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.

- If your data demonstrates kurtosis, it is referring to the shape of your data distribution.
- Median is the middle value, and mode is the most frequent value.
- While kurtosis analyzes the distribution of a dataset, the Sharpe ratio more commonly is used to evaluate investment performance.
- The beta distribution is studied in detail in the chapter on Special Distributions.

Calculating kurtosis by hand is a lengthy endeavor, and takes several steps to get to the results. We’ll use new data points and limit their number to simplify the calculation. Mesokurtic distributions have outliers https://1investing.in/ that are neither highly frequent, nor highly infrequent, and this is true of the elephant birth weights. Occasionally, a female baby elephant will be born weighing less than 180 or more than 240 lbs.

For investors, kurtosis is important in understanding tail risk, or how frequently “infrequent” events occur, given one’s assumption about the distribution of price returns. Open the special distribution simulator, and select the continuous uniform distribution. Vary the parameters and note the shape of the probability density function in comparison with the moment results in the last exercise. For selected values of the parameter, run the simulation 1000 times and compare the empirical density function to the probability density function.

Instead, it approximately follows a uniform distribution (shown by the purple curve). Kurtosis is a statistical term used to quantify distribution that is like skewness. Unlike skewness, which only distinguishes absolute value in one tail from those in the other, kurtosis assesses extreme values in both tails.

## Exploring Continuous Variable

Κ attains its minimal value in a symmetric two-point distribution. In terms of the original variable X, the kurtosis is a measure of the dispersion of X around the two values μ ± σ. Leptokurtosis is sometimes called positive kurtosis, since the excess kurtosis is positive. A leptokurtic distribution is fat-tailed, meaning that there are a lot of outliers. Platykurtosis is sometimes called negative kurtosis, since the excess kurtosis is negative.

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.

Each security or investment has a single beta that indicates whether or not that security is more or less volatile compared to a market benchmark. Again, beta measures the degree of asymmetry of the distribution, while kurtosis measures the peak or flatness of the distribution. While kurtosis measures the nature of the peak or flatness of the distribution, alpha measures the skewness or asymmetry of the distribution. Instead, it approximately follows a Laplace distribution (shown by the blue curve). From the graph, we can see that the frequency distribution (shown by the gray bars) approximately follows a normal distribution (shown by the green curve). With low kurtosis, a distribution can be extremely peaked as well, and with infinite kurtosis, it can be completely normal or flat with no deviation.

## Scientific definitions for kurtosis

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 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.

A bell peak is shown with most data within three standard deviations within + or – variations of the mean and can be seen when normal data is graphed using a histogram. During a strong kurtosis, this bell histogram tends to extend beyond the normal variations’ length. It gives an idea about the shape of a frequency distribution. Basically, the measure of kurtosis is the extent to which a frequency distribution is peaked in comparison with a normal curve. Platykurtic having a thin tail and stretched around the center means most data points are present in high proximity to the mean.

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.

## Data Visulization Libraries

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.

## Kurtosis vs. Other Commonly Used Measurements

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.).

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.

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.