosis with high variance, but distributions with

identical kurtosis can differ in variance, and distributions with identical variances can

differ in kurtosis. Here are some simple distributions that may help you appreciate that

kurtosis is, in part, a measure of tail heaviness relative to the total variance in the

distribution (remember the “

?

4

” in the denominator).

Table 1.

Kurtosis for 7 Simple Distributions Also Differing in Variance

X

05

10

15

Kurtosis

Variance

freq A

20

00

20

-2.0

25

freq B

20

10

20

-1.75

20

freq C

20

20

20

-1.5

16.6

freq D

10

20

10

-1.0

12.5

freq E

05

20

05

0.0

8.3

freq F

03

20

03

1.33

5.77

freq G

01

20

01

8.0

2.27

Platykurtic

Leptokurtic

When I presented these distributions to my colleagues and graduate students

and asked them to identify which had the least kurtosis and which the most, all said A

has the most kurtosis, G the least (excepting those who refused to answer). But in fact

A has the least kurtosis (

?

2 is the smallest possible value of kurtosis) and G the most.

The trick is to do a mental frequency plot where the abscissa is in standard deviation

units. In the maximally platykurtic distribution A, which initially appears to have all its

scores in its tails, no score is more than one

?

away from the mean - that is, it has no

tails! In the leptokurtic distribution G, which seems only to have a few scores in its tails,

one must remember that those scores (5 & 15) are much farther away from the mean

(3.3

?

) than are the 5’s & 15’s in distribution A. In fact, in G nine percent of the scores

are more than three

?

from the mean, much more than you would expect in a

mesokurtic distribution (like a normal distribution), thus G does indeed have fat tails.

If you were you to ask SAS to compute kurtosis on the A scores in Table 1, you

would get a value less than

?

2.0, less than the lowest possible population kurtosis.

Why? SAS assumes your data are a sample and computes the

g

2

estimate of

population kurtosis, which can fall below

?

2.0.

Sune Karlsson, of the Stockholm School of Economics, has provided me with the

following modified example which holds the variance approximately constant, making it

quite clear that a higher kurtosis implies that there are more extreme observations (or

that the extreme observations are more extreme). It is also evident that a higher

kurtosis also implies that the distribution is more ‘single

-

peaked’ (this would be even

more evident if the sum of the frequencies was constant). I have highlighted the rows

representing the shoulders of the distribution so that you can see that the increase in

kurtosis is associated with a movement of scores away from the shoulders.