Excel Skewness & Kurtosis Calculator
Calculate statistical skewness and kurtosis from your Excel data with precision
Comprehensive Guide: How to Calculate Skewness and Kurtosis in Excel
Understanding the shape of your data distribution is crucial for statistical analysis. Skewness and kurtosis are two fundamental measures that describe the asymmetry and “tailedness” of probability distributions. This guide will walk you through calculating these metrics in Excel and interpreting the results.
What Are Skewness and Kurtosis?
Skewness measures the asymmetry of the probability distribution of a real-valued random variable about its mean. Positive skewness indicates a distribution with an asymmetric tail extending towards more positive values, while negative skewness indicates a distribution with an asymmetric tail extending towards more negative values.
Kurtosis measures the “tailedness” of the probability distribution. High kurtosis indicates a distribution with heavier tails (more outliers) than a normal distribution, while low kurtosis indicates lighter tails (fewer outliers).
Excel Functions for Skewness and Kurtosis
Excel provides built-in functions for calculating skewness and kurtosis:
- SKEW() – Returns the skewness of a distribution (sample skewness)
- SKEW.P() – Returns the skewness of a distribution based on a population (population skewness)
- KURT() – Returns the kurtosis of a data set (sample kurtosis)
- KURT.P() – Returns the kurtosis of a data set based on a population (population kurtosis)
Step-by-Step Calculation in Excel
- Prepare your data: Enter your data values in a single column (e.g., A1:A100)
- Calculate sample skewness: In a blank cell, enter =SKEW(A1:A100)
- Calculate population skewness: In a blank cell, enter =SKEW.P(A1:A100)
- Calculate sample kurtosis: In a blank cell, enter =KURT(A1:A100)
- Calculate population kurtosis: In a blank cell, enter =KURT.P(A1:A100)
Interpreting the Results
Skewness interpretation:
- ≈ 0: Symmetrical distribution (like normal distribution)
- > 0: Right-skewed (positive skew) – tail on the right side
- < 0: Left-skewed (negative skew) – tail on the left side
Kurtosis interpretation:
- ≈ 3: Mesokurtic (normal distribution)
- > 3: Leptokurtic (heavier tails, more outliers)
- < 3: Platykurtic (lighter tails, fewer outliers)
Practical Example
Let’s consider a dataset of exam scores: 78, 85, 92, 65, 72, 88, 95, 70, 82, 90
| Metric | Sample Calculation | Population Calculation | Interpretation |
|---|---|---|---|
| Skewness | -0.34 | -0.29 | Slightly left-skewed |
| Kurtosis | 1.93 | 1.67 | Platykurtic (lighter tails than normal) |
Common Mistakes to Avoid
- Using wrong function: Don’t use SKEW() when you should use SKEW.P() for population data
- Ignoring data quality: Outliers can significantly affect skewness and kurtosis values
- Misinterpreting kurtosis: Remember that normal distribution has kurtosis of 3, not 0
- Small sample size: Results may be unreliable with fewer than 30 data points
Advanced Applications
Skewness and kurtosis have important applications in various fields:
- Finance: Assessing risk in investment returns (fat tails indicate higher risk of extreme events)
- Quality Control: Monitoring process capability and identifying non-normal distributions
- Psychometrics: Evaluating test score distributions for fairness and validity
- Biostatistics: Analyzing medical data distributions before applying parametric tests
Comparison of Statistical Software
| Software | Skewness Function | Kurtosis Function | Notes |
|---|---|---|---|
| Excel | SKEW(), SKEW.P() | KURT(), KURT.P() | Easy to use, limited advanced options |
| R | moments::skewness() | moments::kurtosis() | More customization, requires coding |
| Python (SciPy) | scipy.stats.skew() | scipy.stats.kurtosis() | Powerful, integrates with data science workflows |
| SPSS | Analyze → Descriptive Statistics | Analyze → Descriptive Statistics | GUI-based, good for social sciences |
When to Use Sample vs Population Formulas
The choice between sample and population formulas depends on your data context:
- Use sample formulas (SKEW, KURT) when:
- Your data is a subset of a larger population
- You’re making inferences about a population
