Averages and spread, grouped-data estimates, histograms (frequency density), cumulative frequency and quartiles, and scatter graphs. TMUA questions are reasoning questions about what must happen to a mean or median when the data change.
Averages and spread (M6.3)
- Mean \(= \frac{\sum x}{n}\); combined mean \(= \frac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}\); adding a value changes the total, so a new mean fixes the new value.
- Median: middle value of ordered data (mean of the two middle values when \(n\) is even). Mode: most frequent.
- Transforming every value by \(ax + b\): mean \(\to a\bar x + b\); range and IQR \(\to |a| \times\) (unchanged by \(b\)).
- Grouped data: use midpoints for an estimated mean; the modal class is the one with the greatest frequency density.
- An outlier pulls the mean towards it but usually leaves the median alone; changing the smallest value may or may not change the median.
Diagrams (M6.1, M6.2, M6.4)
- Histogram: area = frequency; frequency density \(= \frac{\text{frequency}}{\text{class width}}\).
- Cumulative frequency graph: read the lower quartile at \(\frac n4\), median at \(\frac n2\), upper quartile at \(\frac{3n}{4}\); IQR \(= Q_3 - Q_1\).
- Scatter graphs: correlation describes a trend, not a cause; interpolation within the data range is reasonable, extrapolation is risky.
Must-know (no formula booklet)
- frequency \(=\) density \(\times\) width
- combined mean \(= \frac{n_1\bar x_1 + n_2 \bar x_2}{n_1+n_2}\)
- IQR \(= Q_3 - Q_1\)
- correlation \(\neq\) causation
Classic traps
- Doubling then adding \(3\): the mean becomes \(2m + 3\) but the range becomes \(2R\) (no \(+3\)).
- Estimated means from grouped data are estimates because the exact values are unknown.
- The median of an even-sized list need not be a member of the list.
Worked example
Q. Mean of five numbers is \(12\); adding a sixth makes it \(14\). Sixth number?
A. \(84 - 60 = 24\).