If the data are not distributed normally, as measured by skewness, then the distribution is likely heavy-tailed, as shown by the kurtosis statistic.
Normal distributions are the most prevalent kind of data and probability distributions. A bell-shaped distribution best describes this distribution.
This article will discuss the concepts of skewness and kurtosis, and their implications for the Python programming language. Skewness and kurtosis are two important factors that can lead to a normal distribution becoming skewed. We will look at how these concepts can be used in Python to measure skewness and kurtosis. Furthermore, we will explore the implications of skewness and kurtosis for the use of Python in data analysis.
The Probability Is Normally Distributed
A normal distribution is a continuous probability distribution of random values, which are contingent upon some unpredictable factor. For example, when flipping a coin, there is no way of predicting what the outcome will be – either heads or tails – as the result is completely random.
A probability distribution is a graphical representation of the relative likelihood of different potential outcomes for a given set of random variables. It is typically visualised as a scatter plot, in which one event is plotted against another event for which there is only a small chance of occurrence. A continuous probability distribution is used to describe the distribution of potential outcomes when the random variables may take on any value within a given range.
The number of values that probability can take are infinite, which results in a never-ending curve when graphed. Instead of explicitly writing out the probability terms, one can specify the range in which the values of probability fall.
The normal distribution is characterised by a bell-shaped continuous probability distribution curve, which has a clearly identifiable peak that is close to the mean. The distribution is also symmetric, with the median, mode and mean being in close proximity to each other.
Skewness is a powerful statistical tool used to evaluate and quantify the shape of a given frequency distribution. Rather than simply calculating the data points within the distribution, it can be used to measure any asymmetrical behaviour, which is commonly represented by a numerical value that is either positive or negative.
If the skew is positive, the tail will be on the right. The reach will be aimed at the highest ideals.
If the skew is negative, however, the tail will be on the left and extend farther to the negative side.
The absence of skewness, represented by a value of 0, indicates that the distribution is fully symmetrical.
Below is a table showing the skewness distribution:
- If a distribution is normal, then its skewness value will be 0.
- When the left side of a distribution is given more importance than the right, the skewness is greater than zero.
- When the right side of the distribution is given greater importance than the left, the skewness value is negative.
Analysing the Skewness
The Fisher-Pearson Coefficient of Skewness is considered to be the most widely accepted measure of skewness. In addition to this, there are several other techniques which are used to evaluate skewness, including Kelly’s Measure, Bowley, and Momental. It is worth noting that there are an array of other methods available for assessing skewness.
Skewness is a statistical measure that evaluates the third moment of a distribution, which describes the degree of asymmetry of the data points. It can be a difficult concept to comprehend initially, however, following the step-by-step instructions provided can assist in better understanding the concept.
Take into account the following sequence of ten numbers, which stands for marks on an examination.
X = [54, 73, 59, 98, 68, 45, 88, 92, 75, 96]
If we take X and find its mean, we get:
x = 74.8
Using the skewness formula to get a solution:
m 3 = [(54 – 74.8)3 – (73 – 74.8)3 – …… – (96 – 74.8)3] / 10
The skewness, as measured by the Fisher-Pearson Coefficient, is 0.745631. A positive skew may be seen in the numbers.
The mode, median, and mean may also be looked up to verify these numbers.
This frequency distribution has a measure of kurtosis, a statistical term used to characterise the shape of a frequency distribution. In addition to indicating whether a distribution is heavy-tailed or not, the form of the frequency distribution can be revealed through the kurtosis measure.
It can be stated that the kurtosis of a normal distribution is 3. If the kurtosis is below 3, it is referred to as platykurtic, whereas if the kurtosis is above 3, it is termed as leptokurtic. This leptokurtic distribution deviates from the normal distribution, resulting in the presence of a few extreme values.
Standardised fourth instant provides the formula for calculating kurtosis. The procedure for solving the equation is outlined below.
There are four moments of a distribution, and skewness is the fourth most well-known one.
Once again, let’s imagine a list of ten digits that stands in for test results. X = [54, 73, 59, 98, 68, 45, 88, 92, 75, 96]
If we take X and find its mean, we get:
x = 74.8
This number may be substituted into the kurtosis formula for the final result.
