The time (in minutes) taken by 40 students to reach school is 0-10: 6, 10-20: 10, 20-30: 13, 30-40: 7, 40-50: 4. Find the standard deviation.

1. 1 · Write what is given and what is to be found Five classes of width c = 10, N = 40. To find: σ.
2. 2 · Write the formula σ = c × √(Σfd²/N − (Σfd/N)²), with d = (x − A)/c.
3. 3 · Make the table
   • Mid-values x: 5, 15, 25, 35, 45; take A = 25.
   • d: −2, −1, 0, 1, 2
   • fd: −12, −10, 0, 7, 8, so Σfd = −7
   • fd²: 24, 10, 0, 7, 16, so Σfd² = 57
4. 4 · Substitute
   • σ = 10 × √(57/40 − (−7/40)²)
   • = 10 × √(1.425 − 0.030625)
   • = 10 × √1.394375
5. 5 · Work out the root
   • = 10 × 1.1808 ≈ 11.81 minutes
6. 6 · Check and box the answer σ ≈ 11.81 minutes. Check: N = 6 + 10 + 13 + 7 + 4 = 40, and σ is positive.

Scoring tip: Square the whole (Σfd/N) term with its minus sign: (−7/40)² is positive. Then subtract it.