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Statistics

Nearly all variance: write down Σx and Σx² first, and missing, wrong or combined data become a few lines of arithmetic.

Questions in the bank
88
q/paper in 2025–26
0.62
Numeric answer
34%
Notes pages
5

Tier: Long tail

When you’ll see it

A mean and a variance or standard deviation are given, and some of the data is missing, wrong, shifted or combined with another group.

How this chapter is tested

Nearly every question here is a variance question, and nearly every one rests on two totals: Σx and Σx². The variance is σ² = Σx²/n − x̄², so Σx² = n(σ² + x̄²). Once both totals are known, missing values, corrected data and combined groups are a few lines of arithmetic.

The pages build those totals from different starting points: given sums of shifted values, two unknown observations, a wrongly recorded value, two groups put together. Adding a constant to every value leaves the variance unchanged; multiplying by a multiplies it by a². Mean deviation and frequency tables come last and are more mechanical.

Time goes on arithmetic, not ideas. A numeric-answer question gives no options to check against, so write the two totals down first, before any formula. Sums of an A.P. and of the squares of natural numbers come from Sequences and Series.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Variance from sums and shifts

    σ² = Σx²/n − x̄²; for y = ax + b the mean is ax̄ + b and the variance a²σ².

  • Finding unknown observations

    Two unknowns give their sum from the mean and the sum of their squares from the variance; together these fix both.

  • Correcting data and combining groups

    Rebuild Σx and Σx² from the recorded data, swap the wrong value, then recompute; for two groups add the totals.

  • Mean deviation

    (1/n)Σ|x − c| about the mean, the median or the mode; sort the data before picking the median.

  • Frequency distributions

    Mean Σfx/N and variance Σfx²/N − x̄²; grouped median l + ((N/2 − F)/f) × h.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Dropping the square of the mean

    Σx² = n(σ² + x̄²), not nσ². Without the x̄² term the sum of squares comes out too small, sometimes negative.

  • Group variances do not average

    The combined variance also carries a term for the gap between the two group means; it is zero only when the means are equal.

  • The sign of the scale

    The variance is multiplied by a², so a and −a give the same spread. A fixed variance usually allows two values of a.

  • Mean deviation is not the SD

    The mean deviation takes absolute distances and never exceeds the standard deviation; an option equal to the SD is a distractor.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Statistics notes

Drill every Statistics question

88 questions from the bank, across 5 subtopics.

Drill the 88 questions

Drill one subtopic at a time

The 5 subtopics, in teaching order.

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