Python Functions for Computing Skew and Kurtosis
First, We’ll Need to Bring in the SciPy Library.
The SciPy Toolkit, a free and open-source scientific computing library, provides users with built-in routines for calculating skewness and kurtosis. To use these routines, simply include the following code:
The second stage involves constructing a dataset.
Constructing a dataset is the next stage. Instances of this may be seen in the code below.
# creating a data set
dataset = [10, 25, 14, 26, 35, 45, 67, 90, 40, 50, 60, 10, 16, 18, 20]
Stage 3: Skewness Calculation
For the skewness calculation, the built-in skew() function may be used using the following syntax.
spicy.stats.skew(array, axis = 0, bias = True)
For the input object that contains elements (here referred to as “array”), the skewness value can be determined by specifying the axis along which the calculation is to be performed. Additionally, one can specify whether the statistical bias is to be taken into account (by setting the “bias” argument to either “True” or “False”).
By utilising this response, the skewness value of the dataset will be horizontal, thereby indicating a distribution that is more positively skewed than the norm.
Fourth, kurtosis calculations
Use the kurtosis() built-in function and the following syntax to compute kurtosis:
spicy.stats.kurtosis(array, axis = 0, fisher = True, bias = True)
where the components are stored in an array and the axis denotes the axis and the desired kurtosis value.
When the mean value is equal to zero, Fisher’s equation will be satisfied, yielding a true result. Conversely, if the average is equal to three, then the statement will be false. Statistically speaking, bias can be either true or false depending on the situation.
The kurtosis value of the given dataset will be calculated, indicating whether the distribution of the data is more peaked than a normal distribution. This can be indicative of a distribution that has outputs with a greater range of values than would be expected with a normal distribution.
Statistics that aim to measure the middle ground
It is widely accepted that all measurable elements on the planet are subject to a range of chance influences. However, when significant influences begin to bear on a process, the resulting change in the shape of the distribution can be quantified using a measure such as skewness.
In the event that we do identify an asymmetrical distribution, however, we will need to investigate methods for measuring its scope.
It is essential to understand how central tendency measures are affected when the normal distribution is skewed. As demonstrated in the above illustration, the left-hand graph is negatively skewed, indicated by its tail stretching towards the left, while the right-hand graph is positively skewed, identified by its tail reaching towards the right.
It is necessary to ascertain the extent of the divergence in the horizontal plane between the two main indicators (mode and mean). It is important to bear in mind that, the greater the skewness, the more dissimilar these figures will be.
Below is the skewness formula:
Skewness = (Mean - Mode) / Standard Deviation
By dividing the values of a dataset by its standard deviation, we can make all distributions appear to be the same size for easy comparison. When dealing with datasets that are relatively small in size, it is not essential to perform a mode calculation; instead, it is recommended that a well-defined formula for skewness be used to replace the mode calculation.
Mode = 3*(Median) - 2*(Mean)
By plugging in the median number, we obtain:
Skewness = 3*(Mean - Median) / Standard Deviation
It may be beneficial to examine the impact of inverting the conventional normal distribution curve. The primary characteristics to analyse are the apex and tails of the curve, which will be accurately recorded by the kurtosis statistic.
Due to the complexity of the kurtosis computation, it is essential to maintain conceptual consistency.
It can be restated that the kurtosis of a normal distribution is 3, referred to as mesokurtic. A leptokurtic distribution has a kurtosis greater than 3, while a platykurtic distribution has a kurtosis less than 3. The kurtosis of a distribution can range from a value of 1 to infinity, with the peak of the distribution becoming taller as the kurtosis value increases.
Using zero as a baseline for normality, we can use the following formula to determine the amount of extra kurtosis:
Excess Kurtosis = Kurtosis - 3
The skewness metric will determine the degree to which the distribution of a given dataset deviates from the normal distribution curve due to a horizontal shift. Conversely, the kurtosis statistic will measure the extent of vertical distortion in the data, which is predominantly caused by extreme values.