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.
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.
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.
