Sample Skewness And Kurtosis Formula

Sample Skewness and Kurtosis Formulas:

\[ Skewness = \frac{n}{(n-1)(n-2)} \sum\left(\frac{x_i - \mu}{\sigma}\right)^3 \] \[ Kurtosis = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum\left(\frac{x_i - \mu}{\sigma}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)} \]

Sample Skewness:

Sample Kurtosis:

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1. What Are Sample Skewness And Kurtosis?

Sample skewness measures the asymmetry of a probability distribution, while sample kurtosis measures the "tailedness" or sharpness of the peak of a distribution. These are higher moment statistics that describe the shape of data beyond mean and variance.

2. How Do The Formulas Work?

The calculator uses the sample-adjusted formulas:

Where:

\( n \) — Sample size
\( x_i \) — Individual data points
\( \mu \) — Sample mean
\( \sigma \) — Sample standard deviation

Explanation: These formulas include sample size adjustments (Bessel's correction) to provide unbiased estimates of population skewness and kurtosis from sample data.

3. Importance Of Skewness And Kurtosis

Details: Skewness helps identify distribution asymmetry (positive skew = right-tailed, negative skew = left-tailed). Kurtosis indicates tail heaviness and peak sharpness compared to normal distribution (leptokurtic = heavy tails, platykurtic = light tails).

4. Using The Calculator

Tips: Enter numerical data values separated by commas. Minimum 3 data points required. The calculator computes sample mean, standard deviation, then applies the skewness and kurtosis formulas with sample size corrections.

5. Frequently Asked Questions (FAQ)

Q1: What do positive and negative skewness values mean?
A: Positive skewness indicates a right-skewed distribution (tail extends to right), negative indicates left-skewed. Zero suggests symmetric distribution.

Q2: What is the interpretation of kurtosis values?
A: Normal distribution has kurtosis ≈ 0. Positive kurtosis (leptokurtic) means heavier tails and sharper peak. Negative kurtosis (platykurtic) means lighter tails and flatter peak.

Q3: Why use sample-adjusted formulas instead of population formulas?
A: Sample adjustments reduce bias when estimating population parameters from sample data, especially important for small sample sizes.

Q4: What is the minimum sample size required?
A: Minimum 3 points for skewness, 4 points for kurtosis calculation. Larger samples provide more reliable estimates.

Q5: How are these statistics used in practice?
A: Used in statistical analysis to test normality assumptions, identify outliers, understand data distribution shape, and in various statistical tests and modeling.