CTI must choose between Magasman Technologies and Thap's Technology Solutions. Each company's profit depends on how fast the economy recovers. This model rebuilds the two tables in the model answer one column at a time, then shows where "deviation" comes from and why we square it.
The recipe
Every number in the answer comes from this chain
Read left to right. Each step feeds the next, so a mistake early on flows all the way to the recommendation.
2 · Deviationd = profit − EHow far each outcome lands from the average
→
3 · Deviation²d × dRemoves the sign, magnifies big misses
→
4 · Weightedp × d²The "Variance" column in the answer
→
5 · Std deviationσ = √ Σ (p × d²)Back to rands: the risk figure
→
6 · Coeff. of variationCV = σ ÷ ERisk per R1 of expected return
Build the table
Rebuild the model-answer table, step by step
Recovery level
Probabilityp
Profitanswer calls it "Expected Profit"
p × Profitstep 1working, not shown in answer
Deviationstep 2 · profit − E
Deviation squaredstep 3 · d²
Variancestep 4 · p × d²
Profit and probability cells are editable, so try your own figures.
Step 1 of 6
Why this step
Deviation, explained
Where does a "deviation" come from?
Put every possible profit on a ruler. Mark the expected return E on the same ruler. A deviation is simply the distance, in rands, from E to each outcome. Outcomes to the left of E give negative deviations (a worse result than expected). Outcomes to the right give positive ones.
Below E: negative deviationAbove E: positive deviationE, the expected returnDot size = probability
The problem
Weighted deviations always add up to zero
E is the balance point of the outcomes, like the pivot of a see-saw. The probability-weighted pull on the left exactly cancels the pull on the right. So "average deviation" is always R0, for every company, and tells us nothing about risk.
Level
p
d
p × d
The fix
Square each deviation first
The sign disappears. (−34 300)² and (+34 300)² are both 1 176 490 000. A miss on either side now counts as spread, so nothing cancels.
Big misses weigh far more. Magasman's Exponential deviation (43 700) is about 34 times the Moderate one (1 300), but its square is about 1 130 times bigger. Risk measures should be most sensitive to the outcomes furthest from plan.
The square root undoes the units problem. Squaring rands gives "rand squared", which has no business meaning. √variance returns us to rands, so σ can sit next to E and be compared with it.
Why not simply ignore the minus signs? That gives the mean absolute deviation, which is a valid measure. Finance uses squares because variance feeds the rest of the syllabus: portfolio risk, correlation, and the z-score all build on σ.
Squaring made visible
A deviation squared is literally a square
Each shaded square below has a side equal to one deviation, drawn from E to the outcome. Its area is the deviation squared. Variance is the probability-weighted average of these areas. The gold outline is a square with that average area: its side is the standard deviation. Both companies use the same scale, so compare them directly.
The decision
Return versus risk, side by side
Reading the model answer carefully
The column headed "Expected Profit" lists the profit in each recovery scenario. The expected profit (expected return) is the single weighted figure: R136 300 and R120 750.
The column headed "Variance" holds each scenario's share, p × d². Variance is the total of that column; σ is its square root.
CV is risk ÷ return (σ ÷ E). The answer's figures are calculated that way, although one line of its wording says "return over risk". A lower CV means less risk for each R1 of expected return.
The calculator symbol for standard deviation is σ (sigma). The answer prints it as α.