Kurtosis - The misunderstood Measure

Data will tell us everything. We must be willing to listen!

Exploratory Data Analysis (EDA) takes up a considerable portion of a data scientist's workday. Since each piece of data from different domains is unique, it is necessary to unlearn all you previously knew about each piece of data and re-learn it.

Amongst the 4 moments of business decision namely

• Measures of Central Tendency
• Measure of Dispersion
• Measure of Skewness
• Measure of Kurtosis

The first three are widely employed by data scientists and academics, but the study of kurtosis is one of the most overlooked and undervalued aspects of business choices.

Kurtosis is a statistical metric that describes how divergent a distribution's tails are from a normal distribution in terms of weight. The kurtosis will be able to identify the same extreme values if there are any in the data.

Kurtosis is a measure of how "tailed" the distribution is, according to another definition.

Kurtosis evaluates extreme values in both tails, in contrast to skewness, which only discusses the absolute value in one of the tails.

In many cases, the kurtosis is lowered by 3 to make the value of the normal distribution equal to 0, which acts as a baseline level of the kurtosis. A perfectly normal distribution has a kurtosis value of 3.

To visualize high and low kurtosis

Let us imagine a snowcapped mountain that is quite “sharp” this is high kurtosis as the peak is very thin and narrow.

When the snow melts the peak becomes rounded and the snow and slush flow down to the sides of the mountain. Thereby making the kurtosis lower as the tails are becoming heavier.

While Skewness essentially measures the symmetry of the distribution, kurtosis on the other hand determines the heaviness of the distribution tails and the Peakedness of the distribution

Kurtosis is a measure of how tailed the probability distribution is. A standard normal distribution has a kurtosis of 3 and is notated as mesokurtic. Kurtosis >3 is recognized as leptokurtic and <3 as platykurtic (lepto=thin; platy=broad). There are four different formats of kurtosis, the simplest is the population kurtosis; the ratio between the fourth moment and the variance.

“Quite a few textbooks describe kurtosis as simply a measure of peakedness (positive kurtosis) or flatness (negative kurtosis), with little or no mention of the tails. Kaplansky (1945) referred to the method of describing kurtosis in terms of peakedness alone as a "common error," made in statistics textbooks of the 1940s. As counterexamples to this notion, Kaplansky gave density functions for distribution with positive kurtosis but a lower peak than the normal, and a distribution with negative kurtosis but a higher peak than the normal. The counter-examples illustrate why the definition of kurtosis solely in terms of peakedness or flatness can be misleading”. Click here to learn Data Science Course in Hyderabad

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Kurtosis and the Variance

“One other issue is that many textbooks don't make a distinction between kurtosis and variance. For instance, it's sometimes said that high and negative kurtosis, respectively, indicate significant and little variation. Although it is scale-free, it should be noted that the kurtosis measure 2 is scaled with respect to the variance. In addition to the variance, kurtosis represents the shape of a distribution”.

Kurtosis and Normality

“Part of a complete statistical analysis is an assessment of the assumptions, including any distributional assumptions. When using normal theory methods, the assumption of normality is often checked. Other reasons for assessing normality are because away from normal distribution can affect tests and confidence intervals based on normal theory methods, and because the reduction of multivariate data to covariance matrices may overlook important aspects of the data”

Fundamental to the understanding of statistics is recognizing the distribution of data.

In applied sciences and particularly in those disciplines where experimental work is included, there is a need for robust estimates, e.g., a choice of central tendencies and trimmed datasets.

The concept of uncertainty in measurements used in the accreditation of laboratories is described in some detail and different estimation techniques are described.

A major task in laboratories is to compare different datasets, which come from the comparison of measurement procedures and the comparison of results obtained in different experimental models. This opens a wide field of different statistical procedures including analysis of variance and variance components. Click here to learn Data Science Course

Types of Kurtosis

The excess kurtosis of a specific distribution determines the different forms of kurtosis. The excess kurtosis might have values that are positive, negative, or almost zero.

• Mesokurtic

  The excess kurtosis in a mesokurtic distribution is zero or nearly zero. This suggests that the distribution of the data is normal.

  

• Leptokurtic

  The distribution exhibits an excess of positive kurtosis when it is leptokurtic. The "Heavy Tails" showing the presence of significant outliers make this apparent. Being the source of significant ambiguity, the inference is typically not chosen.

  

• Platykurtic

  It exhibits a disproportionately negative kurtosis in the platykurtic distribution. The "Flat Tails" show minimal or no outliers in the data, which is ideal, indicating this.

  Kurtosis is technically defined as the standardised fourth population moment about the mean, where sigma is the standard deviation, E is the expectation operator, Mu is the mean, and Mu4 is the fourth moment about the mean.

  

  The same may be described for sample data as follows, where n is the number of observations, X bar represents the sample mean, and b2 is the sample kurtosis.

  

Application of Kurtosis

In Finance

The kurtosis measurement is used in the financial application to gauge financial risk. Due to the high likelihood of both extremely big and extremely tiny returns, a high kurtosis is associated with a high level of investment risk. A low kurtosis, on the other hand, suggests a moderate amount of risk because the likelihood of extreme returns is relatively low.