Averages and spread
The three averages
Mean: sum of all values ÷ number of values. Sensitive to extreme values (outliers).
Example: 3, 5, 8, 9, 9. Mean = (3 + 5 + 8 + 9 + 9) ÷ 5 = 34 ÷ 5 = 6.8.
Estimated mean from a frequency table: use midpoints of each class. Estimated mean = Σ(midpoint × frequency) ÷ Σfrequency.
Median: the middle value when data is arranged in order. For n values, the median is at position (n + 1)/2.
- If n is odd: the median is the exact middle value.
- If n is even: the median is the mean of the two middle values.
Example: 3, 5, 8, 9, 9 (n = 5). Median at position 3 = 8. Example: 4, 6, 8, 10 (n = 4). Median = (6 + 8)/2 = 7.
Mode: the most frequently occurring value. A dataset may have no mode, one mode, or more than one mode (bimodal, multimodal).
Example: 3, 5, 8, 9, 9. Mode = 9 (appears twice).
Measures of spread
Range: highest value − lowest value. Simple but affected by outliers.
Interquartile range (IQR): the range of the middle 50% of data.
- Lower quartile (Q1): median of the lower half.
- Upper quartile (Q3): median of the upper half.
- IQR = Q3 − Q1.
Why IQR is better than range: it is not affected by extreme outliers.
Example: 2, 4, 5, 7, 8, 10, 12. Q1 = median of {2, 4, 5} = 4. Q3 = median of {8, 10, 12} = 10. IQR = 10 − 4 = 6.
Choosing the best average
| Situation | Best average |
|---|---|
| Data with outliers | Median |
| Qualitative or categorical data | Mode |
| All numerical data is used | Mean |
| "Most popular" | Mode |
| Symmetric distribution | Mean or median (similar) |
Comparing two distributions
CCEA often asks you to compare two datasets. Always comment on:
- A measure of average (compare means or medians): "the mean of Set A is higher, showing..."
- A measure of spread (compare ranges or IQRs): "Set B has a larger IQR, suggesting..."
Write a conclusion that relates back to the context of the question.
AI-generated · claude-opus-4-7 · v3-ccea-maths