Skewness and Kurtosis: Understanding Two Important Statistical Concepts
Have you ever heard the terms skewness and kurtosis but aren’t quite sure what they mean? These two statistical concepts are important measures of the shape of a data distribution and can provide valuable insights into the data you’re working with.
In this article, we’ll dive into what skewness and kurtosis are, how to calculate them, and what they can tell us.
Skewness
Let’s start with skewness.
Skewness is a measure of the asymmetry of a probability distribution.
A perfectly symmetrical distribution, such as a normal distribution, has a skewness of 0. However, a distribution that is skewed to the right has a positive skewness, while a distribution that is skewed to the left has a negative skewness.
To interpret skewness, you need to look at the direction of the skewness value (positive or negative) and its magnitude. A small positive or negative skewness value indicates a somewhat asymmetrical distribution.
A larger value, on the other hand, suggests a more pronounced skewness. For example, a skewness of 2 indicates that the distribution is highly skewed to the right.
Calculating skewness is relatively simple. The most commonly used formula for skewness is:
Skewness = (3 x (mean – median)) / standard deviation
If the resulting value is positive, the distribution is skewed to the right, and if it’s negative, the distribution is skewed to the left. Keep in mind that values close to 0 don’t necessarily mean that the distribution is perfectly symmetrical, as some distributions may have small deviations from perfect symmetry.
Kurtosis
Moving on to kurtosis, this measures how “peaked” or flat a distribution is in relation to the normal distribution. The normal distribution has a kurtosis value of 3, and distributions that are more peaked than the normal distribution, such as those with many outliers or extreme values, have a positive kurtosis value.
Conversely, distributions that are flatter than the normal distribution, such as those with a lot of data grouped around the mean, have a negative kurtosis value.
Kurtosis is often divided into three categories:
- Mesokurtic: A normal distribution with a kurtosis of 3.
- Leptokurtic: A distribution that is more peaked than the normal distribution, that is, with a kurtosis value greater than 3.
- Playkurtic: A distribution that is flatter than the normal distribution, that is, with a kurtosis value less than 3.
Calculating kurtosis is slightly more complex than skewness.
There are a few different ways to calculate kurtosis, but we’ll focus on Fisher’s definition of kurtosis, which is widely used and gives reliable results for most distributions. The formula for Fisher’s kurtosis is:
Kurtosis = [(n x ((xi – x)^4)) / (s^4 x (n-1) x (n-2))] – (3 x (n-1)^2)/((n-2) x (n-3))
where n is the sample size, x is the sample mean, s is the sample standard deviation, and xi are the individual data points.
Interpretation
So, what can we learn from skewness and kurtosis? In short, quite a bit.
Skewness can tell us if our data is skewed to one side or the other, giving us clues about underlying processes that may be affecting it. For example, a positive skewness value in a dataset of salaries could indicate that the sample includes a few outliers with very high salaries, while a negative skewness value in a dataset of grades could indicate that there are a few students who performed exceptionally well.
Kurtosis, on the other hand, can tell us whether our data is unusually peaked or flat, which can also give us insights into the underlying processes that generated it. For example, a leptokurtic distribution in a dataset of stock prices could indicate a high degree of volatility, with a few extreme price swings, while a platykurtic distribution in a dataset of measures of happiness could indicate that people are fairly consistent in their level of happiness.
Conclusion
In conclusion, understanding skewness and kurtosis can provide valuable insights into the shape of a dataset and the underlying processes that generated it. By calculating these measures and interpreting their values, you can gain a deeper understanding of your data and make more informed decisions based on it.
Python Functions for Skewness and Kurtosis Calculation
When it comes to working with data, Python is one of the most popular programming languages for data analysis and manipulation. Luckily, Python provides powerful libraries and functions for calculating skewness and kurtosis values.
In this article, we’ll dive into these Python functions and how to utilize them for your data analysis.
Numpy and SciPy Libraries
The first libraries that come to mind when it comes to working with statistics using Python are NumPy and SciPy. NumPy is a fundamental package for scientific computing with Python, particularly for numerical operations. SciPy, on the other hand, is built on top of NumPy and provides additional functions for scientific and technical computing, including statistics.
SciPy provides a statistical library for Python called scipy.stats that offers a range of statistical functions, including skewness and kurtosis. The Scipy Stata library contains functions to calculate statistical estimates, publish results in Stata format, and read in Stata data files (.dta).
Syntax and Usage
The SciPy library offers the skew() and kurtosis() functions for calculating skewness and kurtosis values, respectively. These functions are used to calculate the skewness and kurtosis values of a dataset, as well as to determine whether the distribution is positively or negatively skewed or has heavy or light tails.
Both functions offer several arguments that allow you to calculate the skewness and kurtosis values for a sample or population. You can also adjust the bias of the calculation by setting the bias parameter to False for unbiased estimation and to True for biased estimation.
For example, to calculate the sample skewness of a dataset in Python using SciPy, you would use the following code:
import scipy.stats as stats
data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
sample_skew = stats.skew(data, bias=False)
print("Sample Skewness:", sample_skew)
In this example, we first import the SciPy stats library. Then, we define a dataset of ten values and calculate the sample skewness using the skew() function, setting the bias parameter to False.
Finally, we print the output.
Dataset Analysis
Now that we know how to calculate skewness and kurtosis using Python functions, let’s explore the significance of skewness and kurtosis values in a dataset analysis. Using
Skewness and Kurtosis Values
Skewness and kurtosis values provide helpful insights when analyzing a dataset. The skewness value indicates the extent to which the distribution deviates from a normal distribution.
A positive skewness value indicates that the distribution is skewed to the right, while a negative skewness value indicates that the distribution is skewed to the left. A skewness value of 0 indicates that the distribution is symmetric.
Kurtosis values provide insights into the peakedness of the distribution. A value of 3 indicates that the distribution is normally distributed.
Values greater than 3 indicate that the distribution is more peaked, and values less than 3 indicate that the distribution is flatter.
Sample Calculation
Let’s consider the following example. Data representing the time (in milliseconds) it took for participants to complete a reaction time task is as follows:
data = [23, 29, 27, 35, 24, 40, 45, 31, 25, 26, 30, 32, 33, 22, 28, 36, 28, 39, 25, 38]
To calculate the skewness and kurtosis values of this dataset using Python functions, we can use the following code:
import scipy.stats as stats
data = [23, 29, 27, 35, 24, 40, 45, 31, 25, 26, 30, 32, 33, 22, 28, 36, 28, 39, 25, 38]
sample_skew = stats.skew(data, bias=False)
sample_kurt = stats.kurtosis(data, bias=False)
print("Sample Skewness:", sample_skew)
print("Sample Kurtosis:", sample_kurt)
Running this code gives us an output of:
Sample Skewness: 0.673272802691
Sample Kurtosis: 2.59378127939
From these values, we can conclude that the distribution is positively skewed, with more values in the tails and a slightly higher peak than a normal distribution.
In conclusion, Python functions make it easy to calculate skewness and kurtosis values, providing valuable insights into a dataset’s shape and distribution. By analyzing these values, we can better understand and interpret our data and make more informed decisions based on them.
In conclusion, understanding skewness and kurtosis values is important for analyzing and interpreting datasets. These statistical measures provide us with insights into the shape and distribution of our data, giving us a deeper understanding of the underlying processes that generated it.
Python’s powerful libraries and functions, such as SciPy’s skew() and kurtosis() functions, make it easy to calculate these values and gain valuable insights into our data. By analyzing skewness and kurtosis values, we can make more informed decisions based on our data and better understand the real-world implications and applications of our statistics.