Data Handling & Probability — 11+ Topic Guide

Part of the GrammarMock topic guides. The hard maths papers carry chart-based questions with real diagrams — use those for applied practice.

The four averages words (and the rhyme that keeps them straight)

• Mean: add them all, divide by how many
• Median: middle value when sorted
• Mode: the most common value
• Range: biggest minus smallest (a spread, not an average)

The playground mnemonic that works: mode = most, median = middle, mean = the meanie that makes you do all the work.

Mean questions — the three levels

Level 1 — compute it: mean of 4, 9, 12, 15, 20 → total 60 ÷ 5 = 12.

Level 2 — reverse it (this is where papers actually test): "The mean of five numbers is 14. Four of them are 10, 12, 16, 18. Find the fifth." Method: total must be 5 × 14 = 70. The four known sum to 56. Fifth = 70 − 56 = 14. The insight to teach: mean questions are secretly TOTAL questions. Convert to totals immediately and everything becomes addition.

Level 3 — change it: "The mean of six numbers is 21. One number is removed and the mean of the rest is 20. What was removed?" Totals again: 126 − 100 = 26. Or: "a sixth child aged 15 joins five children with mean age 9" → new total 60, new mean 10.

Median traps

• Sort FIRST, always. The exam lists values unsorted on purpose.
• Even count of values → median is halfway between the middle two: for 27, 33, 38, 40, 44, 51 the median is (38 + 40) ÷ 2 = 39. Children who circle 38 or 40 lose the mark.

Reading charts (where the marks actually are)

GL papers rarely ask "what does the bar say" — they ask a computation ABOUT the chart:

• Bar charts: "how many more X than Y" (read two bars, subtract), "mean per month" (read all bars, total, divide), "range" (tallest minus shortest)
• Line graphs: "which day had the biggest increase" — compute the day-to-day differences; the steepest upward segment wins. Falls are negative, not "big changes" — the exam plants a steep fall as a trap option
• Pie charts: connect the slice to the total. "24 of 90 pupils chose tennis — what angle is the slice?" → 24/90 × 360° = 96°. And in reverse: a 90° slice is a quarter of the total
• Pictograms: READ THE KEY. If ■ = 4 books, then 5½ symbols = 22. The half-symbol is always there and always matters
• Frequency tables: "modal class" = the row with the biggest frequency; "how many scored 11 or more" = add the qualifying rows

The habit that earns these marks: before answering, say what ONE unit on the chart represents.

Probability — the 11+ version

Probability at 11+ is counting, dressed up:

Single event: P(outcome) = ways it can happen ÷ total ways. Bag with 6 red, 4 blue, 10 green: P(red) = 6/20 = 3/10. Always simplify to match the options.

"NOT" questions: P(not yellow) = 1 − P(yellow). Or count directly: 25 counters, 7 red, 10 blue, rest yellow → yellow = 8, P = 8/25.

Two events (the stretch): multiply along the branches. Two dice both showing 6: 1/6 × 1/6 = 1/36. Coin twice, both heads: 1/2 × 1/2 = 1/4. The wording "both", "twice", "in a row" is the signal to multiply.

Counting outcomes for two dice: totals are best handled by listing pairs. "P(total = 8)": (2,6),(3,5),(4,4),(5,3),(6,2) = 5 ways out of 36 → 5/36. Note (3,5) and (5,3) count separately — the dice are different objects. This is the single most-missed idea in 11+ probability.

Common errors checklist

1. Forgetting to sort before finding the median
2. Mean answered with the total (did the adding, forgot the dividing)
3. Ignoring the pictogram key or the half-symbols
4. P(total on two dice) counted without ordered pairs
5. Not simplifying fractions to match the answer options

One-week plan

• Day 1: the four M-words, computing each from raw lists
• Day 2: reverse-mean (totals thinking)
• Day 3: bar + line chart computations
• Day 4: pie chart angle conversions both directions
• Day 5: single-event probability incl. NOT
• Day 6: two-event multiplication + dice pairs
• Day 7: Data & Probability mini-test, timed