Measures of dispersion — variance, standard deviation: Worked examples

We only use two engines the parent gave us. Let me restate them in plain words so nothing here is a mystery symbol.

The scenario matrix

Below is the full list of "case classes." Every cell is covered by at least one example — the tag in each [!example] tells you which cell it hits.

Case class	What makes it tricky	Covered by
A. Definition, plain data	just follow steps	Ex 1
B. Shortcut on same data	must match A exactly	Ex 2
C. Degenerate — all equal	$σ$ should be $0$	Ex 3
D. Shift invariance ( $x + c$ )	variance must not move	Ex 4
E. Scaling ( $c x$ , incl. negative $c$ )	variance $\times c^{2}$ , SD $\times	c
F. Sample vs population ( $n$ vs $n - 1$ )	which divisor?	Ex 6
G. Frequency / grouped data	weights, not raw list	Ex 7
H. Word problem, real units	interpret SD in context	Ex 8
I. Exam twist — combine sets	derive from summary stats	Ex 9
J. Limiting behaviour ( $n \to \infty$ , one outlier grows)	how $σ$ reacts	Ex 10

Figure — Measures of dispersion — variance, standard deviation

The figure above is the mental map: every example is a dart-scatter picture, and $σ$ measures how wide the scatter is.

A & B — definition vs shortcut agree

C — the degenerate case (all values equal)

D & E — shifting and scaling

These two are the workhorses of exam questions: they let you re-use a variance you already computed.

F — sample vs population divisor

G — frequency / grouped data

Worked example Example 7 — Case G (weighted data)

Marks and how many students got them:

mark $x_{i}$	$2$	$4$	$6$
frequency $f_{i}$	$1$	$3$	$2$

Forecast: most students sit at $4$ ; guess whether $σ$ is below or above $2$ .

Total count $N = 1 + 3 + 2 = 6$ .
Weighted mean: $\sum f_{i} x_{i} = 1 \cdot 2 + 3 \cdot 4 + 2 \cdot 6 = 2 + 12 + 12 = 26$ , so $\overset{x}{ˉ} = 26/6 \approx 4.33$ . Why weight? A mark that appears $3$ times pulls the centre three times as hard.
Weighted mean of squares: $\sum f_{i} x_{i}^{2} = 1 \cdot 4 + 3 \cdot 16 + 2 \cdot 36 = 4 + 48 + 72 = 124$ ; divide by $N$ : $124/6 \approx 20.67$ .
Shortcut: $σ^{2} = 20.67 - (4.33)^{2} = 20.67 - 18.78 \approx 1.89$ .
$σ \approx 1.37$ .

Verify: exact fractions — $\overset{x}{ˉ} = 13/3$ , $\overline{x^{2}} = 62/3$ , so $σ^{2} = \frac{62}{3} - \frac{169}{9} = \frac{186 - 169}{9} = \frac{17}{9} \approx 1.89$ ✓. Below $2$ , as forecast, because most students cluster at $4$ .

H — real-world word problem

I — exam twist: combine two datasets from summaries

Sometimes you're not given the raw numbers, only summary sums. The shortcut engine shines here.

Worked example Example 9 — Case I (merge using

$\sum x$ and $\sum x^{2}$ ) Group P: $n_{P} = 3$ values with $\sum x = 12$ , $\sum x^{2} = 56$ . Group Q: $n_{Q} = 2$ values with $\sum x = 18$ , $\sum x^{2} = 164$ . Find the variance of the combined population.

Forecast: combining a low-mean group with a high-mean group usually increases spread. Watch for that.

Combined count $N = 3 + 2 = 5$ .
Combined mean: $\overset{x}{ˉ} = \frac{12 + 18}{5} = \frac{30}{5} = 6$ . Why add the $\sum x$ ? Total sum over total count is the overall balance point.
Combined mean of squares: $\overline{x^{2}} = \frac{56 + 164}{5} = \frac{220}{5} = 44$ . Why add the $\sum x^{2}$ ? $\sum x^{2}$ is additive across disjoint groups, just like $\sum x$ .
Shortcut: $σ^{2} = 44 - 6^{2} = 44 - 36 = 8$ .
$σ = 8 \approx 2.83$ .

Verify: consistent example — Group P could be $2, 4, 6$ ( $\sum = 12$ , $\sum x^{2} = 4 + 16 + 36 = 56$ ✓) and Group Q $8, 10$ ( $\sum = 18$ , $\sum x^{2} = 64 + 100 = 164$ ✓). Merged list $2, 4, 6, 8, 10$ : mean $6$ , squared devs $16, 4, 0, 4, 16$ sum $40$ , $σ^{2} = 40/5 = 8$ ✓.

J — limiting behaviour

Recall Self-test: name the case, then solve

For each mini-prompt, first say which matrix cell (A–J) it is. Data $5, 5, 5$ population variance ::: Case C — degenerate, $σ^{2} = 0$ . You already found $σ^{2} = 9$ ; now every value is multiplied by $3$ . New variance? ::: Case E — $\times c^{2} = 9 \times 9 = 81$ . Same data, but now add $7$ to every value. New variance? ::: Case D — unchanged, still $9$ . $n = 3$ sample, sum of squared deviations $= 18$ . Sample variance? ::: Case F — $18/ (3 - 1) = 9$ . Merge: $N = 4$ , $\sum x = 8$ , $\sum x^{2} = 24$ . Variance? ::: Case I — $24/4 - (8/4)^{2} = 6 - 4 = 2$ .

Connections

Measures of Dispersion — Variance & Standard Deviation — the parent note these examples exercise.
Mean, Median, Mode — measures of central tendency — every example starts by locating the centre.
Coefficient of Variation — divide any $σ$ above by its $\overset{x}{ˉ}$ for a unitless spread.
Normal Distribution — the SD values here feed the 68-95-99.7 rule.
Covariance and Correlation — Example 9's "additive $\sum x^{2}$ " trick generalises to two variables.
Least Squares Regression — same square-then-minimise machinery.