Skewness and Kurtosis: Definition, Types, Formula & Examples
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Skewness and Kurtosis: Definition, Types, Formula & Examples
When you describe a dataset, the first things you usually check are the mean, median, and standard deviation. But two numbers alone do not tell you the full story. Two distributions can have the same mean and standard deviation yet look completely different on a graph.
That is where skewness and kurtosis come in. They measure the shape of a distribution: skewness tells you whether the data is symmetric or lopsided, while kurtosis tells you how heavy the tails are and how likely the data is to produce outliers.
In this guide, you will learn the definitions, types, formulas, and real-world examples of skewness and kurtosis everything you need to read and interpret distribution shapes confidently in statistics and data science.
If you are building a foundation in statistics, you may also want to explore data science statistics for a broader view of the concepts.
What is Skewness?
Skewness is a measure of the asymmetry of a probability distribution about its mean. In other words, it tells you whether the left and right sides of the distribution mirror each other, or whether one tail stretches out longer than the other.
- A symmetric distribution has zero skewness.
- A positive skew means the right tail is longer; most values cluster on the left.
- A negative skew means the left tail is longer; most values cluster on the right.
Imagine a histogram of household incomes. Most people earn low-to-moderate salaries, but a small number of very high earners pull the mean upward. That creates a long tail on the right, so the distribution is positively skewed.
Skewness is an essential idea in descriptive statistics and is often used alongside the mean, median, and mode.
Types of Skewness (Positive, Negative, Zero)
Skewness is usually divided into three main types.
1. Positive Skew (Right Skew)
In a positively skewed distribution, the right tail is longer. The bulk of the data lies on the left side, and a few large values stretch the tail to the right.
Typical relationship:
Mean > Median > Mode
The mean is pulled upward by the extreme high values, so it becomes larger than the median. Common examples include income distribution, house prices, and insurance claims.
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2. Negative Skew (Left Skew)
In a negatively skewed distribution, the left tail is longer. Most values are high, but a few very small values pull the tail to the left.
Typical relationship:
Mean < Median < Mode
The mean is pulled downward by the extreme low values. Examples include scores on an easy exam where most students score high, or age at retirement.
3. Zero Skew (Symmetric)
A distribution with zero skewness is symmetric. The left and right halves are mirror images of each other.
Typical relationship:
Mean ≈ Median ≈ Mode
The classic example is the normal distribution, which is perfectly symmetric.
For a deeper explanation of the mean-median-mode relationship, see our guide on mean, median, and mode.
Types of Skewness Table
| Type | Tail Direction | Mean vs Median vs Mode | Example |
|---|---|---|---|
| Positive (Right) Skew | Longer right tail | Mean > Median > Mode | Income, house prices |
| Negative (Left) Skew | Longer left tail | Mean < Median < Mode | Easy exam scores, retirement age |
| Zero (Symmetric) | No tail | Mean ≈ Median ≈ Mode | Normal distribution, heights |
Skewness Formula and Interpretation
There are several ways to calculate skewness. The most common version is the Pearson moment coefficient of skewness, which uses the third standardized moment.
Population skewness formula
γ₁ = μ₃ / σ³
Where:
- γ₁ (gamma-one) = population skewness
- μ₃ = third central moment, i.e., E[(X − μ)³]
- σ = population standard deviation
- σ³ = the cube of the standard deviation, used to standardize the third moment
Sample skewness formula
For a sample of size n, the adjusted Fisher-Pearson coefficient is:
g₁ = [n / ((n − 1)(n − 2))] × Σ[(xᵢ − x̄)³ / s³]
Where:
- n = number of observations
- xᵢ = each individual value
- x̄ = sample mean
- s = sample standard deviation
- Σ = summation over all observations
The multiplier n / ((n − 1)(n − 2)) corrects for small-sample bias.
How to interpret skewness values
| Skewness Range | Interpretation |
|---|---|
| -0.5 to 0.5 | Approximately symmetric |
| -1 to -0.5 or 0.5 to 1 | Moderately skewed |
| Less than -1 or greater than 1 | Highly skewed |
Keep in mind that these thresholds are rules of thumb, not absolute rules. Always look at a histogram or density plot alongside the number.
