Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. There are two common formulae for calculating skewness:

1. Sample Skewness Formula (using third central moment):

The formula for calculating sample skewness is:

$\text{Skewness} = \frac{n}{(n-1)(n-2)} \sum \left( \frac{x_i - \bar{x}}{s} \right)^3$

Where:

• $n$ = sample size
• $x_i$ = individual data points
• $\bar{x}$ = sample mean
• $s$ = sample standard deviation

This formula uses the third central moment and adjusts for small sample sizes. It accounts for how the data points are distributed relative to the mean.

2. Population Skewness Formula (using third central moment):

The population skewness formula is:

$\text{Skewness} = \frac{1}{N \sigma^3} \sum_{i=1}^{N} (x_i - \mu)^3$

Where:

• $N$ = population size
• $x_i$ = individual data points
• $\mu$ = population mean
• $\sigma$ = population standard deviation

This version is used when dealing with an entire population and assumes that the population parameters (mean and standard deviation) are known.

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Summary of Differences:

• The sample skewness formula adjusts for finite sample size (through $n/(n-1)(n-2)$).
• The population skewness formula does not require such adjustments as it assumes the entire population is known.

Would you like a more detailed explanation or an example calculation?

Related Questions:

1. How do you calculate the standard deviation needed in the skewness formula?
2. What does a positive skewness value indicate about a dataset?
3. How is skewness related to the shape of the distribution curve?
4. What happens if skewness equals 0?
5. How do you interpret the difference between sample and population skewness?

Tip: When calculating skewness for sample data, always check for outliers, as they can heavily influence the skewness value.