Understanding Data Dispersion: From Range to Kurtosis

Knowing the central location of a data set, one may want to know whether the observations are far or near that center; this is called a measure of dispersion. In financial analysis, dispersion is commonly used to assess risk.

Range

The range is defined as max - min. A smaller range indicates lower dispersion. Because it only uses the maximum and minimum values, it ignores the distribution of the remaining data and therefore provides an incomplete picture of variability.

Mean Absolute Deviation

The mean absolute deviation (MAD) is defined as \frac{1}{n}\sum_{i=1}^{n}|x_i - \mu|, where \mu denotes the sample mean and n the number of observations.

Population Variance and Population Standard Deviation

The population variance is defined as \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2, where \mu is the population mean and N the population size.

The population standard deviation is the square root of the variance: \sigma = \sqrt{\sigma^2}.

Sample Variance and Sample Standard Deviation

The sample variance is defined as s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2, where \bar{x} is the sample mean and n the sample size.

The sample standard deviation is the square root of the sample variance: s = \sqrt{s^2}.

Coefficient of Variation

The coefficient of variation (CV) is defined as the standard deviation divided by the mean: CV = \frac{\sigma}{\mu}.

Skewness

Skewness measures the asymmetry of a data distribution. A perfectly symmetric distribution has a skewness of 0, and its mean, median, and mode are equal. Right‑skewed distributions have mean > median > mode, while left‑skewed distributions have mean < median < mode.

Kurtosis

Kurtosis measures how peaked a distribution is relative to a normal distribution. A normal distribution has kurtosis equal to 3. Excess kurtosis is defined as kurtosis minus 3. Positive excess kurtosis indicates a leptokurtic (high‑peaked) distribution, while negative excess kurtosis indicates a platykurtic (low‑peaked) distribution.

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