Python Functions for Computing Skewness and Kurtosis

If the data does not follow a normal distribution, as determined by the skewness measurement, this may indicate a heavy-tailed distribution, which can be determined by examining the kurtosis statistic.

Among different types of data and probability distributions, normal distributions are the most common. This distribution can be described as a bell-shaped curve.

This blog post will cover the fundamental concepts of skewness and kurtosis, and how they apply to the Python programming language in software development. When a normally distributed data set becomes skewed, skewness and kurtosis are two key factors to consider. We will explain how these essential concepts can be measured using Python, as well as the impact of skewness and kurtosis on Python’s use in data analysis.

Normal Probability Distribution

A normal distribution refers to a continuous probability distribution of unforeseeable values that may be dependent on a random factor. For instance, when tossing a coin, it is impossible to predict the outcome – heads or tails, since the result is entirely unpredictable.

Probability distribution is a visualized representation of the possible outcomes and their likelihoods within a given set of random variables. It is commonly displayed as a scatter plot, plotting one rare event against another event. Continuous probability distribution is utilized to portray the distribution of possible outcomes when random variables are capable of taking any value within a specific range.

The potential values that probability can assume are infinite, leading to the creation of an infinite curve when plotted. Rather than detailing the probability terms, one can designate the probability value ranges.

A bell-shaped curve signifies the normal distribution, which is a continuous probability distribution having a distinctive peak lying close to the mean. The distribution also exhibits symmetry, as the median, mode and mean are close to each other.


Skewness is a valuable statistical technique utilized to assess and quantify the form of a given frequency distribution. This approach can help measure any asymmetrical patterns in the distribution, which is usually represented by a numerical value that could either be positive or negative, rather than simply computing the data points within the distribution.

If skewness is positive, the tail will appear to the right and extend to the highest values.
On the other hand, if the skewness is negative, the tail will be extended to the left and reaches farther into the negative side.
Meanwhile, when the value is 0, it implies that there is no skewness and that the distribution is entirely symmetrical.

Presented below is a table that displays the distribution of skewness:

  • A skewness value of 0 indicates that a distribution is normal.
  • When the left side of a distribution is more significant than the right side, a positive skewness value results.
  • If the right side of the distribution is more significant than the left side, it results in negative skewness value.

Assessing Skewness

The Fisher-Pearson Coefficient of Skewness is the most commonly used approach for measuring skewness. Apart from this, various other methods like Bowley, Kelly’s Measure, and Momental are also utilized to evaluate skewness. It’s worth mentioning that several other techniques are available for assessing skewness.

Skewness is a statistical technique that examines the third moment of a distribution, measuring the degree of the asymmetry of data points. While it can be challenging to understand at first, following the step-by-step instructions provided can help in gaining a better grasp of the concept.


Consider the following set of ten numbers that represents marks obtained in an exam:
X = [54, 73, 59, 98, 68, 45, 88, 92, 75, 96]

When we calculate the mean of X, we obtain:

x = 74.8

Applying the skewness formula to arrive at a solution:

m 3 = [(54 – 74.8)3 – (73 – 74.8)3 – …… – (96 – 74.8)3] / 10

The Fisher-Pearson Coefficient calculates the skewness as 0.745631, indicating a positive skew in the numbers.
To verify these numbers, one may also refer to the mode, median, and mean.


The frequency distribution possesses a kurtosis measure, which is a statistical technique used to illustrate the shape of a frequency distribution. Besides determining whether a distribution is heavy-tailed or not, the kurtosis measure can also uncover the form of the frequency distribution.

A normal distribution has a kurtosis of 3. If the kurtosis falls under 3, it’s referred to as platykurtic, whereas if the kurtosis value is above 3, it is labelled as leptokurtic. Such leptokurtic distribution deviates from the normal distribution by the presence of a few extreme values.

Analysis of Kurtosis

The standardised fourth moment offers a formula to determine kurtosis. The steps for solving this equation are explained below.

Skewness is one of the four moments of a distribution.


