Kurtosis

Kurtosis
See also
Harmonic mean Confidence level Coefficient of determination Autocorrelation Continuous distribution C chart Central tendency Asymmetrical distribution Probability density function

Kurtosis is a measure of the "peakedness" of a probability distribution. It is a statistical measure that compares the amount of data near the mean with data far from the mean, and can be used to compare different probability distributions. Generally, distributions with high kurtosis have a sharper peak near the mean and longer tails than distributions with low kurtosis.

Kurtosis can be used to compare different probability distributions, and is especially useful for determining whether a distribution has outliers that are skewing the overall data. High kurtosis distributions have a sharper peak and longer tails than distributions with low kurtosis. Thus, high kurtosis can indicate that there are a few extreme values that are pulling the data away from the mean. This can be used to identify outliers in data sets.

Kurtosis is an important measure of a probability distribution and can be used to compare different distributions. It can also be used to identify outliers in data sets and to assess the skewness of a distribution.

Example of Kurtosis

To illustrate the concept of kurtosis, we can look at two probability distributions. The first is a normal distribution with a mean of 0, standard deviation of 1, and kurtosis of 0.

The second is a distribution with the same mean and standard deviation, but with a kurtosis of 5. This distribution has a sharper peak near the mean and much longer tails than the normal distribution.

These two probability distributions illustrate the concept of kurtosis and how it can be used to compare different probability distributions. High kurtosis distributions have a sharper peak and longer tails than distributions with low kurtosis. This can be used to identify outliers in data sets and to assess the skewness of a distribution.

Formula of Kurtosis

Kurtosis is measured using the following formula:

$kurtosis={\frac {N(N+1)}{(N-1)(N-2)(N-3)}}{\frac {\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{4}}{(\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2})^{2}}}$

In this formula, N is the number of data points and xi is the ith value. This formula is used to measure the peakedness of a probability distribution. By comparing the amount of data near the mean with data far from the mean, kurtosis can be used to compare different probability distributions and assess the skewness of a distribution.

In summary, kurtosis is a measure of the peakedness of a probability distribution and is measured using the above formula. It can be used to compare different probability distributions, and is especially useful for determining whether a distribution has outliers that are skewing the overall data.

When to use Kurtosis

Kurtosis can be used to compare different probability distributions and to identify outliers in data sets. It can be used to assess the skewness of a distribution, as high kurtosis distributions have a sharper peak and longer tails than distributions with low kurtosis. Kurtosis can also be used to determine if a distribution is platykurtic, mesokurtic, or leptokurtic, which can be useful for understanding the data.

Types of Kurtosis

Kurtosis comes in two types:

• Positive Kurtosis: This type of kurtosis is seen in distributions that have a high peak and long tails. This is often observed in data sets with a few extreme values that are pulling the data away from the mean.
• Negative Kurtosis: This type of kurtosis is observed in distributions that have a low peak and short tails. This is often observed in data sets with a few extreme values that are pulling the data towards the mean.

Kurtosis is an important measure of a probability distribution that can be used to compare different distributions and to identify outliers in data sets. By understanding the type of kurtosis present in a distribution, it is possible to assess the skewness of the distribution and identify extreme values that are pulling the data away from or towards the mean.

Steps of Kurtosis

• Calculate the mean of the data set:

${\bar {x}}={\frac {\sum _{i=1}^{N}x_{i}}{N}}$

• Calculate the differences between the data points and the mean:

$d_{i}=x_{i}-{\bar {x}}$

• Calculate the fourth power of these differences:

$d_{i}^{4}=(x_{i}-{\bar {x}})^{4}$

• Calculate the sum of these fourth powers:

$\sum _{i=1}^{N}d_{i}^{4}=\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{4}$

• Calculate the second power of the differences:

$d_{i}^{2}=(x_{i}-{\bar {x}})^{2}$

• Calculate the sum of these second powers:

$\sum _{i=1}^{N}d_{i}^{2}=\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}$

• Calculate the kurtosis:

$kurtosis={\frac {N(N+1)}{(N-1)(N-2)(N-3)}}{\frac {\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{4}}{(\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2})^{2}}}$

Kurtosis is a measure of the peakedness of a probability distribution and is a useful tool for comparing different probability distributions. The formula for calculating kurtosis requires calculating the mean of a data set, the differences between the data points and the mean, the fourth power of these differences, the sum of these fourth powers, the second power of the differences, the sum of these second powers, and the kurtosis itself. By using this formula, kurtosis can be used to identify outliers in data sets and to assess the skewness of a distribution.

Advantages of Kurtosis

• Kurtosis is a measure of the peakedness of a probability distribution, and can be used to compare different distributions.
• It can be used to identify outliers in data sets, as high kurtosis can indicate a few extreme values that are skewing the overall data.
• It can also be used to assess the skewness of a distribution.

Limitations of Kurtosis

Kurtosis is a limited measure of a probability distribution and has several drawbacks. These include:

• Kurtosis does not take into account the range of the data, so it can be misleading if the data range is very small.
• Kurtosis does not indicate whether or not the data is normally distributed.
• Kurtosis does not indicate the shape of the probability distribution.

Other approaches related to Kurtosis

Kurtosis is related to other statistical measures, such as skewness, which is a measure of the asymmetry of a distribution. Skewness measures the degree to which a probability distribution deviates from a normal distribution and can be used to compare different distributions.

In addition, Kurtosis is related to the Jarque-Bera test, which is a statistical test used to detect departures from normality. The Jarque-Bera test uses the skewness and kurtosis of a data set to determine whether the data are normally distributed.

In sum, Kurtosis is an important measure of a probability distribution and can be used to compare different distributions. It is related to other statistical measures, such as skewness and the Jarque-Bera test, which are both used to assess the normality of a data set.

Suggested literature

• Balanda, K. P., & MacGillivray, H. L. (1988). Kurtosis: a critical review. The American Statistician, 42(2), 111-119.
• DeCarlo, L. T. (1997). On the meaning and use of kurtosis. Psychological methods, 2(3), 292.