For example, imagine a stock had an average price of $25.85 per share. If the stock’s price swung widely and often enough, the bell curve would have https://1investing.in/ heavy tails (high kurtosis). This means that there is a lot of variation in the stock price—an investor should anticipate wide price swings often.

These types of distributions have short tails (fewer outliers.). Platykurtic distributions have demonstrated more stability than other curves because extreme price movements rarely occurred in the past. The excess kurtosis in a platykurtic distribution is negative that is characterized by a flat-tail distribution. The minor outliers in a distribution are indicated by the flat tails. The platykurtic distribution of investment returns is advantageous for investors in the financial context as this would mean a higher return on investment.

However, the two concepts must not be confused with each other. Skewness essentially measures the symmetry of the distribution, while kurtosis determines the heaviness of the distribution tails. Therefore a kurtosis value of 3 for your data distribution indicates the tails are not statistically different from those of a normal distribution. Kurtosis measures how much of the data in a probability distribution are centered around the middle (mean) vs. the tails. Skewness instead measures the relative symmetry of a distribution around the mean. Beta measures the volatility a stock compared to the broader market.

Since skewness is defined in terms of an odd power of the standard score, it’s invariant under a linear transformation with positve slope (a location-scale transformation of the distribution). On the other hand, if the slope is negative, skewness changes sign. An extreme positive kurtosis indicates a distribution where more of the values are located in the tails of the distribution rather than around the mean. R-squared measures the percent of movement a portfolio or fund has that can be explained by a benchmark. Though r-squared is used in regression analysis to assess the goodness of fit of a regression model, kurtosis is used in descriptive statistics to describe the shape of a distribution.

The sociologist calculates that the kurtosis of the sample is 1.78 and its excess kurtosis is −1.22.

For example, the KURT() function in Excel calculates kurtosis using the above formula.

A distribution or dataset is symmetric if it looks the same to the left and right of the center point.

It is a sort of distribution where the measures are dispersing, unlike symmetrically distributed data where all measures of the central tendency (mean, median, and mode) equal each other.

Note—these numbers do not represent standard deviation; they represent the variance of each data point.

In positively skewed, the mean of the data is greater than the median (a large number of data-pushed on the right-hand side).

The PDF \( f \) is clearly not symmetric about 0, and the mean is the only possible point of symmetry. Over 1.8 million professionals use CFI to learn accounting, financial analysis, modeling and more. Start with a free account to explore 20+ always-free courses and hundreds of finance templates and cheat sheets. For example, suppose the data values are 0, 3, 4, 1, 2, 3, 0, 2, 1, 3, 2, 0, 2, 2, 3, 2, 5, 2, 3, 999. The kurtosis of a sample is an estimate of the kurtosis of the population. A trick to remember the meaning of “platykurtic” is to think of a platypus with a thin tail.

Data Visulization Libraries

A stock with a leptokurtic distribution generally depicts a high level of risk but the possibility of higher returns because the stock has typically demonstrated large price movements. Distributions with a large kurtosis have more tail data than normally distributed data, which appears to bring the tails in toward the mean. Distributions with low kurtosis have fewer tail data, which appears to push the tails of the bell curve away from the mean.

Kurtosis – Definition, Example, Types

Median is the middle value, and mode is the most frequent value. Due to an unbalanced distribution, the median will be higher than the mean. The following exercise gives a simple example of a discrete distribution that is not symmetric but has skewness 0. The converse is not true—a non-symmetric distribution can have skewness 0. When kurtosis is equal to 0, the distribution is platykurtic.

Platykurtic (Kurtosis

Where X is a random variable, μ is the mean and σ is the standard deviation. The astronomers calculate that the kurtosis of the sample is 6.54 and its excess kurtosis is 3.54. The sociologist calculates that the kurtosis of the sample is 1.78 and its excess kurtosis is −1.22. She finds that the kurtosis is 3.09 and the excess kurtosis is 0.09, and she concludes that the distribution is mesokurtic. Pearson’s second coefficient of skewnesssubtract the median from the mean, multiply the difference by 3, and divide the product by the standard deviation.

What is Kurtosis?

This distribution has a kurtosis similar to that of the normal distribution, meaning the extreme value characteristic of the distribution is similar to that of a normal distribution. Therefore, a stock with a mesokurtic distribution generally depicts a moderate level of risk. Kurtosis is sometimes confused with a measure of the peakedness of a distribution. However, kurtosis is a measure that describes the shape of a distribution’s tails in relation to its overall shape.

Vary the shape parameter and note the shape of the probability density function in comparison to the moment results in the last exercise. For selected values of the parameter, run the experiment 1000 times and compare the empirical density function to the true probability density function. Kurtosis is a statistical measure that quantifies the shape of a probability distribution. It provides information about the tails and peakedness of the distribution compared to a normal distribution. In statistics, a positively skewed or right-skewed distribution has a long right tail. It is a sort of distribution where the measures are dispersing, unlike symmetrically distributed data where all measures of the central tendency (mean, median, and mode) equal each other.

Kurtosis excess is commonly used because of a normal distribution is equal to 0, while the kurtosis proper is equal to 3. Unfortunately, Abramowitz and Stegun (1972) confusingly refer to as the “excess or kurtosis.” For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see variance. Note that in these cases the platykurtic densities have bounded support, whereas the densities with positive or zero excess kurtosis are supported on the whole real line. The kurtosis can now be seen as a measure of the dispersion of Z2 around its expectation. Alternatively it can be seen to be a measure of the dispersion of Z around +1 and −1.

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. A distribution with kurtosis greater than three is leptokurtic and a distribution with kurtosis less than three is platykurtic.

There is no upper limit to the kurtosis of a general probability distribution, and it may be infinite. Excess kurtosis is the tailedness of a distribution relative to a normal distribution. The excess kurtosis of a given distribution determines the forms of kurtosis. Excess kurtosis can be either positive or negative, as well as near to zero.

Leptokurtic distributions are named by the prefix “lepto” meaning “skinny.” 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.

Therefore, this tool calculates and focusses more on the “tailendness” instead of peak. A. A distribution with a negative kurtosis value indicates that the distribution has lighter tails than the normal distribution. Some statistical models are robust to outliers like Tree-based models, but it will limit the possibility of trying other models. So define kurtosis there is a necessity to transform the skewed data to be close enough to a Normal distribution. In a symmetrically distributed dataset, both the left-hand side and the right-hand side have an equal number of observations. (If the dataset has 90 values, then the left-hand side has 45 observations, and the right-hand side has 45 observations.).

Indicator variables are the building blocks of many counting random variables. The corresponding distribution is known as the Bernoulli distribution, named for Jacob Bernoulli. One of the most well known leptokurtic distributions is Student’s t distribution. Besides normal distributions, binomial distributions for which p is close to 1/2 are considered to be mesokurtic.

Since kurtosis is defined in terms of an even power of the standard score, it’s invariant under linear transformations. Prior to doing some statistical analysis on a manufacturing process, the company Six Sigma Black Belt (BB) tested her data for normality. Below is her result showing the current process data has a significant degree of kurtosis. You can see that with the long tails and a kurtosis value of over 4. If you use the above equation, the kurtosis for a normal distribution is 3. And is commonly denoted (Abramowitz and Stegun 1972, p. 928) or .

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