For a more mathematical treatment, you can refer to the Wikipedia page on skewness.
If you are interested in applying statistical concepts in real-world projects, the Scaler Data Science Course covers distributions, hypothesis testing, and data modeling in depth.
What is Kurtosis?
Kurtosis measures the tailedness of a distribution that is, how heavy or light the tails are compared with a normal distribution. It also tells you how prone the distribution is to producing outliers.
Many people think kurtosis measures "peakedness," but that is a common misconception. The correct interpretation is about the tails and the combined influence of outliers.
- A distribution with high kurtosis has heavy tails and more outliers.
- A distribution with low kurtosis has light tails and fewer outliers.
- A normal distribution has a reference kurtosis of 3.
Kurtosis is closely related to probability distributions, which are covered in our guide on probability distributions in data science.
Types of Kurtosis (Lepto, Platy, Meso)
Kurtosis is classified into three types based on the value of the fourth standardized moment relative to the normal distribution.
1. Mesokurtic
A distribution with kurtosis equal to 3 (or excess kurtosis equal to 0) is called mesokurtic. It has tails similar to the normal distribution.
- Kurtosis = 3
- Excess kurtosis = 0
The normal distribution itself is the standard example of mesokurtic.
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A distribution with kurtosis greater than 3 (or excess kurtosis greater than 0) is called leptokurtic. It has heavier tails and a sharper peak than the normal distribution, meaning it produces more outliers.
- Kurtosis > 3
- Excess kurtosis > 0
Examples include the t-distribution with small degrees of freedom and financial return data, which often has extreme losses or gains.
3. Platykurtic
A distribution with kurtosis less than 3 (or excess kurtosis less than 0) is called platykurtic. It has lighter tails and a flatter peak, meaning it produces fewer outliers.
- Kurtosis < 3
- Excess kurtosis < 0
The uniform distribution is a classic platykurtic distribution.
Types of Kurtosis Table
| Type | Kurtosis Value | Excess Kurtosis | Tail Characteristic | Example |
|---|---|---|---|---|
| Mesokurtic | = 3 | = 0 | Normal-like tails | Normal distribution |
| Leptokurtic | > 3 | > 0 | Heavy tails, more outliers | t-distribution, stock returns |
| Platykurtic | < 3 | < 0 | Light tails, fewer outliers | Uniform distribution |
Kurtosis Formula and Interpretation
Kurtosis is based on the fourth standardized moment of the distribution.
Population kurtosis formula
β₂ = μ₄ / σ⁴
Where:
- β₂ (beta-two) = population kurtosis
- μ₄ = fourth central moment, i.e., E[(X − μ)⁴]
- σ = population standard deviation
- σ⁴ = the fourth power of the standard deviation, used to standardize the fourth moment
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Sample kurtosis formula
For a sample, the formula is:
b₂ = [n(n + 1) / ((n − 1)(n − 2)(n − 3))] × Σ[(xᵢ − x̄)⁴ / s⁴] − [3(n − 1)² / ((n − 2)(n − 3))]
Where:
- n = sample size
- xᵢ = individual observations
- x̄ = sample mean
- s = sample standard deviation
The subtraction at the end is a bias correction, and the result is sometimes called excess kurtosis.
Excess kurtosis
Many software packages report excess kurtosis, which is simply:
Excess kurtosis = Kurtosis − 3
This makes the normal distribution the reference point at 0 instead of 3.
| Excess Kurtosis | Interpretation |
|---|---|
| 0 | Normal-like tails (mesokurtic) |
| > 0 | Heavy tails, more outliers (leptokurtic) |
| < 0 | Light tails, fewer outliers (platykurtic) |
The NIST/SEMATECH Engineering Statistics Handbook provides detailed guidance on kurtosis and its use in exploratory data analysis: NIST Handbook on Skewness and Kurtosis.
