Probability & Statistics 1

Mathematics9709

Probability & Statistics 1

Mathematics9709

Coded Data and Totals

Coding replaces awkward data values with simpler ones. It does not change which observations belong to the data set, and every step must be reversible. A scale factor multiplies every value; a shift adds or subtracts the same amount from every value.

Coding is a reversible machine

encode

x → multiply by 2 → subtract 3 → y

decode in reverse

y → add 3 → divide by 2 → x

check

x = (y + 3) ÷ 2

Undo the last operation first.

Decode in reverse order. For y = ax − b, first add b, then divide by a.

Decode a mean in reverse order

y=ax-b

Symbols

$x$ = original value (original unit)
$y$ = coded value (coded scale)
$a$ = scale factor (none)
$b$ = shift (coded scale)

The scale factor $a$ must not be zero; otherwise every original value would be sent to the same coded value and the coding could not be reversed.

The same transformation acts on the mean:

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

Reverse the coding to recover the original mean:

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

Worked example

Undo a scale and a shift

Data are coded by $y=2x-3$ . If the coded mean is 17, reverse the code: add 3, then divide by 2.

Answer

\bar{x}=(17+3)/2=10

Common mistake

Decoding in the wrong order changes the answer. The inverse of “multiply by

a

, then subtract

b

” is “add

b

, then divide by

a

”.

A shift appears once for every value

\sum(ax-b)=a\sum x-nb

The subtraction of $b$ happens to every one of the $n$ values. That is why the total changes by $nb$ , not by only $b$ .

With a frequency table, the total number of observations is $\sum f$ . The same idea becomes

\begin{aligned}\sum(ax-b)f&=a\sum xf\\&\quad-b\sum f\end{aligned}

Divide the coded total by $\sum f$ to get the coded mean, then undo the shift and scale in reverse order.

Expand the coded total. Write $\sum(ax-b)=a\sum x-nb$ .
Solve for the requested total or count. Keep $n$ attached to the shift.
Check the scale. Substitute the result back into the coded-total equation.

Examiner note

A calculator can confirm the final arithmetic, but writing the decoded total equation is usually the method-bearing step.

Now you try

Your turnIndependent check [4 marks]

For $n$ values, $\sum x=944$ and $\sum(x-50)=144$ . Find $n$ and the mean of the original values.

Show worked answer

Since $\sum(x-50)=\sum x-50n$ , $144=944-50n$ , so $n=16$ .

Answer

n=16

\bar{x}=944/16=59

PreviousCombined Means and Changing Data NextThe Median

Coded Data and Totals

Coding is a reversible machine

encode

x → multiply by 2 → subtract 3 → y

decode in reverse

y → add 3 → divide by 2 → x

check

x = (y + 3) ÷ 2

Undo the last operation first.

Decode in reverse order. For y = ax − b, first add b, then divide by a.

Decode a mean in reverse order

y=ax-b

Symbols

$x$ = original value (original unit)
$y$ = coded value (coded scale)
$a$ = scale factor (none)
$b$ = shift (coded scale)

The scale factor $a$ must not be zero; otherwise every original value would be sent to the same coded value and the coding could not be reversed.

The same transformation acts on the mean:

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

Reverse the coding to recover the original mean:

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

Worked example

Undo a scale and a shift

Data are coded by $y=2x-3$ . If the coded mean is 17, reverse the code: add 3, then divide by 2.

Answer

\bar{x}=(17+3)/2=10

Common mistake

Decoding in the wrong order changes the answer. The inverse of “multiply by

a

, then subtract

b

” is “add

b

, then divide by

a

”.

A shift appears once for every value

\sum(ax-b)=a\sum x-nb

The subtraction of $b$ happens to every one of the $n$ values. That is why the total changes by $nb$ , not by only $b$ .

With a frequency table, the total number of observations is $\sum f$ . The same idea becomes

\begin{aligned}\sum(ax-b)f&=a\sum xf\\&\quad-b\sum f\end{aligned}

Divide the coded total by $\sum f$ to get the coded mean, then undo the shift and scale in reverse order.

Expand the coded total. Write $\sum(ax-b)=a\sum x-nb$ .
Solve for the requested total or count. Keep $n$ attached to the shift.
Check the scale. Substitute the result back into the coded-total equation.

Examiner note

A calculator can confirm the final arithmetic, but writing the decoded total equation is usually the method-bearing step.

Now you try

Your turnIndependent check [4 marks]

For $n$ values, $\sum x=944$ and $\sum(x-50)=144$ . Find $n$ and the mean of the original values.

Show worked answer

Since $\sum(x-50)=\sum x-50n$ , $144=944-50n$ , so $n=16$ .

Answer

n=16

\bar{x}=944/16=59

PreviousCombined Means and Changing Data NextThe Median