Kurtosis Calculator
Select Sample if the data is a sample of a large or unlimited population and you wish to make a statement about the entire population.
Calculating Kurtosis (Formula)
The Kurtosis Calculator is a helpful shortcut that speeds up your work. But you can also calculate a population's or a sample’s kurtosis by hand. The formulas, however, are slightly different.
The sample excess kurtosis is calculated as follows:
In this formula, n denotes the number of observations in the sample, ˉX is the sample mean, whereas s is the sample standard deviation.
Meanwhile, population excess kurtosis is calculated using the following formula:
In this formula, n denotes the number of observations in the population, μ is the population mean, whereas σ is the population standard deviation.
A positive kurtosis value means that a distribution is more peaked than a normal distribution, while a negative one means it is less peaked. It is important to note here that a normal distribution has a kurtosis value of 0.
Please note that you need at least four observations in your dataset. Otherwise, the denominator would be equal to 0.
How to Calculate Sample and Population Kurtosis: A Step-by-Step Guide
To calculate the kurtosis of a dataset, you can follow these steps:
Calculate the sample or population mean of the dataset:
ˉX=∑ni=1Xin
μ=∑ni=1Xin
Find the sample or population standard deviation of the dataset:
s=√n∑i=1(Xi−ˉX)2n−1
σ=√n∑i=1(Xi−μ)2n
Standardize each data point by subtracting the mean and dividing by the sample or population standard deviation:
Z=Xi−ˉXs
Z=Xi−μσ
Raise each standardized value to the power of 4:
(Xi−ˉXs)4
(Xi−μσ)4
Sum all the standardized values raised to the power of 4:
n∑i=1(Xi−ˉXs)4
n∑i=1(Xi−μσ)4
Multiply the result by n(n+1)/(n-1)(n-2)(n-3) and subtract 3(n-1)^3/(n-2)(n-3) to obtain the sample kurtosis, or multiply it by 1/N, where n is the number of data points in the population, and subtract 3 to obtain the population kurtosis:
K={n(n+1)(n−1)(n−2)(n−3)n∑i=1(Xi−ˉXs)4}−3(n−1)2(n−2)(n−3)γ2=1Nn∑i=1(Xi−ˉμσ)4−3
Note that the subtraction of 3 from the population formula is to make the kurtosis of a normal distribution equal to 0.
Kurtosis Calculator
What Is Kurtosis?
Kurtosis measures the degree of tailedness—the weight of the tails relative to the rest of the distribution. In simpler words, it shows how much of the data is in the tails compared to the center.
The term is derived from the Greek word "kurtos", which stands for curve. This refers to the shape of the probability distribution that kurtosis measures, which can be more peaked or more flattened than a normal distribution .
Located farthest from center, the tails represent the regions where data points are more dispersed, which in turn suggest the presence of more extreme values. If a distribution is heavy-tailed, there is more data in the tails, then it exhibits high kurtosis. Meanwhile a low kurtosis occurs when the data is more evenly distributed between the tails and the center (or the distribution is light-tailed).
Based on the level of kurtosis, distributions can be classified as:
- Leptokurtic - the tails are fatter compared to a normal distribution.
- Platykurtic - the tails are thinner compared to a normal distribution.
- Mesokurtic - the tails are the same as a normal distribution.
What Is Excess Kurtosis?
In a normal distribution, the kurtosis is equal to 3. However, many statistical packages report estimates of excess kurtosis, which is the difference between the kurtosis and 3.
Put simply, kurtosis measures the weight of a distribution’s tails in absolute terms, whereas excess kurtosis characterizes kurtosis relative to the normal distribution.
A distribution with excess kurtosis of 0 is considered normal or mesokurtic. Meanwhile, if the excess is greater than 0, then the distribution is leptokurtic. Finally, if it has excess kurtosis less than 0, then it is platykurtic.
Types of Kurtosis
The table below shows how the three types of kurtosis differ from each other based on their tailedness, outlier frequency, kurtosis value, excess kurtosis, and example distribution.
Notably, leptokurtic distributions have heavy tails and a high frequency of outliers, while platykurtic ones have light tails and a low frequency of outliers. Mesokurtic distributions, on the other hand, have a moderate level of tail thickness and a moderate peak, similar to that of a normal distribution.
What Is a Leptokurtic Distribution?
A leptokurtic distribution has fatter tails than a normal distribution. Put simply, this type has a higher frequency of extreme values or outliers.