Difference Between Skewness and Kurtosis
Skewness and kurtosis are both shape measures, but they answer different questions. Use the following table to compare them side by side.
| Feature | Skewness | Kurtosis |
|---|---|---|
| What it measures | Asymmetry of the distribution | Tailedness and outlier tendency |
| Based on | Third standardized moment | Fourth standardized moment |
| Value for a normal distribution | 0 | 3 (or 0 excess kurtosis) |
| Positive value | Longer right tail | Heavier tails than normal (leptokurtic) |
| Negative value | Longer left tail | Lighter tails than normal (platykurtic) |
| Zero value | Symmetric distribution | Normal-like tails (mesokurtic) |
| Main concern | Direction of the tail | Weight of the tails |
| Example question | Are high values pulling the mean up? | Are extreme values common? |
Together, skewness and kurtosis give a much richer picture of a distribution than the mean and standard deviation alone.
Kurtosis is particularly useful when you are dealing with outliers. For techniques on handling them, see our article on handling outliers in data science.
Why Skewness and Kurtosis Matter in Data Science
Understanding skewness and kurtosis is not just an academic exercise. These measures have real consequences for data science work.
1. Model assumptions
Many statistical models assume that data is normally distributed. Linear regression, ANOVA, and many hypothesis tests rely on this assumption. If skewness or kurtosis is too far from normal, the results may be unreliable.
2. Outlier detection
High kurtosis often signals the presence of outliers. Before building a model, you should investigate whether those outliers are errors or genuine extreme values.
3. Feature transformations
Skewed features can sometimes be normalized using transformations such as:
- Log transformation for right-skewed data
- Square-root transformation for moderate right skew
- Box-Cox transformation for more flexible normalization
Applying the right transformation can improve model performance and interpretation.
4. Risk and finance
In finance, leptokurtic return distributions indicate a higher probability of extreme losses or gains. Risk managers use kurtosis alongside variance and Value at Risk (VaR) to assess portfolio risk.
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Conclusion
Skewness and kurtosis are two essential shape measures in statistics. Skewness tells you whether a distribution is symmetric or lopsided, while kurtosis tells you how heavy the tails are and how likely the data is to contain outliers.
By learning the types, formulas, and interpretation rules, you can quickly assess whether a dataset is approximately normal, whether it needs transformation, and whether your model assumptions are reasonable.
Practice by plotting histograms, calculating skewness and kurtosis for real datasets, and interpreting the results. The more distributions you examine, the faster your intuition will grow.
For a structured journey from statistics fundamentals to advanced data science, explore the Scaler Data Science Course or the Scaler Data Analytics Course.
FAQs
Q1. What is skewness in statistics?
Skewness measures the asymmetry of a data distribution around its mean. It shows if the data is spread out more on the left or the right side, highlighting the direction of extreme values.
Q2. What is kurtosis?
Kurtosis measures the "tailedness" of a probability distribution—how heavy its tails are and its likelihood of producing outliers. Higher kurtosis means more data falls in the extreme tails rather than around the center.
Q3. What are the types of skewness?
The three types are positive (right) skew (a longer tail stretching to the right), negative (left) skew (a longer tail to the left), and zero skew (perfect symmetry where both sides are balanced).
Q4. What are the types of kurtosis?
The three categories are leptokurtic (heavy tails and a sharp peak), platykurtic (lighter tails and a flatter peak), and mesokurtic (standard, bell-curve data comparable to a normal distribution).
Q5. What is the difference between skewness and kurtosis?
Skewness describes the direction of a distribution's asymmetry (where the longer tail lies). Kurtosis focuses entirely on the heaviness of the distribution's tails and the frequency of outliers.
Q6. What is a good value for skewness?
A skewness value between -0.5 and 0.5 is generally considered excellent for indicating an approximately symmetric distribution. Values outside this range suggest the data is significantly skewed and may need transformation for certain models.
Q7. What does excess kurtosis mean?
Excess kurtosis measures a distribution's tailedness relative to a standard normal distribution, which is calibrated to 0 . A positive value means fatter tails (more outliers), while a negative value means thinner tails.