Allow us to consider another scenario where a set of ten numbers represent test scores. Let X = [54, 73, 59, 98, 68, 45, 88, 92, 75, 96]

Upon determining the mean of X, we obtain:

x = 74.8

This figure can be integrated into the kurtosis formula to obtain the ultimate outcome.

Functions in Python for Calculating Skewness and Kurtosis

To Begin, the SciPy Library Must Be Imported.

The SciPy Toolkit is a cost-free and open-source scientific computing library that offers built-in routines to users for the computation of skewness and kurtosis. The code below can be used to employ these routines:

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# importing
import SciPy

The second step requires creation of a dataset.

The following code illustrates an example of creating a dataset, which is the subsequent step.

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# creating a data set
dataset = [10, 25, 14, 26, 35, 45, 67, 90, 40, 50, 60, 10, 16, 18, 20]

Step 3: Computing Skewness

To calculate skewness, the built-in skew() function can be utilized as illustrated below.

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spicy.stats.skew(array, axis = 0, bias = True)

The skewness value for an input object, also known as an “array,” is obtained by specifying the axis along which the calculation is to be performed. Furthermore, one can indicate whether the statistical bias should be considered (by setting the “bias” argument to either “True” or “False”).

Using this output will result in a horizontal skewness value for the dataset, which indicates a distribution that is more positively skewed than usual.

Step 4: Computing Kurtosis

To calculate kurtosis, utilize the built-in kurtosis() function with the following syntax:

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spicy.stats.kurtosis(array, axis = 0, fisher = True, bias = True)

Here, the components are stored in an array with the axis indicating the desired kurtosis value.

When the average is zero, Fisher’s equation is satisfied, resulting in an accurate output. On the other hand, if the mean is three, the statement will be incorrect. The determination of whether bias is present or not is context-dependent and can be either true or false in statistical terms.

Computing the kurtosis value for the provided dataset will reveal if the data’s distribution is more peaked than usual. This could demonstrate that the distribution’s outputs have a more extensive range of values than anticipated with a normal distribution.

Measures of Central Tendency in Statistics

It is generally recognized that all measurable entities on earth are impacted by various chance factors. However, if a process is subject to significant influences, a measure like skewness can be used to quantify the resulting changes in the distribution’s shape.

If we notice an uneven distribution, we must examine methods to determine its range.

It’s crucial to comprehend how measures of central tendency are impacted when the normal distribution is skewed, as shown in the preceding example. The graph on the left is skewed negatively, with a tail stretching to the left. The graph on the right, on the other hand, is positively skewed, having a tail extending towards the right.

It’s crucial to determine the amount of deviation in the horizontal axis between the primary measures of central tendency (mode and mean). It’s worth noting that, as skewness increases, the difference between these figures becomes more pronounced.

The formula for skewness is shown below:

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Skewness = (Mean - Mode) / Standard Deviation

To facilitate comparison, we can divide the values of a dataset by its standard deviation, making all distributions appear to be of equal size. If working with datasets that are relatively small, it may not be necessary to calculate the mode; instead, it’s wise to replace the mode calculation with a well-defined formula for skewness.

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Mode = 3*(Median) - 2*(Mean)

By inserting the median value, we obtain:

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Skewness = 3*(Mean - Median) / Standard Deviation

It can be advantageous to investigate the repercussions of flipping the traditional normal distribution curve. The primary features to assess are the peak and tails of the curve, which will be precisely measured by the kurtosis statistic.

Given the intricacy of the kurtosis calculation, it’s vital to maintain a consistent conceptual approach.

To reiterate, the kurtosis of a normal distribution is 3, known as mesokurtic. A leptokurtic distribution has a kurtosis greater than 3, while a platykurtic distribution has a kurtosis lower than 3. The kurtosis of a distribution can range from 1 to infinity, and as the kurtosis value rises, the distribution’s peak becomes higher.

By using zero as a reference point for normality, we can utilise the following formula to calculate the degree of additional kurtosis:

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Excess Kurtosis = Kurtosis - 3

Skewness is a metric that will gauge how much a dataset’s distribution deviates from the normal curve, caused by a horizontal shift. Conversely, the kurtosis statistic quantifies the degree of vertical distortion in the data primarily due to outlier values.

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