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# Topic: Kurtosis

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 Kurtosis - Wikipedia, the free encyclopedia In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real-valued random variable. This is because the kurtosis as we have defined it is the ratio of the fourth cumulant and the square of the second cumulant of the probability distribution. Given a sub-set of samples from a population, the sample kurtosis above is a biased estimator of the population kurtosis. en.wikipedia.org /wiki/Kurtosis   (563 words)

 KURTOSIS(<>)   (Site not responding. Last check: 2007-10-08) Kurtosis is used in distribution analysis to describe how big the tails are for a distribution. Kurtosis indicates the likelihood of an event far away from the average. Kurtosis is based on the size of a distribution's tails. www.oledb.org /products/connx/InfoNaut/connxcdd32f/kurtosis_numeric_value_.htm   (206 words)

 Kurtosis, Leptokurtosis and Platykurtosis Kurtosis is a parameter that describes the shape of a random variable’s probability density function (PDF). A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is said to be leptokurtic. www.riskglossary.com /articles/kurtosis.htm   (393 words)

 Kurtosis -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-08) Higher kurtosis means more of the (The second moment around the mean; the expected value of the square of the deviations of a random variable from its mean value) variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations. This is because the kurtosis as we have defined it is the ratio of the fourth (Click link for more info and facts about cumulant) cumulant and the square of the second cumulant of the probability distribution. The most prominent example of a mesokurtic distribution is the (A theoretical distribution with finite mean and variance) normal distribution family, regardless of the values of its parameters. www.absoluteastronomy.com /encyclopedia/k/ku/kurtosis.htm   (678 words)

 Kurtosis Both are symmetrical "bell shaped" curves, the coefficient of kurtosis for the normal distribution is constant with a value of 3.0, whilst for the logistic it is 4.2, for this reason, the logistic distribution is preferred for datasets where the "tails" have a greater significance. Kurtosis is based on the ratio of the fourth moment about the mean to the standard deviation: Kurtosis is also referred to as the coefficient of excess. www.brighton-webs.co.uk /Statistics/kurtosis.asp   (164 words)

 1.3.5.11. Measures of Skewness and Kurtosis Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. The skewness is 0.06 and the kurtosis is 5.9. www.itl.nist.gov /div898/handbook/eda/section3/eda35b.htm   (812 words)

 ED230A Kurtosis   (Site not responding. Last check: 2007-10-08) Kurtosis is one of the least discussed indices of normality. Or, kurtosis can be viewed as a measure of tail heaviness, that is, a leptokurtic distribution has heavier tails than a normal. Thus, the index of kurtosis is based on ratio of the fourth moment about the mean divided by the variance squared. www.gseis.ucla.edu /courses/ed230a2/kurtosis.html   (401 words)

 Estimating Independence in ICA   (Site not responding. Last check: 2007-10-08) Kurtosis is the classical method of measuring nongaussianity. When data is preprocessed to have unit variance, kurtosis is equal to the fourth moment of the data. Kurtosis is extremely simple to calculate, however, it is very sensitive to outliers in the data set. www.oursland.net /tutorials/ica/ica-estimate.html   (240 words)

 kurtosis (Statistics Toolbox)   (Site not responding. Last check: 2007-10-08) is the kurtosis of the elements in the vector Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. The kurtosis of a distribution is defined as www.eecs.umich.edu /dco/faq/matlab-6.5/help/toolbox/stats/kurtosis.html   (173 words)

 Questions and answers about Statistics: Skewness and kurtosis As the kurtosis statistic departs further from zero, a positive value indicates the possibility of a leptokurtic distribution (that is, too tall) or a negative value indicates the possibility of a platykurtic distribution (that is, too flat, or even concave if the value is large enough). Yet another alternative would be that the kurtosis statistic might fall within the range between - 1.7888 and + 1.7888, in which case, you would have to assume that the kurtosis was within the expected range of chance fluctuations in that statistic. The existence of flat or peaked distributions as indicated by the kurtosis statistic is important to you as a language tester insofar as it indicates violations of the assumption of normality that underlies many of the other statistics like correlation coefficients, t-tests, etc. used to study the validity of a test. www.jalt.org /test/bro_1.htm   (1683 words)

