Measures of dispersion: range, interquartile range, standard deviation

Dispersion tells us how spread out a set of numbers is. Think of it as the “spread of the crowd” in a classroom. A tight crowd (low dispersion) vs. a scattered crowd (high dispersion). Below we explore three key measures: Range, Interquartile Range (IQR), and Standard Deviation (σ).

Definition: The difference between the largest and smallest values in a data set.

Formula: $R = \max(x_i) - \min(x_i)$

Analogy: Imagine a race track. The range is the distance between the fastest and slowest runners.

Example: Data: 5, 12, 9, 3, 15 $R = 15 - 3 = 12$

Definition: The spread of the middle 50% of the data, i.e., between the first quartile ($Q_1$) and the third quartile ($Q_3$).

Formula: $IQR = Q_3 - Q_1$

Analogy: Think of a classroom where the middle 50% of students are the “core group.” The IQR tells us how spread out that core group is.

Example: Data (sorted): 2, 4, 5, 7, 8, 9, 12, 14, 18 $Q_1 = 5$, $Q_3 = 12$ $IQR = 12 - 5 = 7$

Definition: A measure of how far each number in the set is from the mean, on average.

Formula (population): $$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\bar{x})^2}$$

Formula (sample): $$s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2}$$

Analogy: Picture a group of friends standing at different distances from a central point. The standard deviation tells us how far, on average, they are from that central point.

Example: Data: 4, 8, 6, 5, 9 $\bar{x} = \frac{4+8+6+5+9}{5} = 6.4$ $s = \sqrt{\frac{(4-6.4)^2+(8-6.4)^2+(6-6.4)^2+(5-6.4)^2+(9-6.4)^2}{4}} \approx 1.93$

📝 Examination Tips

• Always check whether the data set is a sample or a population before choosing the correct formula.
• For the IQR, remember that you need to find the median of the lower half and the median of the upper half.
• When calculating standard deviation by hand, use the shortcut formula: $\sum x_i^2 - \frac{(\sum x_i)^2}{n}$ for population data.
• Practice rounding to the correct number of decimal places as specified in the question.
• Use the range to spot potential outliers quickly before calculating more detailed measures.