I am surprised to get different results for skewness and kurtosis from different application or library. Take for example, with moments library package in R we have the following result:
skewness: -0.1450712
Kurtosis: 1.464111
with library e1071 in R
skewness: -0.123864
kurtosis: -1.81407
With excel and SAS, I obtain the same result as:
skewness: -0.1720333
kurtosis: -1.7509470
I think going for the last result is better. The same result from two different applications. What do you think?
Although Marek rightly writes that kurtosis has a unique definition without any variant, note that different formulas are being used. When you are refering to a "majority of scholars", you obviously have in mind or are refering to the adjusted version of Pearson's kurtosis (so called excess kurtosis) that is 0 if data are following the Gaussian normal distribution. Using this formula the kurtosis of any data must be -2 or more with no upper limit. However, without this adjustment, the kurtosis of normally distributed data is 3. See http://en.wikipedia.org/wiki/Kurtosis and search for "minus 3" within the text.
Although you do not know the algorithm behind the software, this could basically the difference in the formula used. ie can calculate asymmetry using Pearson, Bowley or Fisher and different results.
I have understood that EXCEL calculated by Fisher
As an elementary exercise, you can show that Excel cannot produce the book defined beta_{3}, try it.
And to add a word, you can create your own package for computing those values with the if controls that you want!
I would strongly suggest to Read the Reference Manual or help.
library(e1071)
help(skewness), or help(kurtosis)
kurtosis(x, na.rm = FALSE, type = 3)
skewness(x, na.rm = FALSE, type = 3)
type: an integer between 1 and 3 selecting one of the algorithms for computing skewness detailed below
Type 1: g_1 = m_3 / m_2^(3/2). This is the typical definition used in many older textbooks.
Type 2: G_1 = g_1 * sqrt(n(n-1)) / (n-2). Used in SAS and SPSS.
Type 3: b_1 = m_3 / s^3 = g_1 ((n-1)/n)^(3/2). Used in MINITAB and BMDP.
I suggest you calculate the skewness and kurtosis once by hand using excel as a simple "calculator". That ensures you that you will have the right values at the end. Do not use the suggested formulas from excel, as these are faulty. May be some formulas for R are faulty as well, you never know until you calculate it back by yourself. If you need help I can send you a spreadsheet to do it in excel.
I recommend reading the article by Joanes and Gill (1998):
Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183-189.
They describe three types of measures (each with the variant of subtracting 3 from the kurtosis measure or not - if not subtracted, normally distributed variables have a kurtosis of 3):
Type 1 (3 subtracted): used in many textbooks and by using option type=1 of the R library "e1071"
Type 1 (3 not subtracted): used by the R library "moments" and by Stata
Type 2 (3 subtracted): used by SAS and SPSS, by using option type=2 of the R library "e1071", and obviously also by EXCEL
Type 2 (3 not subtracted): rarely used
Type 3 (3 subtracted): used by MINITAB, BMDP and by using option type=3 (default) of the R library "e1071"
Type 3 (3 not subtracted): rarely used.
For the fomulas see the article of Joanes and Gill (1998). Some argue that type 2 should be prefered because it calculates the "unbiased" (estimated population) values from sample data. There is a Stata .ado-file -moments2- (in Stata, see -findit moments2-) which allows to calculate all of these types and variants.
Dirk,
one thing is to estimate skewness and kurtosis and the other - compare them between different data. Those two moments of probability distributions have unique definitions, without any "variants". Sandra's advise seems therefore optimal.
Otherwise one should carefully study the manuals of the said packages just to be sure of what they really compute.
I agree that calculation by hand is the most reliable: Then you accept what package agree with it.
Majorities of scholars in methodology said skewness not more +-2.0 and kurtosis +-3.0. To me it is the same.
Although Marek rightly writes that kurtosis has a unique definition without any variant, note that different formulas are being used. When you are refering to a "majority of scholars", you obviously have in mind or are refering to the adjusted version of Pearson's kurtosis (so called excess kurtosis) that is 0 if data are following the Gaussian normal distribution. Using this formula the kurtosis of any data must be -2 or more with no upper limit. However, without this adjustment, the kurtosis of normally distributed data is 3. See http://en.wikipedia.org/wiki/Kurtosis and search for "minus 3" within the text.
The issue of subtracting 3 for kurtosis is no more clear in the output of Stata for 100,000 realizations of z-scores from the standard normal distribution:
. clear
. set obs 100000
//obs was 0, now 100000
. gen zscore=rnormal()
. summ zscore,det
//zscore
-------------------------------------------------------------
Percentiles Smallest
1% -2.324222 -4.633008
5% -1.64566 -4.162048
10% -1.288536 -4.14397 Obs 100000
25% -.6762114 -4.064388 Sum of Wgt. 100000
50% .0001107 Mean -.0029699
Largest Std. Dev. 1.000935
75% .6691694 3.80799
90% 1.281106 3.945858 Variance 1.00187
95% 1.64328 4.293149 Skewness .0053897
99% 2.351803 4.490108 Kurtosis 3.005589
So, Stata does not subtract 3 from kurtosis, whereas other programs can, and do. Thus, be aware of the particular software being used. LP
Right. The formula and methods used by Stata are documented in Stata'a Base Reference Manual [R] under the entry -summarize- which does not subtract 3 from kurtosis. That is why I wrote -moments2- which you can install using the Stata command -ssc install moments2- . This .ado-program subtracts 3 by default - and allows to calculate all other measures of skewness and kurtosis as discussed in Joanes & Gill (1998).
Joanes, D. N. & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183-189.
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