The term is a combination of the words "leptos", which is Greek for thin or small, and "kurtosis," which refers to the tails’ weight. Consequently, "leptokurtic" stands for a distribution that has a thin or small central part (peakedness) and heavy tails.
Both leptokurtic and normal distributions have the same mean, standard deviation, and skewness. However, the former is more likely to generate observations in the tails (or the areas of about ±2.5 standard deviations.) It also tends to produce more observations that are close to the mean (the region within one standard deviation of the mean).
Compared to a normal distribution (with a kurtosis of 3), the leptokurtic distribution has a kurtosis greater than 3, and an excess kurtosis greater than 0. As a result, practitioners often refer to those type as positive kurtosis.
The Student’s t-distribution is one example of an example of a leptokurtic distribution. It’s also known as one of the biggest breakthroughs in statistics as it allows you to estimate population parameters when the sample size is small or the population standard deviation is unknown. This setting can be applied to a big part of the statistical problems we face today.
Visually, the Student’s T distribution looks symmetric and bell-shaped much like a normal distribution. However, its tails are fatter, which means more of its observations are far from the mean.
Another example of leptokurtic distribution is the Laplace distribution. Also called a double exponential distribution, it has a pointed shape in the middle, resembling a pole holding up a circus tent.
What Is a Mesokurtic Distribution?
A mesokurtic distribution has the same kurtosis as a normally distributed dataset. In other words, the probability of extreme values is close to 0.
The word "meso" comes from the Greek word for "middle" and denotes a distribution with a moderate peak.
The normal distribution is the most common example of mesokurtic distribution as it has an excess kurtosis equal to 0.
What Is a Platykurtic Distribution?
A platykurtic distribution has a lower peak and flatter tails compared to a normal distribution. It is characterized by a negative kurtosis value, indicating fewer outliers and a lower probability of extreme values. The term is derived from the Greek words "platus", meaning flat.
In a platykurtic distribution, the data is more spread out than in a normal distribution, however, there is less variability, making it easier to model and predict.
Compared to the normal distribution, the platykurtic one has a kurtosis of less than 3 and an excess kurtosis of less than 0.
The uniform distributions is one example; it refers to a type of probability distribution in which each potential result has an equal chance of occurring. As each variable has an equal chance of being the outcome, the probability remains constant.
How to Interpret the Kurtosis Coefficient
You can determine the kurtosis of your dataset using the Kurtosis Calculator or the step-by-step guide above. But either way, the value is meaningless unless you know how to interpret it.
The kurtosis coefficient, also known as excess kurtosis, determines whether a distribution is more or less peaked around its central point than a normal distribution.
To interpret the kurtosis coefficient, follow these general guidelines:
- A positive excess kurtosis coefficient (greater than 0) indicates that the distribution has heavier tails and more extreme values. This suggests that there are more outliers in the data, and the distribution may be more peaked in the center.
- A negative excess kurtosis coefficient (less than 0) indicates that the distribution has lighter tails and fewer extreme values. This suggests that there are fewer outliers in the data, and the distribution may be more spread out.
- An excess kurtosis coefficient of 0 indicates that the distribution has the same shape as a normal distribution.
Remember that kurtosis is only one measure of a distribution's shape. For this reason, it should be used together with other measures, like skewness and visual inspection of the data, in order to fully understand the distribution. Furthermore, the kurtosis interpretation may vary depending on the context and the dataset analyzed.
What Is the Difference Between Skewness and Kurtosis?
Kurtosis and skewness both assess the shape of a probability distribution .
Kurtosis measures the degree of tailedness or flatness. This tells us what the weight of the tails is, relative to the rest of the distribution. Or put simply, how much of the data is in the tails, compared to the center. A high-kurtosis distribution indicates more data in the tails, whereas a low-kurtosis distribution shows that it is more evenly distributed.
Skewness , on the other hand, is a measure of a probability distribution’s asymmetry . It indicates whether the observations in a dataset are concentrated on one side. In other words, we see how the distribution deviates from normality and more easily understand the data’s fundamental structure as a result.
In fact, skewness plays a significant role in statistical modeling and hypothesis testing: if the data is skewed, the statistical test may not be accurate, and the results—misleading. As such, it is crucial to know how to calculate and interpret this value. You can obtain it using our Skewness Calculator and learn more about it in the related article.