Calculate Skewness and Kurtosis in R

R Functions for Moments:

\[ \text{skew} = \text{skewness}(x); \quad \text{kurt} = \text{kurtosis}(x) \]

Skewness:

Kurtosis:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Skewness and Kurtosis?

Skewness and kurtosis are statistical measures that describe the shape of a probability distribution. Skewness measures the asymmetry of the distribution, while kurtosis measures the "tailedness" or peakiness of the distribution compared to a normal distribution.

2. How Does the Calculator Work?

The calculator uses R-style moment calculations:

\[ \text{skewness} = \frac{1}{n}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^3 \] \[ \text{kurtosis} = \frac{1}{n}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^4 - 3 \]

Where:

\( n \) — Number of data points
\( x_i \) — Individual data values
\( \bar{x} \) — Mean of the data
\( s \) — Standard deviation of the data

Explanation: Skewness values indicate distribution symmetry (0 = symmetric, positive = right-skewed, negative = left-skewed). Kurtosis values indicate tail behavior (0 = normal, positive = heavy tails, negative = light tails).

3. Importance of Skewness and Kurtosis

Details: These moments are crucial for understanding data distribution characteristics, testing normality assumptions, identifying outliers, and selecting appropriate statistical models. They help determine if data meets assumptions for parametric tests.

4. Using the Calculator

Tips: Enter numerical values separated by commas. The calculator will compute both skewness and kurtosis using the same formulas as R's e1071 package. Ensure you have sufficient data points for meaningful results.

5. Frequently Asked Questions (FAQ)

Q1: What do different skewness values mean?
A: Skewness = 0 indicates symmetric distribution; >0 indicates right-skew (tail to right); <0 indicates left-skew (tail to left).

Q2: How to interpret kurtosis values?
A: Kurtosis = 0 indicates normal distribution; >0 indicates leptokurtic (heavy tails, more outliers); <0 indicates platykurtic (light tails, fewer outliers).

Q3: What sample size is needed for reliable results?
A: Generally, n ≥ 30 provides reasonable estimates, but larger samples give more stable moment calculations.

Q4: Are there different types of skewness/kurtosis calculations?
A: Yes, different packages may use biased vs unbiased estimators. This calculator uses the same approach as R's e1071 package.

Q5: When should I be concerned about skewness/kurtosis?
A: When |skewness| > 2 or |kurtosis| > 7, data may significantly deviate from normality, affecting parametric test validity.