Coding

Skill 1: Coding With Raw Data

To code a data point $x$, we turn it into $y=dfrac{x-a}{b}$ where $a$ and $b$ are of our choosing. Then we can work with our new data set $y$ to get information about our original $x$.

Recall notation: $text{mean of }x=bar{x}$ and $text{variance of }x= text{var}(x)$

To get information about $x$ from our coded set $y$, we have the following formulas:

$bar{y}=dfrac{bar{x}-a}{b}$

$text{var}(y)=dfrac{text{var}(x)}{b^{2}}$

Example: Find the mean and variance of $14010,14030,14040,14060$

If we use the coding $y=dfrac{x-14000}{10}$ we get $y=1,3,4,6$, which is much easier to work with.

So, find the mean and variance of the $y$ values:

$bar{y}=dfrac{1+3+4+6}{4}=3.5$

$text{var}(y)=dfrac{1^{2}+3^{2}+4^{2}+6^{2}}{4}-3.5^{2}=dfrac{1+9+16+36}{4}-12.25=dfrac{62}{4}-12.25=15.5-12.25=3.25$

Then, find the mean of the original values:

$3.5=dfrac{bar{x}-14000}{10}$

$bar{x}-14000=35$

$bar{x}=14035$

and the variance of the original values:

$3.25=dfrac{text{var}(x)}{10^{2}}$

$text{var}(x)=325$

So the mean and variance of $14010,14030,14040,14060$ is $14035$ and $325$

Skill 2: Coding With Summarised Data

We can also apply coding to simplify summarised data to calculate the mean and variance.

Example: Suppose we have $sum{(x-100)}=18$ and $sum{(x-100)^{2}}=45$ with ten data points.

The obvious coding to try is $y=x-100$

This gives $sum{y}=18$ and $sum{y^{2}}=45$

So, the mean and variance of the $y$ values are:

$bar{y}=dfrac{18}{10}=1.8$

$text{var}(y)=dfrac{45}{10}-1.8^2=4.5-3.24=1.26$

This gives results of:

$bar{x}= bar{y} +100 = 1.8 + 100 = 101.8$

and

$text{var}(x)= text{var} (y) = 1.26$

(the variance of $x$ is the same as the variance of $y$, since we only subtracted $100$ from each number).

Skill 3: Coding With Grouped Data

We can code grouped data by coding the midpoint and calculating all our sums with the coded midpoint, then converting back at the end.

Example: Estimate the mean and variance of the heights of the teachers at a school from the following table.

Step 1: Find the midpoints.

Step 2: Choose a suitable coding for the midpoints. Here we should go for $y=dfrac{x-1.55}{0.1}$. Now add the coded midpoints to the table.

Step 3: Proceed normally with the coded values, by adding an $fy$ column and an $fy^{2}$ column to the table.

Step 4: Calculate the mean and variance estimates for $y$.

$bar{y}=dfrac{87}{50}=1.74$

$text{var}(y) =dfrac{191}{50}-1.74^{2}=3.82-3.0276=0.7924$

Step 5: Use the coding formulas to turn these into values for $x$.

$1.74=dfrac{bar{x}-1.55}{0.1}$

$bar{x}-1.55=0.174$

$bar{x}=1.724$

$0.7924=dfrac{text{var}(x)}{0.1^{2}}$

$text{var}(x)=0.007924$