Summary statistics (also called descriptive statistics) are single numbers that describe key features of a dataset. In Foundation Mathematics, four statistics are used: mode, median, mean, and range.
KEY TAKEAWAY: No single statistic tells the whole story. Mean, median, mode and range each reveal a different aspect of the data — use them together.
The value that appears most frequently in the dataset.
Example: \$3, 5, 5, 7, 8, 5, 9, 3$
$$\text{Mode} = 5 \quad (\text{appears 3 times})$$
The middle value when data is arranged in order. It is not affected by extreme values (outliers).
Odd number of values: The middle value.
$\$1, 3, 5, 7, 9 \quad \text{Median} = 5$$
Even number of values: The mean of the two middle values.
$\$2, 4, 6, 8 \quad \text{Median} = \frac{4 + 6}{2} = 5$$
The arithmetic average: divide the sum of all values by the count.
$$\bar{x} = \frac{\text{sum of all values}}{\text{number of values}} = \frac{\sum x}{n}$$
Example: \$4, 7, 2, 9, 3$
$$\bar{x} = \frac{4 + 7 + 2 + 9 + 3}{5} = \frac{25}{5} = 5$$
EXAM TIP: Always add up all values first, then divide. A common error is dividing too early.
The spread of the data — the difference between the maximum and minimum values.
$$\text{Range} = \text{Maximum} - \text{Minimum}$$
Example: Dataset: \$4, 7, 2, 9, 3$
$$\text{Range} = 9 - 2 = 7$$
A large range indicates the data is widely spread; a small range indicates the data is clustered.
| Situation | Best Measure | Reason |
|---|---|---|
| Categorical data | Mode | Only valid average for categories |
| Data with outliers | Median | Not affected by extreme values |
| Symmetric data, no outliers | Mean | Uses all values, most precise |
| Comparing spread | Range | Shows how spread out the data is |
Example — outlier effect:
Salaries: $\$42000, \$45000, \$43000, \$44000, \$180000$
$$\text{Mean} = \frac{42000+45000+43000+44000+180000}{5} = \$70800$$
$$\text{Median} = \$44000$$
The mean ($\$70800$) is much higher than most salaries due to the outlier ($\$180000$). The median ($\$44000$) better represents the typical salary.
COMMON MISTAKE: Forgetting to sort the data before finding the median. Always arrange values from smallest to largest first.
Test scores: \$65, 72, 58, 80, 72, 91, 58, 72$
Step 1 — Sort: \$58, 58, 65, 72, 72, 72, 80, 91$
Step 2 — Mode: $72$ (appears $3$ times)
Step 3 — Median: $n = 8$ (even) → middle two values are $72$ and $72$:
$$\text{Median} = \frac{72 + 72}{2} = 72$$
Step 4 — Mean:
$$\bar{x} = \frac{58+58+65+72+72+72+80+91}{8} = \frac{568}{8} = 71$$
Step 5 — Range:
$\$91 - 58 = 33$$
VCAA FOCUS: VCAA tasks may give you a dataset and ask for one or more summary statistics, or ask you to compare two datasets using these statistics. Show all working steps clearly.