- You want to account for sampling variability
- Use population formulas (SKEW.P, KURT.P) when:
- Your data includes the entire population
- You’re only describing this specific dataset
- You don’t need to generalize beyond your data
Mathematical Formulas Behind the Calculations
The sample skewness formula used by Excel is:
G₁ = [n/(n-1)(n-2)] * Σ[(xᵢ – x̄)/s]³
Where:
- n = sample size
- xᵢ = individual data points
- x̄ = sample mean
- s = sample standard deviation
The sample kurtosis formula is:
G₂ = [n(n+1)/((n-1)(n-2)(n-3))] * Σ[(xᵢ – x̄)/s]⁴ – 3(n-1)²/((n-2)(n-3))
Limitations and Considerations
While skewness and kurtosis are valuable statistical measures, they have some limitations:
- Sensitivity to outliers: Both measures can be heavily influenced by extreme values
- Sample size requirements: Reliable estimates typically require at least 30-50 observations
- Interpretation challenges: The practical significance of small deviations from normality can be difficult to assess
- Multimodal distributions: These measures may be misleading for distributions with multiple peaks
Alternative Measures of Distribution Shape
In some cases, you might consider alternative approaches:
- Quantile-quantile (Q-Q) plots: Visual comparison to normal distribution
- Shapiro-Wilk test: Formal test for normality
- Anderson-Darling test: Another normality test, more sensitive to tails
- Histogram analysis: Visual inspection of distribution shape
Real-World Case Study: Financial Returns
A 2021 study by the Federal Reserve analyzed S&P 500 returns from 1950-2020 and found:
| Period | Skewness | Kurtosis | Implications |
|---|---|---|---|
| 1950-1980 | -0.42 | 4.1 | Negative skew (more large negative returns), fat tails |
| 1980-2000 | -0.18 | 3.2 | Near normal distribution |
| 2000-2020 | -0.75 | 5.8 | Strong negative skew, very fat tails (more extreme events) |
This analysis demonstrates how market conditions can significantly affect the distribution shape of financial returns over time.
Best Practices for Reporting Results
- Always specify whether you’re reporting sample or population measures
- Include the sample size with your results
- Provide visualizations (histograms, Q-Q plots) alongside numerical measures
- Discuss the practical implications of your findings
- Consider reporting confidence intervals for sample measures
Learning Resources
For deeper understanding, consider these authoritative resources:
- NIST Engineering Statistics Handbook – Comprehensive guide to statistical methods
- UC Berkeley Statistics Department – Advanced statistical education resources
- U.S. Census Bureau – Practical applications of statistical methods in government data
Frequently Asked Questions
Why does Excel’s kurtosis sometimes show negative values?
Excel’s KURT() function calculates “excess kurtosis” which is the kurtosis minus 3 (the kurtosis of a normal distribution). This means:
- 0 = normal distribution kurtosis
- > 0 = more peaked than normal
- < 0 = less peaked than normal
Can I calculate skewness and kurtosis for grouped data?
Yes, but you’ll need to use different formulas that account for the frequency distribution. Excel doesn’t have built-in functions for this, so you would need to:
- Calculate the midpoint of each group
- Multiply by frequency to get “expanded” data
- Apply the regular skewness/kurtosis functions
How do I handle missing values in my data?
Excel’s SKEW and KURT functions automatically ignore empty cells and text values. For more control:
- Use data cleaning functions to remove missing values
- Consider whether missing data might bias your results
- Document how you handled missing values in your analysis
What’s the relationship between skewness and the mean/median?
In skewed distributions:
- Positive skew: Mean > Median (tail pulls mean to the right)
- Negative skew: Mean < Median (tail pulls mean to the left)
- Symmetric: Mean ≈ Median
Can I calculate skewness and kurtosis for non-numeric data?
No, these measures require numerical data. For categorical data, you would need to:
- Convert to numerical codes (being careful about implied numerical relationships)
- Or use alternative measures appropriate for categorical data