Summary Statistics: Location and Spread

Measures of Location (Centre)

Mean ($\bar{x}$)

$$\bar{x} = \frac{\sum x}{n}$$

• Sum of all values divided by the count
• Sensitive to outliers
• Best for symmetric distributions

Median ($M$)

• The middle value when data is sorted in order
• For $n$ values: position = $\frac{n+1}{2}$
• If $n$ is even, average the two middle values
• Resistant to outliers
• Best for skewed distributions

Mode

• The most frequently occurring value
• A dataset can have no mode, one mode, or multiple modes
• Most useful for categorical data

Measures of Spread

Range

$$\text{Range} = \text{Maximum} - \text{Minimum}$$
- Simple but heavily affected by outliers

Interquartile Range (IQR)

$$\text{IQR} = Q_3 - Q_1$$

Where:
- $Q_1$ = lower quartile (median of lower half)
- $Q_3$ = upper quartile (median of upper half)

The IQR covers the middle 50% of the data. Resistant to outliers.

Standard Deviation ($s$)

$$s = \sqrt{\frac{\sum(x - \bar{x})^2}{n-1}}$$

• Measures average distance of values from the mean
• Sensitive to outliers
• Used with the mean (both assume symmetric data)
• Always $s \geq 0$; $s = 0$ only if all values are identical

Choosing the Right Pair

KEY TAKEAWAY: Mean and standard deviation go together; median and IQR go together. Never mix them.

Worked Example

Data: 12, 15, 14, 10, 18, 14, 22, 13

Sorted: 10, 12, 13, 14, 14, 15, 18, 22 (n = 8)

EXAM TIP: On VCAA exams, always show which formula/method you used. For the median with even $n$, show the two middle values and their average.

COMMON MISTAKE: When finding $Q_1$ and $Q_3$, exclude the median value(s) from the two halves. Different calculators/textbooks use slightly different conventions — CAS calculators use the standard VCAA method.