Mathematics 9709 · Probability & Statistics 1

Coded data: shift & scale

Sliding every value by a constant moves the data, but not its spread.
So $\operatorname{Var}(x-b)=\operatorname{Var}(x)$ .
Stretching every value by $a$ stretches the spread.
So $\operatorname{Var}(ax)=a^2\operatorname{Var}(x)$ and $\text{SD}(ax)=|a|\text{SD}(x)$ .
Shift = move the ruler's zero. Scale = change the ruler's units.

Shift does nothing; scale squares

$\operatorname{Var}(ax-b)=a^2\operatorname{Var}(x),\qquad \text{SD}(ax-b)=|a|\,\text{SD}(x)$

So you can find the SD straight from coded totals: $\operatorname{Var}(x)=\dfrac{\sum(x-b)^2}{n}-\Big(\dfrac{\sum(x-b)}{n}\Big)^2$ , useful when the raw numbers are huge.

Sliding (−b) leaves spread untouched; stretching (×a) scales SD by |a| and variance by a². Shift = move zero; scale = change units.

Let $u=x-b$ . Then $\operatorname{Var}(x)=\operatorname{Var}(u)$ (the −b does nothing).
$\bar u=\sum u/n\Rightarrow \bar x=\bar u+b$ .
$\operatorname{Var}=\sum u^2/n-\bar u^2$ ; $\text{SD}(x)=\sqrt{\operatorname{Var}}$ .
If scaled too ( $u=ax-b$ ): $\operatorname{Var}(x)=\operatorname{Var}(u)/a^2$ .

Worked example

Scale rule (×a squares the variance)

Brand prices have SD $\$24$ . Imitations cost $0.25x$ . Variance of imitation prices?

$\operatorname{Var}(0.25x)=0.25^2\times24^2=0.0625\times576$

Answer

=36\ (\$^2)

Worked example

SD from coded totals

Eight values: $\sum(x-20)=40$ , $\sum(x-20)^2=1000$ . Let $u=x-20$ : $\bar u=5$ so $\bar x=25$ , and $\operatorname{Var}(x)=\operatorname{Var}(u)=\tfrac{1000}{8}-5^2=100$ .

Answer

\text{SD}=10\ (\text{the }-20\text{ never touched it})

Worked example

SD from a coded frequency table

Code $u=x-1$ . With $u=0,1,2,3,4$ and $f=2,4,8,5,1$ : $\sum f=20,\ \sum uf=39,\ \sum u^2f=97$ . Because the shift does nothing, $\operatorname{Var}(x)=\operatorname{Var}(u)$ :

$\operatorname{Var}=\tfrac{97}{20}-\big(\tfrac{39}{20}\big)^2=4.85-3.8025=1.0475$

Answer

\text{SD}=\sqrt{1.0475}=1.023

Worked example

Scale + shift: recover Σx² from coded totals

For $u=3x-1$ over $n=10$ values: $\sum u=53,\ \sum u^2=9136$ . First the mean: $\bar u=5.3$ and $\bar u=3\bar x-1$ , so $\bar x=2.1$ .

$\operatorname{Var}(x)=\tfrac1{3^2}\operatorname{Var}(u)=\tfrac19\big(\tfrac{9136}{10}-5.3^2\big)=\tfrac19(885.51)=98.39$ .

Then $\tfrac{\sum x^2}{10}=\operatorname{Var}(x)+\bar x^2=98.39+4.41$ .

Answer

\sum x^2=10\times102.8=1028

Transform	Mean	Variance	SD
$x-b$	$\bar x-b$	unchanged	unchanged
$a\,x$	$a\bar x$	$a^2\operatorname{Var}$	$\|a\|\,\text{SD}$
$a\,x-b$	$a\bar x-b$	$a^2\operatorname{Var}$	$\|a\|\,\text{SD}$

Examiner note

State

\operatorname{Var}(x)=\operatorname{Var}(x-b)

(the coding identity) before substituting — that equality is the method evidence — then show

\sum(x-b)^2/n-(\sum(x-b)/n)^2

in full. When the code scales too, divide by

a^2

and recover the mean from

\bar u=a\bar x-b

Common mistake

Scaling the variance by

a

instead of

a^2

, or thinking the shift

b

changes the spread (it never does). In

{}^\circ\!F=1.8\,{}^\circ\!C+32

, only the

\times1.8

acts on the SD; the

+32

is ignored.

