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How do you find the kurtosis of a normal distribution?

How do you find the kurtosis of a normal distribution?

The normal distribution has skewness equal to zero. The kurtosis of a probability distribution of a random variable x is defined as the ratio of the fourth moment μ4 to the square of the variance σ4, i.e., μ 4 σ 4 = E { ( x − E { x } σ ) 4 } E { x − E { x } } 4 σ 4 . κ = μ 4 σ 4 −3 .

What is the kurtosis of normal distribution?

A standard normal distribution has kurtosis of 3 and is recognized as mesokurtic. An increased kurtosis (>3) can be visualized as a thin “bell” with a high peak whereas a decreased kurtosis corresponds to a broadening of the peak and “thickening” of the tails.

What is the formula for calculating kurtosis?

x̅ is the mean and n is the sample size, as usual. m4 is called the fourth moment of the data set. m2 is the variance, the square of the standard deviation. The kurtosis can also be computed as a4 = the average value of z4, where z is the familiar z-score, z = (x−x̅)/σ.

What is the kurtosis of a normal distribution Mesokurtic?

Mesokurtic distributions have a kurtosis of zero, meaning that the probability of extreme, rare, or outlier data is zero or close to zero. Mesokurtic distributions are known to match that of the normal distribution, or normal curve, also known as a bell curve.

Why do we take 3 in kurtosis?

The kurtosis of any univariate normal distribution is 3. It is common to compare the kurtosis of a distribution to this value. An example of a platykurtic distribution is the uniform distribution, which does not produce outliers. Distributions with kurtosis greater than 3 are said to be leptokurtic.

Why does a normal distribution have a kurtosis of 3?

The sample kurtosis is correspondingly related to the mean fourth power of a standardized set of sample values (in some cases it is scaled by a factor that goes to 1 in large samples). As you note, this fourth standardized moment is 3 in the case of a normal random variable.

What are the three types of kurtosis?

There are three types of kurtosis: mesokurtic, leptokurtic, and platykurtic.

What is an acceptable kurtosis value?

The values for asymmetry and kurtosis between -2 and +2 are considered acceptable in order to prove normal univariate distribution (George & Mallery, 2010). (2010) and Bryne (2010) argued that data is considered to be normal if skewness is between ‐2 to +2 and kurtosis is between ‐7 to +7.

What does kurtosis indicate?

Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values.

Is negative kurtosis good?

A negative kurtosis implies platykurtosis. For the normal distribution the moment measure is equal to 3. This means your distribution is platykurtic or flatter as compared with normal distribution with the same M and SD.

How much kurtosis is acceptable?

A kurtosis value of +/-1 is considered very good for most psychometric uses, but +/-2 is also usually acceptable. Skewness: the extent to which a distribution of values deviates from symmetry around the mean.

What is too much kurtosis?

Excess kurtosis means the distribution of event outcomes have lots of instances of outlier results, causing fat tails on the bell-shaped distribution curve. Normal distributions have a kurtosis of three. Excess kurtosis can, therefore, be calculated by subtracting kurtosis by three.

What is the kurtosis of a perfect normal distribution?

With this definition a perfect normal distribution would have a kurtosis of zero. The second formula is the one used by Stata with the summarize command. This definition of kurtosis can be found in Bock (1975). The only difference between formula 1 and formula 2 is the -3 in formula 1.

What’s the formula with the different formulas for kurtosis?

Thus, with this formula a perfect normal distribution would have a kurtosis of three. The third formula, below, can be found in Sheskin (2000) and is used by SPSS and SAS proc means when specifying the option vardef=df or by default if the vardef option is omitted.

When to use skewness and excess kurtosis in statistics?

One application is testing for normality: many statistics inferences require that a distribution be normal or nearly normal. A normal distribution has skewness and excess kurtosis of 0, so if your distribution is close to those values then it is probably close to normal.

How to calculate the moment coefficient of kurtosis?

The moment coefficient of kurtosisof a data set is computed almost the same way as the coefficient of skewness: just change the exponent 3 to 4 in the formulas: kurtosis: a4= m4/ m22 and excess kurtosis: g2= a4−3 . (5)where. m4= ∑(x−x̅)4 / n and m2= ∑(x−x̅)2 / n.