 Histograms: Snapshots of Process Variation Kurtosis is a measure of the pointiness of a distribution. Positive kurtosis is usually more of a problem for quality control, since, with "big" tails, the process may well be wider than the spec limits. Kurtosis is also a measure of the length of the tails of a distribution. www.skymark.com /resources/tools/histograms.asp   (1340 words)

 Kurtosis The classical measure of nongaussianity is kurtosis or the fourth-order cumulant. Kurtosis, or rather its absolute value, has been widely used as a measure of nongaussianity in ICA and related fields. Below we shall consider negentropy whose properties are rather opposite to those of kurtosis, and finally introduce approximations of negentropy that more or less combine the good properties of both measures. www.cis.hut.fi /aapo/papers/IJCNN99_tutorialweb/node13.html   (714 words)

 Skewness and Kurtosis of the Grating LSF The second deviation from symmetry is known as kurtosis and compares the population of the tails of the dataset to that of the central region. The skewness and kurtosis for the HEG and MEG data are shown in figure 3 for the dispersion direction. Similarly the skewness and kurtosis for the cross-dispersion direction is shown in figure 4. space.mit.edu /ASC/docs/techref_psf/node10.html   (730 words)

 [No title] 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. 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 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 Z2 = 1, that is, Z = (1. core.ecu.edu /psyc/wuenschk/docs30/Skew-Kurt.doc   (1188 words)

 Spectrum: Get kurtosis...   (Site not responding. Last check: 2007-10-08) The (normalized) kurtosis of a spectrum is the fourth central moment of this spectrum, divided by the square of the second central moment, minus 3. The kurtosis is a measure for how much the shape of the spectrum around the centre of gravity is different from a Gaussian shape. For a white noise, the kurtosis is Ð6/5. fonsg3.let.uva.nl /praat/manual/Spectrum__Get_kurtosis___.html   (83 words)

 Number 2 Pencil: Statistics term of the day: Kurtosis Kurtosis is the fourth moment of the distribution, and is the peakedness (that's three syllables, not two) of the distribution. The distribution on the right has greater kurtosis - more peaked, less flat - but it's possible that it has about the same SD as the graph on the left, which is more spread out but is thinner at the tails. Normal distributions are likely have a skew of 0 and a kurtosis of 3.The graph on the right is more likely to be leptokurtic (defined as a kurtosis value of greater than 3), while the graph on the left is platykurtic (kurtosis value less than 3). www.kimberlyswygert.com /archives/002812.html   (515 words)

 Determining if skewness and kurtosis are significantly non-normal In general, kurtosis is not very important for an understanding of statistics, and we will not be using it again. Note, that these numerical ways of determining if a distribution is significantly non-normal are very sensitive to the numbers of scores you have. With small sets of scores (say less than 50), measures of skewness and kurtosis can vary widely from negative to positive skews to perfectly normal and the parent population from which the scores have come from could still be quite normal. www.une.edu.au /WebStat/unit_materials/c4_descriptive_statistics/determine_skew_kurt.html   (534 words)

 HMW - Skewness and Kurtosis   (Site not responding. Last check: 2007-10-08) Standard Error of Kurtosis The ratio of kurtosis to its standard error can be used as a test of normality (that is, you can reject normality if the ratio is less than -2 or greater than +2). A large positive value for kurtosis indicates that the tails of the distribution are longer than those of a normal distribution; a negative value for kurtosis indicates shorter tails (becoming like those of a box-shaped uniform distribution). A > large positive value for kurtosis > indicates that the tails of the > distribution are longer than those > of a normal distribution; a negative > value for kurtosis indicates shorter > tails (becoming like those of a > box-shaped uniform distribution). www.indiana.edu /~jkkteach/p553_bbs/p553.cgi?read=93   (429 words)

 stats::kurtosis -- the kurtosis (excess) of a data sample   (Site not responding. Last check: 2007-10-08) The kurtosis measures whether a distribution is ``flat'' or ``peaked''. If the distribution function of the data has a flatter top than the normal distribution, then the kurtosis is negative. The kurtosis is positive, if the distribution function has a high peak compared to the normal distribution. www.sciface.com /STATIC/DOC25/de/stats/kurtosis.shtml   (282 words)

