Concept of Kurtosis, Business Mathematics & Statistics

# Concept of Kurtosis, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com

## What Is Kurtosis?

Distributions of data and probability distributions are not all the same shape. Some are asymmetric and skewed to the left or to the right. Other distributions are bimodaland have two peaks. Another feature to consider when talking about a distribution is the shape of the tails of the distribution on the far left and the far right.  Kurtosis is the measure of the thickness or heaviness of the tails of a distribution.

The kurtosis of a distributions is in one of three categories of classification:

• Mesokurtic
• Leptokurtic
• Platykurtic

We will consider each of these classifications in turn. Our examination of these categories will not be as precise as we could be if we used the technical mathematical definition of kurtosis.

### Mesokurtic

Kurtosis is typically measured with respect to the normal distribution. A distribution that has tails shaped in roughly the same way as any normal distribution, not just the standard normal distribution, is said to be mesokurtic. The kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications.

Besides normal distributions, binomial distributions for which p is close to 1/2 are considered to be mesokurtic.

### Leptokurtic

A leptokurtic distribution is one that has kurtosis greater than a mesokurtic distribution.

Leptokurtic distributions are sometimes identified by peaks that are thin and tall. The tails of these distributions, to both the right and the left, are thick and heavy. Leptokurtic distributions are named by the prefix "lepto" meaning "skinny."

There are many examples of leptokurtic distributions.

One of the most well known leptokurtic distributions is Student's t distribution.

### Platykurtic

The third classification for kurtosis is platykurtic. Platykurtic distributions are those that have slender tails.  Many times they possess a peak lower than a mesokurtic distribution. The name of these types of distributions come from the meaning of the prefix "platy" meaning "broad."

All uniform distributions are platykurtic. In addition to this the discrete probability distribution from a single flip of a coin is platykurtic.

### Calculation of Kurtosis

These classifications of kurtosis are still somewhat subjective and qualitative. While we might be able to see that a distribution has thicker tails than a normal distribution, what if we don’t have the graph of a normal distribution to compare with? What if we want to say that one distribution is more leptokurtic than another?

To answer these kinds of questions we need not just a qualitative description of kurtosis, but a quantitative measure. The formula used is μ44 where μ4 is Pearson’s fourth moment about the mean and sigma is the standard deviation.

### Excess Kurtosis

Now that we have a way to calculate kurtosis, we can compare the values obtained rather than shapes.

The normal distribution is found to have a kurtosis of three. This now becomes our basis for mesokurtic distributions. A distribution with kurtosis greater than three is leptokurtic and a distribution with kurtosis less than three is platykurtic.

Since we treat a mesokurtic distribution as a baseline for our other distributions, we can subtract three from our standard calculation for kurtosis. The formula μ44 - 3 is the formula for excess kurtosis. We could then classify a distribution from its excess kurtosis:

• Mesokurtic distributions have excess kurtosis of zero.
• Platykurtic distributions have negative excess kurtosis.
• Leptokurtic distributions have positive excess kurtosis.

### A Note on the Name

The word "kurtosis" seems odd on the first or second reading. It actually makes sense, but we need to know Greek to recognize this.

Kurtosis is derived from a transliteration of the Greek word kurtos. This Greek word has the meaning "arched" or "bulging," making it an apt description of the concept known as kurtosis.

The document Concept of Kurtosis, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com is a part of the B Com Course Business Mathematics and Statistics.
All you need of B Com at this link: B Com

115 videos|142 docs

## FAQs on Concept of Kurtosis, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com

 1. What is the concept of kurtosis in business mathematics and statistics? Ans. Kurtosis is a statistical measure that quantifies the shape of a distribution's tails relative to its peak, or the degree of peakedness or flatness of a probability distribution. It helps in determining whether a dataset has heavy tails or outliers compared to a normal distribution. In business mathematics and statistics, kurtosis is used to analyze and interpret the distribution of data and assess the risk associated with extreme events.
 2. How is kurtosis different from skewness? Ans. Skewness and kurtosis are both measures of the shape of a distribution, but they focus on different aspects. Skewness measures the asymmetry of a distribution, indicating whether it is skewed to the left or right. On the other hand, kurtosis measures the concentration of data in the tails of the distribution, indicating whether it has heavy tails or is more peaked than a normal distribution. While skewness tells us about the lack of symmetry, kurtosis provides insight into the tails of the distribution.
 3. What are the different types of kurtosis? Ans. There are three types of kurtosis: mesokurtic, leptokurtic, and platykurtic. - Mesokurtic distributions have a kurtosis of zero, indicating that their tails are similar to those of a normal distribution. - Leptokurtic distributions have positive kurtosis, meaning their tails are heavier than a normal distribution. These distributions have a higher probability of extreme events. - Platykurtic distributions have negative kurtosis, indicating that their tails are lighter than a normal distribution. These distributions have a lower probability of extreme events.
 4. How is kurtosis interpreted in business and finance? Ans. In business and finance, kurtosis is often used to assess the risk associated with extreme events or outliers. A higher kurtosis value indicates a higher risk of extreme events or heavy tails, which means that there is a greater chance of unusual or extreme outcomes occurring. This can be important in the fields of investment analysis, risk management, and insurance, where understanding the distribution of returns or losses is crucial.
 5. Can kurtosis be negative? What does negative kurtosis indicate? Ans. Yes, kurtosis can be negative. Negative kurtosis, or platykurtic distribution, indicates that the tails of a distribution are lighter or less extreme than those of a normal distribution. It implies that the dataset has fewer outliers or extreme values compared to a normal distribution. Negative kurtosis suggests a lower risk of extreme events or outliers, which can be relevant in certain business scenarios where stability or consistency is desired.

115 videos|142 docs Explore Courses for B Com exam Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Track your progress, build streaks, highlight & save important lessons and more!
Related Searches

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;