AS & A Level Maths

📘 Notes

📊 Coded Data (Addition/Subtraction) — Year 12/13 Notes

Coding a dataset means transforming every value by adding (or subtracting) the same constant. If the original data are $x_1, x_2, \dots, x_n$, choose a convenient constant $a$ and define

\[ y_i = x_i - a \quad\text{(or } y_i = x_i + c \text{ with } c=-a\text{).} \]

We often pick $a$ close to the centre of the data (e.g. a round number near the mean) to make arithmetic simpler.

Key Properties (Addition/Subtraction only)

Mean: If $\bar x$ is the mean of $x$’s and $\bar y$ is the mean of $y$’s, then \[ \bar y = \bar x - a \quad\Longleftrightarrow\quad \bar x = \bar y + a. \]
Median & Quartiles: All shift by the same amount $a$. $\text{median}(y)=\text{median}(x)-a$, similarly for $Q_1, Q_3$.
Spread (variance, s.d., IQR): Unchanged by adding/subtracting a constant. \[ s_y^2 = s_x^2,\qquad s_y = s_x,\qquad \text{IQR}_y=\text{IQR}_x. \]
Mode: Value shifts by $a$; the frequency pattern does not change.

Why code?

To make mental/hand calculations faster (e.g., subtract 100 from prices around \$100).
To centre data around 0 for tidy sums (often $\sum y_i = 0$ if $a$ is the mean or a midpoint).
To simplify grouped-data computations (choose class midpoints minus a convenient origin).

📝 Examples (with “Show answer”)

Example 1 (uncoded → coded → mean): Compute the mean of 98, 101, 103, 97, 101 by coding with $a=100$.

Code with $y_i=x_i-100$:

$x:\; 98,\,101,\,103,\,97,\,101$ $y:\; -2,\, 1,\, 3,\,-3,\, 1$

$\sum y = (-2)+1+3+(-3)+1 = 0 \Rightarrow \bar y = 0/5 = 0.$ Decode: $\bar x = \bar y + 100 = \boxed{100}.$

Note: Standard deviation is unchanged by the shift.

Example 2 (median & spread): For data $x:\; 12, 15, 17, 19, 22$. Code with $a=15$. Find median and IQR before/after coding.

$y = x - 15:\; -3, 0, 2, 4, 7$.

Median($x$) = 17 → Median($y$) = $17-15=2$. Quartiles shift by 15; IQR is unchanged.

Original $Q_1=15$, $Q_3=22$ so IQR($x$) = $7$. Coded $Q_1=0$, $Q_3=7$ so IQR($y$) = $7$ (same).

Example 3 (grouped data mean via coding): Classes centred near 50

Suppose midpoints $m$: 40, 50, 60 with frequencies $f$: 8, 14, 8. Choose origin $a=50$, code $y=m-a$: $-10, 0, +10$.

$\sum f = 30$, $\sum fy = 8(-10) + 14(0) + 8(10) = 0\Rightarrow \bar y = 0.$ Decode: $\bar x = \bar y + a = 50.$

Example 4 (fast variance check): Shifting does not change s.d.

For $x: 7, 9, 10$, take $a=9$ so $y=-2,0,1$. Compute $s_y$ quickly from small numbers; then $s_x = s_y$.

🎯 Workflow for coded calculations (addition/subtraction)

Pick $a$ (a convenient centre such as a round number or a class origin).
Compute $y_i = x_i - a$.
Do the arithmetic with the $y_i$ (means, frequency sums, etc.).
Decode results: add $a$ back to means/locations (not to spreads).

🔎 Extension (not examined here): Multiplying / Full linear coding

More generally you can use linear coding: \[ y_i = \frac{x_i - a}{b}\qquad (b\neq 0). \]

Mean: $\bar y = \dfrac{\bar x - a}{b}\;\Rightarrow\; \bar x = a + b\,\bar y.$
Variance: $s_y^2 = \dfrac{s_x^2}{b^2}\;\Rightarrow\; s_x^2 = b^2 s_y^2.$
Standard deviation: $s_y = \dfrac{s_x}{|b|}\;\Rightarrow\; s_x = |b|\, s_y.$
Medians & quartiles: subtract $a$, then divide by $b$ (so they scale and shift accordingly).

Example (extension): $x:$ prices near 200; take $a=200$, $b=5$

Code $y=\dfrac{x-200}{5}$ to make tiny integers. Compute $\bar y$, $s_y$, then decode: $\bar x = 200 + 5\bar y,\; s_x = 5 s_y.$

⚠️ Common pitfalls

Don’t add the coding constant to the standard deviation or variance—they do not change under addition.
When decoding after multiplying (extension), remember spreads scale by $|b|$ (or $b^2$ for variance).
When working with grouped data, choose a sensible origin $a$ (e.g., a class midpoint) to make $\sum fy$ simple.