Hi Mr Ebart i Would say that "kur·to·sis /kərˈtōsəs/ noun Statistics noun: kurtosis is Defined as The sharpness of the peak of a frequency-distribution curve." OR

In probability theory and statistics "kurtosis (from Greek: κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable". In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population.

Depending on the particular measure of kurtosis that is used, there are various interpretations of kurtosis, and of how particular measures should be interpreted.

The kurtosis is the fourth standardized moment, defined as (Check Diagram 1). Where μ4 is the fourth central moment and σ is the standard deviation. Several letters are used in the literature to denote the kurtosis. A very common choice is κ, which is fine as long as it is clear that it does not refer to a cumulant. Other choices include γ2, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis.

The kurtosis is bounded below by the squared skewness plus 1: (Check Diagram 2). Where μ3 is the third central moment. The lower bound is realized by the Bernoulli distribution. There is no upper limit to the excess kurtosis of a general probability distribution, and it may be infinite. A reason why some authors favor the excess kurtosis is that cumulants are extensive. Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. For example, let X1, ..., Xn be independent random variables for which the fourth moment exists, and let Y be the random variable defined by the sum of the Xi.

What is Kurtosis? For symmetric, unimodal distributions:

•A positive value tells you that you have heavy-tails (i.e. a lot of data in your tails).

•A negative value means that you have light-tails (i.e. little data in your tails). This heaviness or lightness in the tails usually means that your data looks flatter (or less flat) compared to the normal distribution. The standard normal distribution has a kurtosis of 3, so if your values are close to that then your graph is nearly normal. These nearly normal distributions are called mesokurtic. Kurtosis is the fourth moment in statistics.

Mesokurtic: Mesokurtic distributions are technically defined as having a kurtosis of zero, although the distribution doesn’t have to be exactly zero in order for it to be classified as mesokurtic. The most common mesokurtic distributions are:

•The normal distribution.

•Any distribution with a Gaussian (normal) shape and zero probability at other places on the real line.

•The binomial distribution is mesokurtic for some values (i.e. for p = 1/2±√(1/12).

Excess kurtosis?: Excess kurtosis is usually defined as kurt – 3 (see Important note about equations). It is a measure of how the distribution’s tails compare to the normal (Aldrich, E, 2014).

•Excess kurt for the normal distribution is 0 (i.e. 3 -3 = 0).

•Negative excess equals lighter tails than a normal distribution.

•Positive excess equals heavier tails than the normal.

Calculating Kurtosis: Important Note about formulas: There’s no real consensus for exactly what the correct equation is for calculating kurt. Which definition/equation you use is a matter of convention in your field, the particular software you’re working with, and sometimes the preference of the author. Therefore, it’s a good idea to check which formula you’re working with. This Cross Validated thread has an excellent rundown of the different equations and which software uses which equation.

For Minitab and SPSS, you can find the option in the “Descriptive Statistics” tab.

Kurtosis Excel 2013: KURT function:

Step 1: Type your data into columns in an Excel worksheet.

Step 2: Click a blank cell.

Step 3: Type “=KURT(A1:A99)” where A1:99 is the cell locations for your data.

Kurtosis Excel 2013: Data Analysis:

Step 1: Click the “Data” tab and then click “Data Analysis.”

Step 2: Click “Descriptive Statistics” and then click “OK.”

Step 3: Click the Input Range box and then type the location for your data. For example, if you typed your data into cells A1 to A10, type “A1:A10” into that box.

Step 4: Click the radio button for Rows or Columns, depending on how your data is laid out.

Step 5: Click the “Labels in first row” box if your data has column headers. Step 6: Click the “Descriptive Statistics” check box.

Step 7: Select a location for your output. For example, click the “New Worksheet” radio button.

Step 8: Click “OK.”

I didnot mean to Teach How to do or Calculate Kurtosis by Excel or SPSS OR Minitab or "R". The point I am Making is as it says in Paragraph up "Depending on the particular measure of kurtosis that is used, there are various interpretations of kurtosis, and of how particular measures should be interpreted.". Might the Softwares you have used R and Excel their Interpretations are Different than each other or they used the Formula which have Different words and Symbols. That is why your Results are 1 Point different not a big Variation though. I hope i have made my point. Regards

Dear Ahsan, Thank you for your detailed answer. I had three formulae for kurtosis, but your answer showed that two of them were the same. I deleted one.

I checked skewness. Excel gives a value of 2.772706 while R gives 2.287366. So there is some difference in formulae there too (used the data provided in original question).

Is there a discussion of when (or why) one would use one formula over another?

Is there a name for the different formulae where it is easy to distinguish one from another without resorting to writing out each equation?

It seems like it is user beware: if you use multiple software programs you might get odd results because part of the time you are using one formula and part of the time you are using another formula.