Range & Interquartile Range

• Range: The simplest measure of spread, calculated as the difference between the maximum and minimum values in a dataset. It represents the total span of the observations.

• Quartiles: Values that divide a sorted dataset into four equal parts. The Lower Quartile ($Q_1$) marks the 25th percentile, the Median ($Q_2$) marks the 50th percentile, and the Upper Quartile ($Q_3$) marks the 75th percentile.

• Interquartile Range (IQR): The difference between the upper and lower quartiles ($Q_3 - Q_1$). It measures the spread of the central 50% of the data, effectively ignoring the tails of the distribution.

Calculating the Range

• Step 1: Identify the maximum and minimum values in the dataset.
• Step 2: Subtract the minimum from the maximum: $Range = x_{max} - x_{min}$.

Finding Quartiles and IQR

• Step 1: Sort the data in ascending order. This is a critical prerequisite for all percentile-based calculations.
• Step 2: Find the median ($Q_2$), which splits the data into a lower half and an upper half.
• Step 3: Find the median of the lower half to determine $Q_1$ and the median of the upper half to determine $Q_3$.
• Step 4: Calculate the IQR using the formula: $IQR = Q_3 - Q_1$

• When to use Range: Use when you need a quick, simple overview of the total boundaries of the data and there are no known outliers.

• When to use IQR: Use when the dataset contains outliers or is skewed, as it provides a more reliable measure of the 'typical' spread.

• Confusing Index with Value: Students often find the position of a quartile (e.g., the 5th item) and write '5' as the quartile value instead of the actual data value at that position.

• Handling Odd vs. Even $n$: When finding quartiles for an odd number of data points, the median itself is typically excluded from both the lower and upper halves when calculating $Q_1$ and $Q_3$.

• Negative Numbers: When calculating the range with negative numbers, remember that subtracting a negative is equivalent to adding (e.g., $5 - (-3) = 8$).