S2.5StatisticsStretch

Coding

Coding simplifies awkward numbers by transforming the data with $y = \dfrac{x - a}{b}$ . You then find the mean and standard deviation of the easy coded values and reverse the coding to recover the originals.

25 min Measures of Location and Spread

Edexcel AS Level Maths: 2.5 Coding (Mean and Standard Deviation)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Apply a coding transformation
Find the mean of the original data from the coded mean
Find the standard deviation from the coded data
Understand which statistics shift vs scale

What coding does

Coding replaces each value $x$ with $y = \dfrac{x - a}{b}$ — subtracting $a$ then dividing by $b$ to get smaller, friendlier numbers to work with.

y = \dfrac{x - a}{b}

Subtract

a

, divide by

b

Reversing for the mean

The mean codes exactly like a data value: find $\bar{y}$ , then reverse the transformation to get $\bar{x}$ .

\bar{x} = b\,\bar{y} + a

Undo the coding on the mean.

Example 1Coding

y = \dfrac{x - 100}{10}

gives

\bar{y} = 3

. Find

\bar{x}

\bar{x} = 10(3) + 100

Answer

130

Reversing for standard deviation

Standard deviation is affected by the $\textbf{scaling}$ only, not the shift — subtracting $a$ doesn’t change spread, but dividing by $b$ does. So multiply the coded standard deviation by $b$ .

\sigma_x = b\,\sigma_y

Only the scale factor

b

affects spread.

Tip — Adding/subtracting shifts data (mean changes, spread doesn’t); multiplying/dividing scales both.

Formula recap

y = \dfrac{x - a}{b}

The coding transformation.

\bar{x} = b\bar{y} + a

Reverse coding for the mean.

\sigma_x = b\sigma_y

Reverse coding for σ (scale only).

Common mistakes to avoid

Adding a back when reversing the standard deviation.

The shift a does not affect spread; only multiply σ by b.

Forgetting to reverse the coding at the end.

The coded mean/σ must be converted back to the original scale.

Key takeaways

Coding: y = (x − a)/b to simplify numbers.
Mean reverses fully: x̄ = b·ȳ + a.
Standard deviation reverses by scale only: σₓ = b·σᵧ.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.