The °F = 1.8 °C + 32 case in full

$\operatorname{Var}(1.8C+32)=1.8^2\operatorname{Var}(C)=3.24\operatorname{Var}(C)$ , so $\text{SD}_F=1.8\,\text{SD}_C$ . The $+32$ shifts every reading equally, so the spread between them is unchanged — only the $\times1.8$ unit-change stretches it.

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Coded data: shift & scale

Sliding every value by a constant moves the data, but not its spread.
So $\operatorname{Var}(x-b)=\operatorname{Var}(x)$ .
Stretching every value by $a$ stretches the spread.
So $\operatorname{Var}(ax)=a^2\operatorname{Var}(x)$ and $\text{SD}(ax)=|a|\text{SD}(x)$ .
Shift = move the ruler's zero. Scale = change the ruler's units.

Shift does nothing; scale squares

$\operatorname{Var}(ax-b)=a^2\operatorname{Var}(x),\qquad \text{SD}(ax-b)=|a|\,\text{SD}(x)$

So you can find the SD straight from coded totals: $\operatorname{Var}(x)=\dfrac{\sum(x-b)^2}{n}-\Big(\dfrac{\sum(x-b)}{n}\Big)^2$ , useful when the raw numbers are huge.

Sliding (−b) leaves spread untouched; stretching (×a) scales SD by |a| and variance by a². Shift = move zero; scale = change units.

Let $u=x-b$ . Then $\operatorname{Var}(x)=\operatorname{Var}(u)$ (the −b does nothing).
$\bar u=\sum u/n\Rightarrow \bar x=\bar u+b$ .
$\operatorname{Var}=\sum u^2/n-\bar u^2$ ; $\text{SD}(x)=\sqrt{\operatorname{Var}}$ .
If scaled too ( $u=ax-b$ ): $\operatorname{Var}(x)=\operatorname{Var}(u)/a^2$ .

Worked example

Scale rule (×a squares the variance)

Brand prices have SD $\$24$ . Imitations cost $0.25x$ . Variance of imitation prices?

$\operatorname{Var}(0.25x)=0.25^2\times24^2=0.0625\times576$

Answer

=36\ (\$^2)

Worked example

SD from coded totals

Eight values: $\sum(x-20)=40$ , $\sum(x-20)^2=1000$ . Let $u=x-20$ : $\bar u=5$ so $\bar x=25$ , and $\operatorname{Var}(x)=\operatorname{Var}(u)=\tfrac{1000}{8}-5^2=100$ .

Answer

\text{SD}=10\ (\text{the }-20\text{ never touched it})

Worked example

SD from a coded frequency table

Code $u=x-1$ . With $u=0,1,2,3,4$ and $f=2,4,8,5,1$ : $\sum f=20,\ \sum uf=39,\ \sum u^2f=97$ . Because the shift does nothing, $\operatorname{Var}(x)=\operatorname{Var}(u)$ :

$\operatorname{Var}=\tfrac{97}{20}-\big(\tfrac{39}{20}\big)^2=4.85-3.8025=1.0475$

Answer

\text{SD}=\sqrt{1.0475}=1.023

Worked example

Scale + shift: recover Σx² from coded totals

For $u=3x-1$ over $n=10$ values: $\sum u=53,\ \sum u^2=9136$ . First the mean: $\bar u=5.3$ and $\bar u=3\bar x-1$ , so $\bar x=2.1$ .

$\operatorname{Var}(x)=\tfrac1{3^2}\operatorname{Var}(u)=\tfrac19\big(\tfrac{9136}{10}-5.3^2\big)=\tfrac19(885.51)=98.39$ .

Then $\tfrac{\sum x^2}{10}=\operatorname{Var}(x)+\bar x^2=98.39+4.41$ .

Answer

\sum x^2=10\times102.8=1028

Transform	Mean	Variance	SD
$x-b$	$\bar x-b$	unchanged	unchanged
$a\,x$	$a\bar x$	$a^2\operatorname{Var}$	$\|a\|\,\text{SD}$
$a\,x-b$	$a\bar x-b$	$a^2\operatorname{Var}$	$\|a\|\,\text{SD}$

Examiner note

State

\operatorname{Var}(x)=\operatorname{Var}(x-b)

(the coding identity) before substituting — that equality is the method evidence — then show

\sum(x-b)^2/n-(\sum(x-b)/n)^2

in full. When the code scales too, divide by

a^2

and recover the mean from

\bar u=a\bar x-b

Common mistake

Scaling the variance by

a

instead of

a^2

, or thinking the shift

b

changes the spread (it never does). In

{}^\circ\!F=1.8\,{}^\circ\!C+32

, only the

\times1.8

acts on the SD; the

+32

is ignored.

The °F = 1.8 °C + 32 case in full