How to Calculate Skewness and Kurtosis Manually

Skewness and Kurtosis Formulas:

\[ Skewness = \frac{\sum(x_i - \mu)^3}{n \sigma^3} \] \[ Kurtosis = \frac{\sum(x_i - \mu)^4}{n \sigma^4} \]

Dataset Size (n):

Mean (μ):

Standard Deviation (σ):

Skewness:

Kurtosis:

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1. What Are 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.

2. How to Calculate Step-by-Step

The formulas for calculating skewness and kurtosis:

\[ Skewness = \frac{\sum(x_i - \mu)^3}{n \sigma^3} \] \[ Kurtosis = \frac{\sum(x_i - \mu)^4}{n \sigma^4} \]

Step-by-step calculation:

Calculate the mean (μ) of the dataset
Calculate the standard deviation (σ)
For each data point, calculate (xi - μ)
Sum the cubed deviations for skewness
Sum the fourth-power deviations for kurtosis
Divide by n × σ³ for skewness and n × σ⁴ for kurtosis

Variables:

\( x_i \) — Individual data points
\( \mu \) — Mean of the dataset
\( \sigma \) — Standard deviation
\( n \) — Number of data points

3. Interpretation of Results

Skewness Interpretation:

Skewness = 0: Symmetrical distribution
Skewness > 0: Right-skewed (positive skew)
Skewness < 0: Left-skewed (negative skew)

Kurtosis Interpretation:

Kurtosis = 3: Mesokurtic (normal distribution)
Kurtosis > 3: Leptokurtic (heavy tails)
Kurtosis < 3: Platykurtic (light tails)

4. Using the Calculator

Instructions: Enter your dataset as comma-separated values (e.g., 1,2,3,4,5). The calculator will compute all statistics automatically and display the step-by-step results.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between population and sample skewness/kurtosis?
A: This calculator uses population formulas. For sample data, divide by (n-1) for variance and adjust degrees of freedom accordingly.

Q2: What range of values is considered normal for skewness?
A: For practical purposes, skewness between -0.5 and 0.5 is considered approximately symmetrical, while values beyond ±1 indicate substantial skewness.

Q3: Why is kurtosis compared to 3?
A: The normal distribution has kurtosis = 3, so this serves as the reference point. Some formulas subtract 3 to create "excess kurtosis."

Q4: Can skewness and kurtosis be calculated for small datasets?
A: Yes, but results may be less reliable with small sample sizes (n < 20-30). Larger samples provide more accurate estimates.

Q5: What are the limitations of these measures?
A: They are sensitive to outliers and may not fully capture distribution shape in multimodal or irregular distributions.