It is inversely proportional; a larger standard deviation produces a small kurtosis (smaller peak, more spread out data) and a smaller standard deviation produces a larger kurtosis (larger peak, data more centrally located).
I will answer your question in a couple of ways. First as a concept: 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. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case. Now as a mathematical formula: For univariate data Y1, Y2, ..., YN, the formula for kurtosis is:where is the mean, is the standard deviation, and N is the number of data points. You may find more information at this website: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
mesokurtic leptokurtic platykurtic
No, a distribution can have infinitely many moments: the first is the mean, the second variance. Then there are skewness (3), kurtosis (4), hyperskewness (5), hyperflatness (6) and so on.If mk represents the kth moment, thenmk = E[(X - m1)k] where E is the expected value.It is, therefore, perfectly possible for m1 and m2 to be the same but for the distribution to differ at the higher moments.
A variable that shows serious departure from the classic bell-shaped, or "Gaussian", curve is described as being not normally distributed. This departure could take the form of skew and/or kurtosis and/or multi modality.An example might be weekly wages. If you drew a histogram of a population's earnings you would most likely see a distribution skewed significantly toward the right. That is, toward the higher incomes.Another example is height. If you drew a histogram of a population's height you would see a bimodal distribution. One peak for males and another peak for females. The distribution of height for males and females might be normal when looked at individually, but not normal when you combine them.
It means distribution is flater then [than] a normal distribution and if kurtosis is positive[,] then it means that distribution is sharper then [than] a normal distribution. Normal (bell shape) distribution has zero kurtosis.
Kurtosis is a measure of the "peakedness" or thickness of the tails of a distribution compared to a normal distribution. A positive kurtosis indicates a distribution with heavier tails and a sharper peak, while a negative kurtosis indicates lighter tails and a flatter peak. Kurtosis helps to understand the shape of a distribution and the likelihood of extreme outcomes.
For N(0, 1) it is 3.
The kurtosis of a distribution is defined as the fourth central moment divided by the square of the second central moment. Unfortunately, this browser converts Greek characters to the Roman alphabet so I cannot use standard forms of equations but: Suppose that for a random variable X, E(X) = m (mu) and E[(X - E(X))2] = V = s2 (sigma-squared) then Kurtosis = E[(X - E(X))4]/s4. Excess Kurtosis is then Kurtosis - 3. If excess kurtosis < 0 the distribution is platykurtic. They have a peak that is lower than the Normal: the peak is flat and broad. The tails of the distribution are narrow. Uniform distributions are platykurtic. A mesokurtic distibution has excess kurtosis = 0. The Gaussian (Normal) distribution - whatever its parameters - is mesokurtic. The binomial with probability of success close to 1/2 is also considered to be mesokurtic. If excess kurtosis is > 0 the distribution is leptokurtic. Leptokurtic distributions have a high and narrow peak. A good example is the Student's t distribution.
There are many, many formulae:for different probability distribution functions,for cumulative distribution functions,for moment generating functions,for means, variances, skewness, kurtosis and higher moments.There are many, many formulae:for different probability distribution functions,for cumulative distribution functions,for moment generating functions,for means, variances, skewness, kurtosis and higher moments.There are many, many formulae:for different probability distribution functions,for cumulative distribution functions,for moment generating functions,for means, variances, skewness, kurtosis and higher moments.There are many, many formulae:for different probability distribution functions,for cumulative distribution functions,for moment generating functions,for means, variances, skewness, kurtosis and higher moments.
While skewness is the measure of symmetry, or if one would like to be more precise, the lack of symmetry, kurtosis is a measure of data that is either peaked or flat relative to a normal distribution of a data set. * Skewness: A distribution is symmetric if both the left and right sides are the same relative to the center point. * Kurtosis: A data set that tends to have a distant peak near the mean value, have heavy tails, or decline rapidly is a measure of high kurtosis. Data sets with low Kurtosis would obviously be opposite with a flat mean at the top, and a distribution that is uniform.
It is inversely proportional; a larger standard deviation produces a small kurtosis (smaller peak, more spread out data) and a smaller standard deviation produces a larger kurtosis (larger peak, data more centrally located).
In probability theory and statistics, kurtosis (from the Greek word κυρτός, kyrtos or kurtos, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly sized deviations. Sometimes kurtosis gets confused with skewness, so I have added links to both these terms.
Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. See related link. By doing a search on the internet, you can find more examples.
I will answer your question in a couple of ways. First as a concept: 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. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case. Now as a mathematical formula: For univariate data Y1, Y2, ..., YN, the formula for kurtosis is:where is the mean, is the standard deviation, and N is the number of data points. You may find more information at this website: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
The Greek word "kurtosis", when translated to English, means the probability theory of any measure of the "peakedness" of a real valued random variable.
mesokurtic leptokurtic platykurtic