Measures of location and spread

Mean. Arithmetic average.

$\bar{x} = \dfrac{\sum x}{n}, \qquad \bar{x} = \dfrac{\sum fx}{\sum f}.$

For grouped data, $x$ is the midpoint of each class.

• Strengths: uses every value, mathematically tractable, links to variance.
• Weaknesses: pulled by outliers; can be misleading when data is skewed.

Median. Middle value of the ordered data.

• Raw data of size $n$: median = $((n+1)/2)$-th value (interpolating between two values when $n$ is even).
• Grouped data: locate position $n/2$ in the cumulative frequency, then linear-interpolate within the median class.

Strengths: robust to outliers; meaningful for skewed data. Weakness: ignores most of the data.

Mode. Most frequent value (or modal class for grouped data).

• Strength: works for categorical/qualitative data.
• Weakness: may not exist or may be multiple. Least informative for continuous data.

Choosing. Symmetric, no outliers $\Rightarrow$ mean. Skewed or outliers present $\Rightarrow$ median. Categorical $\Rightarrow$ mode.

Quartile positions for grouped data.

Quartile	Position	What it means
$Q_1$	$n/4$	$25\%$ of data below
$Q_2$ (median)	$n/2$	$50\%$ of data below
$Q_3$	$3n/4$	$75\%$ of data below

Linear interpolation. Inside the relevant class with lower boundary $L$, width $w$, cumulative frequency $C_{\text{before}}$ before the class and $f$ within it, the quartile at position $p$ is

$Q = L + \dfrac{p - C_{\text{before}}}{f} \cdot w.$

Worked example. From cumulative frequencies $4, 12, 24, 38, 44, 48, 50$ at marks $\le 20, 30, 40, 50, 60, 70, 80$:

• $n = 50$, $n/4 = 12.5$. Class $30 < x \le 40$ (CF $12 \to 24$): $Q_1 = 30 + (12.5 - 12)/12 \cdot 10 \approx 30.42$.
• $n/2 = 25$. Class $40 < x \le 50$: $Q_2 = 40 + (25 - 24)/14 \cdot 10 \approx 40.71$.
• $3n/4 = 37.5$. Same class: $Q_3 = 40 + (37.5 - 24)/14 \cdot 10 \approx 49.64$.

So IQR $\approx 19.2$.

Percentiles. $P_k$ is the value below which $k\%$ of the data lies. Position is $kn/100$ for grouped data; same interpolation method.

Definitions. Variance is the mean squared deviation from the mean.

$\sigma^{2} = \dfrac{\sum (x - \bar{x})^{2}}{n}.$

Shortcut formula (in the booklet). Algebraic manipulation gives

$\sigma^{2} = \dfrac{\sum x^{2}}{n} - \bar{x}^{2}, \qquad \sigma^{2} = \dfrac{\sum fx^{2}}{\sum f} - \bar{x}^{2}.$

Always use the shortcut for calculator-friendly arithmetic.

Standard deviation $\sigma = \sqrt{\sigma^{2}}$ — same units as the data.

Worked example. $n = 20$, $\sum x = 850$, $\sum x^{2} = 36{,}450$.

• $\bar{x} = 850/20 = 42.5$.
• $\sigma^{2} = 36{,}450/20 - 42.5^{2} = 1822.5 - 1806.25 = 16.25$.
• $\sigma = \sqrt{16.25} \approx 4.03$.

Population vs sample. S1 uses the population formula (divide by $n$). The sample variance $s^{2} = \sum(x - \bar{x})^{2}/(n-1)$ appears in higher-level statistics but not in S1 — your calculator's $\sigma_n$ button gives the S1 answer.

Interpretation. $\sigma$ is the typical distance of a data point from the mean. For approximately normal data, $\approx 68\%$ of values lie within $\bar{x} \pm \sigma$, $\approx 95\%$ within $\bar{x} \pm 2\sigma$.

Why code? When data values are large or have an awkward common shift (e.g. heights all around $175$ cm), define

$y = \dfrac{x - a}{h}.$

Compute $\bar{y}$ and $\sigma_y$ on the simpler $y$ values, then back-transform.

Back-transformation rules. From $x = hy + a$:

$\bar{x} = h\bar{y} + a, \qquad \sigma_x = |h|\,\sigma_y.$

Worked example. $y = (x - 200)/5$ gives $\bar{y} = 4$, $\sigma_y = 2$.

• $\bar{x} = 5 \times 4 + 200 = 220$.
• $\sigma_x = 5 \times 2 = 10$.

Key fact. Shifts do not change spread. Adding $a$ to every value moves the mean by $a$ but leaves $\sigma$ unchanged. Only the scale factor $h$ affects $\sigma$.

Common application. Calculator-friendly coding for a table of heights: $y = x - 170$. The $y$-values are around $0$, easy to square and sum, then $\bar{x} = \bar{y} + 170$ and $\sigma_x = \sigma_y$.

Step 1 — choose statistics.

Situation	Centre	Spread
Roughly symmetric, no outliers	mean	SD
Skewed or outliers present	median	IQR

If one data set has outliers but the other doesn't, use the median + IQR for both so the comparison is on the same footing.

Step 2 — write two sentences.

• One sentence on centre (in context): 'Class A has a higher mean mark ($64$) than Class B's median ($62$), so typical performance is similar.'
• One sentence on spread (in context): 'Class B is much more variable (IQR $= 22$) than Class A (SD $= 8$), so Class A is more consistent.'

Step 3 — add interpretation. Use the variable name from the question (marks, heights, times). Examiners reward 'Class B has a wider range of mark', NOT 'Class B has a higher value'.

Pitfall. Comparing mean of one set against median of the other only works if both sets are reasonably symmetric so that mean $\approx$ median. Otherwise re-quote both statistics on the same scale.