Learn9709Paper 5 · Probability & Statistics 1Representation of dataCoded and combined data

Representation of data · Subtopic 4 of 4

Coded and Combined Data

Shifting every value moves the mean but leaves the spread alone; scaling every value scales both. Knowing this, you can work with simpler "coded" numbers and translate back, and you can merge two groups using their totals.

Cambridge 9709 Paper 5Edexcel YMA01 WST01Edexcel 9MA0 Paper 3AQA 7357 Paper 3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Find the mean and standard deviation from $\Sigma(x-a)$ and $\Sigma(x-a)^2$ .
Know the effect of $y=ax+b$ on the mean and standard deviation.
Recover $\Sigma x$ and $\Sigma x^2$ from a mean and standard deviation.
Combine two data sets into one mean and standard deviation.

Pause and think

Every student's test mark is increased by $5$ . What happens to the mean? What happens to the standard deviation?

Section 2 of 14

Core idea

Start hereMake the numbers small

Heights of adults are all around $170$ cm. Squaring numbers like $172$ , $168$ , $175$ for a variance is tedious. Subtract $170$ first: $2,-2,5$ . The spread is unchanged and the arithmetic is easy. This is coding.

The big idea

If $y=\frac{x-a}b$ : $\bar y=\frac{\bar x-a}b$ and $\sigma_y=\frac{\sigma_x}b$ . Equivalently $\bar x=b\bar y+a$ , $\sigma_x=b\sigma_y$ . Adding/subtracting shifts the mean only; multiplying scales both.

1. Shifting and scaling

Adding a constant slides the whole shape along

shift a = 0.0

The mean moves by

a

; the width of the shape (the standard deviation) does not change at all.

Effect of

y=x-a

$\bar y=\frac{\sum(x-a)}n=\frac{\sum x-na}n=\bar x-a$
Mean shifts.
$y-\bar y=(x-a)-(\bar x-a)=x-\bar x$
Deviations unchanged.
$\sigma_y=\sigma_x$
So the spread is unchanged.

Dividing by

b

divides every deviation by

b

, so the SD is divided by

b

(variance by

b^2

2. Summary statistics in coded form

Worked example $n=20$ , $\sum(x-50)=120$ , $\sum(x-50)^2=2000$

Find the mean and SD of $x$ , then $\sum x$ and $\sum x^2$ .

Goal: Four values.
Strategy: Work with $y=x-50$ .

$\bar y=6\Rightarrow\bar x=56$
Add $50$ back.
$\sigma_y^2=\tfrac{2000}{20}-6^2=64\Rightarrow\sigma_x=8$
SD unchanged.
$\sum x=\sum(x-50)+20\times50=1120$
Undo the coding.
$\sum x^2=20(\sigma^2+\bar x^2)=20(64+3136)=64000$
Rearrange $\sigma^2=\frac{\sum x^2}n-\bar x^2$ .

Answer

\bar x=56

\sigma=8

\sum x=1120

\sum x^2=64000

✓ Check

\frac{64000}{20}-56^2=3200-3136=64

✓.

Common misconceptionShifting the SD Subtracting

50

does not change the SD.

Pause and think

°C has mean 20, SD 5. In °F ( $F=1.8C+32$ )?

Summary

Shift: mean only.
Scale by $b$ : mean and SD × $b$ .

Coding: Linear transformation of data.

HistoryCoding before calculators. Until the 1970s, statistics was done by hand or with mechanical calculators. Squaring numbers like

1047

hundreds of times was slow and error-prone, so statisticians subtracted a convenient value first (coding) and worked with small numbers like

-3

and

7

. Exam questions on coding preserve this clever shortcut.

Fun factCelsius to Fahrenheit is coding.

F=1.8C+32

is a linear code. If a week's temperatures have mean

20^\circ

C and standard deviation

5^\circ

C, in Fahrenheit the mean is

1.8(20)+32=68^\circ

F, but the standard deviation is just

1.8\times5=9^\circ

F: the

+32

never touches the spread.

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