 Investment Performance Analysis - Kurtosis   (Site not responding. Last check: 2007-10-08) Kurtosis ('4th moment of a distribution') describes the peakedness and tails of a return distribution. Normal-distributed returns have a kurtosis of 3, irrespective their mean or standard deviation. If a distribution’s kurtosis is greater than 3, it is said to be leptokurtic. www.andreassteiner.net /performanceanalysis?Risk_Measurement:Return_Distributions:Kurtosis   (93 words)

 PQ Systems - Knowledge Base Kurtosis is a measure of the combined weight of the tails in relation to the rest of the distribution. If the distribution is peaked (tall and skinny), it will have a kurtosis greater than 0 and is said to be leptokurtic. If the distribution is flat, it will have a kurtosis value less than zero and is said to be platykurtic. www.pqsystems.com /kb/vote.php?a=no&qstId=488   (160 words)

 [No title] But the kurtosis measures the 'tail-heaviness' relative to the variance (or variance squared to be precise), hence we should really keep the variance constant when comparing the kurtosis. If you are using kurtosis as a standardised fourth moment like this, it is *not* necessarily true that height in the middle (lets stick with symmetric distributions for now, so we know what 'middle' means) implies anything about kurtosis, even when the spreads are equal (in some sense, say s.d.'s). I suggest something like this: Kurtosis is related to the relative concentrations of the scores in the tails (upper and lower ends), shoulders (between the tails and the center), and center of a distribution. core.ecu.edu /psyc/wuenschk/StatHelp/KURTOSIS.txt   (4442 words)

 PQ Systems - Knowledge Base   (Site not responding. Last check: 2007-10-08) Kurtosis is a measure of how peaked the distribution is. It is based on the fourth moment about the mean, and is therefore always positive and bound by zero on the lower end. The higher the kurtosis, the closer the data (parts) are to the mean value, making the peak of the histogram in the center higher. Thus, the bigger the kurtosis, the better you like it - even though, the distribution will not be normal. www.pqsystems.com /kb/question.php?qstId=382   (210 words)

 [No title] Non-normal skewness and kurtosis in option-implied distributions are found to contribute significantly to the phenomenon of volatility smile. On the average, the implied standard deviation is 0.1162, the implied skewness is (-1.68), and the implied kurtosis is 2.39 (Corrado and Su, 1997). Although the Gram-Charlier (and Edgeworth) expansion allows for additional flexibility over the normal probability density function because it introduces the skewness and kurtosis of the empirical distribution as parameters, this expansion has the drawback of yielding negative values for certain skewness-kurtosis parameters because it is a polynomial approximation. www.fintools.com /docs/GramCharlier.doc   (976 words)

 KURTOSIS   (Site not responding. Last check: 2007-10-08) "Excess Kurtosis" is defined as K-3 because a standard normal distribution has a kurtosis of 3. If K-3>0 the distribution is peaked and has relatively small tails, if K-3<0, the distribution is flat and has larger tails than expected in a normal distribution. Thus we reject the hypothesis that the latter two are normal. www.sbe.csuhayward.edu /~acassuto/mgmt6110/kurtosis.htm   (253 words)

 Business Wire: Elanix Teams With Kurtosis to Offer Industry's ... @ HighBeam Research   (Site not responding. Last check: 2007-10-08) Kurtosis has been involved for three years in xDSL algorithm design for leading manufacturers in the industry. One of Kurtosis' key accomplishments in this area is a unique and patented channel characterization technology developed and integrated into a Smart ADSL Line Tester (SALT) in partnership with Alcatel. Kurtosis Ingniere Ltd. (www.kurtosis.fr), based in France, is a private SME specialized in digital signal processing and communications. www.highbeam.com /library/doc0.asp?DOCID=1G1:61423064&refid=holomed_1   (693 words)

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