Measures of Variability

• The Range is the simplest measure of spread, calculated as the difference between the maximum and minimum values in a dataset: $Range = Max - Min$. While easy to compute, it is highly sensitive to outliers and does not describe the distribution of the middle values.

• The Interquartile Range (IQR) measures the spread of the middle 50% of the data. It is calculated by subtracting the first quartile ($Q_1$) from the third quartile ($Q_3$): $IQR = Q_3 - Q_1$.

• Unlike the range, the IQR is a resistant measure, meaning it is not influenced by extreme outliers. This makes it the preferred measure of spread for skewed distributions or datasets with known extreme values.

• Standard Deviation measures the average distance of each data point from the mean. A small standard deviation indicates that the data points are generally close to the mean, while a large one indicates they are far from it.

• The Variance is the average of the squared differences from the mean. It is the square of the standard deviation ($s^2$ or $\sigma^2$). While variance is useful in higher-level statistical proofs, standard deviation is more commonly used for description because it is expressed in the same units as the original data.

• For a sample, the standard deviation ($s$) uses $n-1$ in the denominator to provide an unbiased estimate: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$ For a population, the standard deviation ($\sigma$) uses $N$: $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$

Comparison of Variability Measures

• Units of Measurement: Range, IQR, and Standard Deviation all share the same units as the original data (e.g., meters). Variance is expressed in squared units (e.g., meters squared), which often makes it less intuitive for direct interpretation.

• Identify the Distribution Shape: If a distribution is skewed or contains outliers, always prioritize the IQR over the standard deviation. Standard deviation is heavily pulled by extreme values, which can provide a misleading sense of the 'typical' spread.

• Check the Units: Ensure that your final answer for standard deviation includes the correct units. If the question asks for variance, remember to square the standard deviation and adjust the units accordingly.

• Outlier Detection: Use the $1.5 \times IQR$ rule to mathematically justify outliers. A value is an outlier if it is less than $Q_1 - (1.5 \times IQR)$ or greater than $Q_3 + (1.5 \times IQR)$.

• Sample vs. Population: Pay close attention to whether the data represents a whole population or just a sample. Using $n$ instead of $n-1$ for a sample standard deviation is a common error that results in lost marks.