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# Kurtosis

Kurtosis is a characteristic used in probability to measure the peakedness of a probability distribution. This measure represents a way of describing a data set which should be normally distributed. This concept of kurtosis is one which helps to describe the shape of a probability distribution, which under the normal distribution should take on a bell-shaped curve. The measure of kurtosis indicates whether the distribution is flat, highly peaked or normal.

There are specific terms which are used in statistics to describe the kurtosis of a distribution. The term platykurtic is used to when the distribution is flat, leptokurtic describes when the distribution is peaked and mesokurtic describes a normal distribution1. In terms of visual appearance, a platykurtic distribution is one in which the peak is very pointed and tall and the tails are thick. On the other hand, a leptokurtic distribution has very thin tails and a very flat, broad peak. Of course, a mesokurtic distribution is representative of a nice bell-shaped curve.

Evaluating the shape of the curve is qualitative in nature and thus, the problem of subjectivity can arise. Therefore, to counter this problem, there is also a numerical way to determine the kurtosis value associated with a distribution. Kurtosis is known to be the fourth value around the mean and can be evaluated using the following formula2:

Î²2 = E(x â€“ Î¼)4 /Ïƒ4

Variables:
Î²2 = kurtosis
Î¼ = mean
Ïƒ = standard deviation
x = random variable

Reference:

1. http://www.r-tutor.com/elementary-statistics/numerical-measures/kurtosis

2. http://knowledge.sagepub.com.proxy.queensu.ca/view/researchdesign/n208.xml?rskey=jQx84c&row=2

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### Expectations - kurtosis

Kurtosis= E[(X-Âµ)]^4]/Î£^4 Find the measure of kurtosis for f(x)=1/2, -1 <x <1, zero elsewhere.