3.4 · Statistics and probability

OxfordAQA 9260 unit 3.4 (specification Version 5.1) — the smallest unit at 13% of the combined specimen marks. Core rows S1–S5 cover data types, reading tables, two-way tables, the full chart bank and the basic averages; the Extension additions add histograms with UNEQUAL class intervals, cumulative frequency diagrams, box plots, and quartiles with the inter-quartile range and percentiles. The 1E specimen carries a histogram and a cumulative-frequency correction to upper boundaries; the 2E specimen's statistics questions include the 40-men and 38% reads. Lines the specimens never sample are labelled specification originals.

Must-know for unit 3.4:

Qualitative 定性 data categorises (favourite fruit); discrete 离散 data counts (number of goals); continuous 连续 data measures (height), grouped or ungrouped. Extract data from printed tables and lists — read the row and column headers first, and quote units. Two-way tables 双向表 cross-tabulate two variables; every row and column total checks the interior cells: 30 students, 18 cricket, 15 tennis and 7 both fills a 2×2 whose cells must sum to 30.

Worked example: completing a two-way table. Start from the cell you know: 7 play both. Then cricket but not tennis is $18 - 7 = 11$, tennis but not cricket is $15 - 7 = 8$, and neither is $30 - 7 - 11 - 8 = 4$.

Every row and column adds up, which is the check. Adding $18 + 15 = 33$ and calling it "more than 30, impossible" forgets the 7 counted twice.

The Core chart bank: scatter graphs, stem-and-leaf, tally charts, pictograms, bar charts, dual and composite bar charts, pie charts, line graphs, frequency polygons and histograms with EQUAL class intervals. Choose by data type — categories take bars, comparisons take dual bars, parts-of-a-whole take pie charts, trends take line graphs, correlation takes scatter graphs.

For a pie chart, each item's angle is its fraction of $360°$: if 14 of 40 students choose apples, the apple sector is $\tfrac{14}{40} \times 360° = 126°$. The angles must add to $360°$.

A histogram 直方图 with unequal class intervals 不等组距 draws bar HEIGHT as the frequency density 频率密度 $=$ frequency over class width — the AREA is the frequency (classes of widths 2, 3, 5, 10, 15 holding 5, 12, 18, 14, 9 give heights 2.5, 4, 3.6, 1.4, 0.6). A cumulative frequency diagram 累积频率图 plots each running total at its UPPER class boundary as a smooth increasing curve (the 1E specimen's correction is exactly this upper-boundary point placement); the median reads at $\tfrac n2$, quartiles at $\tfrac n4$ and $\tfrac{3n}4$. A box plot 箱线图 (five-number summary: minimum, $Q_1$, median, $Q_3$, maximum) shows spread at a glance — the box spans the IQR, the whiskers the range, outliers by judgement.

The mean 平均数 divides the total by the count (grouped data multiplies each MIDPOINT by its frequency first); the median 中位数 orders and takes the middle; the mode 众数 is most frequent and the modal class 众数组 is the class with the highest frequency DENSITY when widths differ. The Extension additions add quartiles 四分位数 and the inter-quartile range 四分位距 $= Q_3 - Q_1$ (the median of the lower and upper halves), and percentiles 百分位数 read like the median at any percentage — the $p$th percentile is the value $\tfrac p{100}n$ of the way up the ordered data or the cumulative-frequency curve.

Worked example: an estimated mean. Masses $m$ (kg) of 20 parcels:

The midpoints are 2, 6, 10 and 16, so $\sum fx = 10 + 48 + 40 + 48 = 146$ and the estimated mean is $146 \div 20 = 7.3$ kg. It is an estimate because the exact masses are not known. The last class is wider (width 8), so check the modal class by frequency density: $\tfrac54 = 1.25$, $\tfrac84 = 2$, $\tfrac44 = 1$, $\tfrac38 = 0.375$; the modal class is $4 < m \le 8$.

Worked example: reading a cumulative frequency curve. 60 workers' weekly hours have cumulative frequencies 6, 17, 27, 42, 60 at upper boundaries 10, 15, 20, 25, 30 (hours). The median reads at worker 30: between 20 (27) and 25 (42), so $20 + 5 \times \tfrac{30 - 27}{42 - 27} = 21$ hours. $Q_1$ at worker 15: $10 + 5 \times \tfrac{15 - 6}{17 - 6} = 14.1$; $Q_3$ at worker 45: $25 + 5 \times \tfrac{45 - 42}{60 - 42} = 25.8$; IQR $\approx 11.7$ hours. Draw the reading lines — the marks are for the reads.

Comparing two distributions. Always make TWO comparisons, each in context: one of an average (median) and one of spread (IQR). For example, "Group A's median time (21 h) is higher than Group B's (17 h), so A typically works longer; A's IQR (11.7 h) is larger than B's (6 h), so A's times are less consistent." Comparing only the ranges is weak, because one extreme value can change a range.

• Data types qualitative/discrete/continuous; grouped and ungrouped (S1).
• Extract from printed tables and lists, quoting units (S2).
• Design and use two-way tables with row/column checks (S3).
• The Core chart bank, chosen by data type (S4).
• Unequal-interval histograms by frequency density (area is frequency); cumulative frequency diagrams at UPPER boundaries with median/quartile reads; box plots (S4 Extension; 1E histogram and CF-boundary anchors).
• Mean (grouped via midpoints), median, mode and modal class (S5).
• Quartiles, inter-quartile range and percentiles from data or the curve (S5 Extension).

Boundary notes: probability vocabulary sits in this specification's Number/Algebra work and earlier units' sets; the AO split is asserted by the specification. Lines not sampled by the specimens (stem-and-leaf, pie charts and several chart types on the Extension papers) are taught as specification originals; the specimens are Extension-tier only, so Core-tier demand has no local evidence. Worked examples are adapted from the 2016 specimen pair or original.