Coded data: shift and scale | CAIE 9709 Probability & Statistics 1 Notes

Probability & Statistics 1

Mathematics9709

Shift leaves spread; scale changes it

Adding shifts every value by the same amount. Multiplying changes every gap by the scale factor, so it changes the spread.

\operatorname{Var}(ax+b)=a^2\operatorname{Var}(x)

Symbols

$a$ = scale factor (none)
$b$ = shift applied to every value (data unit)

Standard deviation is the square root of variance, so it takes one factor of the scale. Distance cannot be negative, hence the absolute value.

\operatorname{SD}(ax+b)=|a|\operatorname{SD}(x)

Add b to every value: add b to the mean; SD and variance do not change.
Multiply every value by a: multiply the mean by a, the SD by |a| and the variance by a².

Shift moves the centre without changing gaps. Scale stretches or compresses every gap.

Use coded totals as a complete data set

Let $u=(x-100)/5$ . First find the mean and variance of $u$ from its totals. Then undo the scale and shift.

Worked example

Decode mean and standard deviation

For 10 values, $\sum u=12$ and $\sum u^2=46$ .

\bar u=1.2

The coded variance is:

\frac{46}{10}-1.2^2=3.16

Since $x=5u+100$ , the mean is multiplied by 5 then shifted by 100. SD is only multiplied by 5.

Answer

5(1.2)+100=106

\bar x=106

5\sqrt{3.16}=8.89

\operatorname{SD}(x)=8.89\ \text{(3 s.f.)}

Common mistake

Do not add 100 to a standard deviation. The +100 shifts every value equally, so it cannot change their gaps.

A pure shift can be left coded

If $u=x-b$ , then $x=u+b$ . The mean needs the shift restored, but the variance and SD of $u$ are already the variance and SD of $x$ .

Worked example

Standard deviation from shifted totals

Eight values satisfy $\sum(x-20)=40$ and $\sum(x-20)^2=1000$ . Let $u=x-20$ .

\bar u=\frac{40}{8}=5

The coded variance is:

\frac{1000}{8}-5^2=100

Answer

\operatorname{SD}(x)=\operatorname{SD}(u)=10

Write the inverse before decoding

Name the coded variable. Write the given relationship exactly.
Rearrange for the original variable. This exposes the scale and shift in the correct order.
Decode centre and spread differently. Mean uses both scale and shift. SD uses only the absolute scale; variance uses its square.

Your turnIndependent check [5 marks]

For $v=4x-7$ , 20 coded values have mean 25 and variance 144. Find the mean, variance and standard deviation of $x$ .

Show worked answer

Rearrange first: $x=(v+7)/4$ .

Answer

\bar x=(25+7)/4=8

\operatorname{Var}(x)=144/4^2=9

\operatorname{SD}(x)=3

Shift leaves spread; scale changes it

Adding shifts every value by the same amount. Multiplying changes every gap by the scale factor, so it changes the spread.

\operatorname{Var}(ax+b)=a^2\operatorname{Var}(x)

Symbols

$a$ = scale factor (none)
$b$ = shift applied to every value (data unit)

Standard deviation is the square root of variance, so it takes one factor of the scale. Distance cannot be negative, hence the absolute value.

\operatorname{SD}(ax+b)=|a|\operatorname{SD}(x)

Add b to every value: add b to the mean; SD and variance do not change.
Multiply every value by a: multiply the mean by a, the SD by |a| and the variance by a².

Shift moves the centre without changing gaps. Scale stretches or compresses every gap.

Use coded totals as a complete data set

Let $u=(x-100)/5$ . First find the mean and variance of $u$ from its totals. Then undo the scale and shift.

Worked example

Decode mean and standard deviation

For 10 values, $\sum u=12$ and $\sum u^2=46$ .

\bar u=1.2

The coded variance is:

\frac{46}{10}-1.2^2=3.16

Since $x=5u+100$ , the mean is multiplied by 5 then shifted by 100. SD is only multiplied by 5.

Answer

5(1.2)+100=106

\bar x=106

5\sqrt{3.16}=8.89

\operatorname{SD}(x)=8.89\ \text{(3 s.f.)}

Common mistake

Do not add 100 to a standard deviation. The +100 shifts every value equally, so it cannot change their gaps.

A pure shift can be left coded

If $u=x-b$ , then $x=u+b$ . The mean needs the shift restored, but the variance and SD of $u$ are already the variance and SD of $x$ .

Worked example

Standard deviation from shifted totals

Eight values satisfy $\sum(x-20)=40$ and $\sum(x-20)^2=1000$ . Let $u=x-20$ .

\bar u=\frac{40}{8}=5

The coded variance is:

\frac{1000}{8}-5^2=100

Answer

\operatorname{SD}(x)=\operatorname{SD}(u)=10

Write the inverse before decoding

Name the coded variable. Write the given relationship exactly.
Rearrange for the original variable. This exposes the scale and shift in the correct order.
Decode centre and spread differently. Mean uses both scale and shift. SD uses only the absolute scale; variance uses its square.

Your turnIndependent check [5 marks]

For $v=4x-7$ , 20 coded values have mean 25 and variance 144. Find the mean, variance and standard deviation of $x$ .

Show worked answer

Rearrange first: $x=(v+7)/4$ .

Answer

\bar x=(25+7)/4=8

\operatorname{Var}(x)=144/4^2=9

\operatorname{SD}(x)=3