Summary Statistics: Location and Spread

Measures of Location (Centre)

Mean (\(\bar{x}\))

• 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)

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 (\(s\))

• 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.