Statistics and Probability - Measures of dispersion (range, IQR, standard deviation)

Range: This is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values. Visually, if data points are plotted on a number line, the range represents the total horizontal span or 'width' that the data covers from the leftmost point to the rightmost point.

Quartiles ($Q_1, Q_2, Q_3$): These values divide an ordered data set into four equal parts. The first quartile ($Q_1$) is the $25^{th}$ percentile, the second quartile ($Q_2$) is the median ($50^{th}$ percentile), and the third quartile ($Q_3$) is the $75^{th}$ percentile. In a box-and-whisker plot, these correspond to the start of the box, the line inside the box, and the end of the box respectively.

Interquartile Range (IQR): The IQR measures the spread of the middle $50\%$ of the data by calculating $Q_3 - Q_1$. Visually, the IQR is represented by the length of the central 'box' in a box plot. It is more robust than the range because it is not affected by extreme outliers at the ends of the distribution.

Standard Deviation ($\sigma$): This measure indicates the average distance of each data point from the mean ($\bar{x}$). A low standard deviation means the data points are clustered closely around the mean, while a high standard deviation indicates the data is spread out over a wider range. On a frequency curve, a smaller standard deviation creates a taller, narrower peak, while a larger one creates a flatter, wider curve.

Variance ($\sigma^2$): This is the square of the standard deviation. It provides a measure of how far a set of numbers are spread out from their average value. While standard deviation is in the same units as the data, variance is in squared units.

Outliers: These are extreme values that fall significantly outside the rest of the data. A common mathematical boundary for outliers is any value less than $Q_1 - 1.5 \times IQR$ or greater than $Q_3 + 1.5 \times IQR$. Visually, outliers are often marked with individual dots or asterisks beyond the 'whiskers' of a box-and-whisker plot.

Effect of Transformations: If a constant $k$ is added to every data value (translation), the measures of dispersion (Range, IQR, SD) remain unchanged. However, if every data value is multiplied by a constant $k$ (scaling), the Range, IQR, and Standard Deviation are all multiplied by $|k|$.