Understanding Skewness And Kurtosis And How to Plot Them

What is Kurtosis and its Types?

While skewness focuses on the spread (tails) of normal distribution, kurtosis focuses more on the height. It tells us how peaked or flat our normal (or normal-like) distribution is. The term, which means curved or arched from Greek, was first coined by, unsurprisingly, from the British mathematician Karl Pearson (he spent his life studying probability distributions).

High kurtosis indicates:

• Sharp peakedness in the distribution’s center.
• More values concentrated around the mean than normal distribution.
• Heavier tails because of a higher concentration of extreme values or outliers in tails.
• Greater likelihood of extreme events.

On the other hand, low kurtosis indicates:

• Flat peak.
• Fewer values concentrated around the mean but still more than normal distribution.
• Lighter tails.
• Lower likelihood of extreme events.

Depending on the degree, distributions have three types of kurtosis:

1. Mesokurtic distribution (kurtosis = 3, excess kurtosis = 0): perfect normal distribution or very close to it.
2. Leptokurtic distribution (kurtosis > 3, excess kurtosis > 0): sharp peak, heavy tails
3. Platykurtic distribution (kurtosis < 3, excess kurtosis < 0): flat peak, light tails

Note that here, excess kurtosis is defined as kurtosis - 3, treating the kurtosis of normal distribution as 0. This way, kurtosis scores are more interpretable.

How to Calculate Kurtosis in Python

You can calculate kurtosis in Python in the same way as skewness using pandas or SciPy:

from scipy.stats import kurtosis

kurtosis(diamond_prices)
2.177382669056634

Pandas offers two functions for kurtosis: kurt and kurtosis. The first one is exclusive to Pandas Series, while you can use the other on DataFrames.

diamond_prices.kurt()
2.17769575924869

# Select numeric features and calculate kurtosis
diamonds.select_dtypes(include="number").kurtosis()
carat     1.256635
depth     5.739415
table     2.801857
price     2.177696
x        -0.618161
y        91.214557
z        47.086619
dtype: float64

Again, the numbers differ for the distribution because pandas and SciPy use different formulas.

If you want a manual calculation of kurtosis, you can use the following formula:

Here:

• n represents the number of observations in the dataset
• x_i denotes each individual data point

We will implement the formula inside a function again:

def moment_based_kurtosis(distribution):
   n = len(distribution)
   mean = np.mean(distribution)
   std = np.std(distribution)

   kurtosis = (1 / n) * sum(((distribution - mean) / std) ** 4) - 3

   return kurtosis

>>> moment_based_kurtosis(diamond_prices)
2.1773826690576463

And we find out that diamond prices have an excess kurtosis of 2.18, which means if we plot the distribution, it will have a sharper peak than a normal distribution.

So, let’s do that!

Visualizing Skewness and Kurtosis in Python

One of the best visuals to see the shape and, thus, the skewness and kurtosis of distributions is a kernel density estimate (KDE) plot. It is available to use through Seaborn:

import matplotlib.pyplot as plt

sns.kdeplot(diamond_prices)

plt.title("KDE plot of diamond prices")
plt.xlabel("Price ($)")

This plot is in accordance with the numbers we saw up to this point: the distribution has a long right tail, indicating a positive skewness, and it has a very sharp peak, which corresponds with high kurtosis.

KDE isn’t the only plot to see the shape. We can use histograms as well:

sns.histplot(diamonds["carat"])

plt.xlabel("Carat")
plt.title("A histogram of the carat of diamonds")

The disadvantage of histograms is that you have to choose the number of bins (the count of bars) yourself. Here, there are too many bars creating noise in the visual — we can’t clearly define the shape. So, let’s decrease the number of bins:

sns.histplot(diamonds["carat"], bins=25)

plt.xlabel("Carat")
plt.title("A histogram of the carat of diamonds")

Now, the shape is more well-defined but we can improve it more. By setting kde=True inside histplot, we can plot a KDE of the distribution on top of the bars:

sns.histplot(diamonds["carat"], bins=25, kde=True)

plt.xlabel("Carat")
plt.title("A histogram of the carat of diamonds")

The overlayed KDE line looks jagged, not the smooth curve that enables us to see the general shape. The reason for the jaggedness is that the carat distribution is naturally spiky and far from the normal distribution.

But, we can decrease the KDE’s sensitivity to these fluctuations by adjusting the bandwidth. This is done using the bw_adjust parameter, which defaults to 1:

# Change the bandwidth from 1 to 3
sns.kdeplot(diamonds["carat"], bw_adjust=3, color="red")

plt.title("KDE of diamond carats");

This version looks much less spiky than the overlaid KDE plot. To adjust the KDE bandwidth when using a histogram overlaid with a KDE, you can use the kde_kws parameter:

ax = sns.histplot(
   diamonds["carat"],
   kde=True,
   kde_kws=dict(bw_adjust=3),
   bins=25,
)

plt.title("An overlaid histogram of diamond carats");

kde_kws accepts any parameter that is acceptable by kdeplot function that controls the KDE computation.

One trick you can use when plotting KDEs is to remove everything except the KDE line. Since the main point of a KDE is to see the distribution shape, other details of the plot like the axis ticks, the spines, and labels are sometimes unnecessary:

sns.kdeplot(diamond_prices, color="red")

# Remove the spine from three sides
sns.despine(top=True, right=True, left=True)

# Remove the ticks and ticklabels
plt.xticks([])
plt.yticks([])
plt.ylabel("")
plt.xlabel("")

# Set a title
plt.title("Diamond prices", fontdict=dict(fontsize=20));

This plot is much tidier. You can further improve the plot by adding lines to denote the position of mean, median, and mode:

sns.kdeplot(diamond_prices, color="red")

sns.despine(top=True, right=True, left=True)
plt.xticks([])
plt.yticks([])
plt.ylabel("")
plt.xlabel("")
plt.title("Diamond prices", fontdict=dict(fontsize=20))

# Find the mean, median, mode
mean_price = diamonds["price"].mean()
median_price = diamonds["price"].median()
mode_price = diamonds["price"].mode().squeeze()

# Add vertical lines at the position of mean, median, mode
plt.axvline(mean_price, label="Mean")
plt.axvline(median_price, color="black", label="Median")
plt.axvline(mode_price, color="green", label="Mode")

plt.legend();

This plot verifies what we discussed in the types of skewness section: in a positively skewed distribution, the mean is higher than the median, and the mode is lower than both mean and median.

Conclusion

Skewness and kurtosis, often overlooked in Exploratory Data Analysis, reveal significant insights about the nature of distributions.

Skewness hints at data tilt, whether leaning left or right, revealing its asymmetry (if any). Positive skew means a tail stretching right, while negative skew veers in the opposite direction.

Kurtosis is all about peaks and tails. High kurtosis sharpens peaks and weighs down the tails, while low kurtosis spreads data, lightening the tails.

If you want to learn more about skewness and kurtosis, you can check out these excellent quantitative analysis courses taught by industry experts on DataCamp:

• Portfolio risk management in Python
• Introduction to Portfolio Analysis in Python