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Class 12 Statistics Notes · GSEB

Areas Under the Curve

Areas Under the Curve — read probabilities directly as areas under the normal curve using the 68-95-99.7 rule and half-band values. GSEB Class 12 Commerce Statistics notes with draggable calculator.

Last updated: 22 Sep 2026

Notes

Areas Under the Curve = Probabilities

For a continuous distribution, probability is area. The whole curve encloses area 1 — so the chance of a value falling inside any slice equals that slice's area. The 68–95–99.7 rule gives these areas without any table.
The rule as probabilities — every value is a measured area under the normal curve.
RangeIn X termsProbability
μ ± 1σP(μ − σ < X < μ + σ)0.6827
μ ± 2σP(μ − 2σ < X < μ + 2σ)0.9545
μ ± 3σP(μ − 3σ < X < μ + 3σ)0.9973
Z-form (same curve)P(−1 < Z < 1) / P(−2 < Z < 2) / P(−3 < Z < 3)0.6827 / 0.9545 / 0.9973
Each half of the curveP(X < μ) or P(X > μ)0.5000
Everything combined−∞ to +∞1.0000

Try It: Shaded Area Calculator

Drag the two red handles to move the boundaries, or use a preset. Watch the area — the probability — change in real time.

Shaded Area Calculator — drag the two red handles
Area = 0.6827−3−2−10123mirror Z = 1.00Za = -1.00Zb = 1.00x

Za (left drag)

-1.00

Zb (right drag)

1.00

Shaded area

0.6827

As percentage

68.27%

P(-1.00 ≤ Z ≤ 1.00) = Φ(1.00) − Φ(-1.00) = 0.6827

Symmetric about 0

Za and Zb are equal distances on opposite sides of 0. By symmetry the left half equals the right half — so this area is exactly 2 × P(0 to 1.00).

Tip: drag either red handle along the curve, or tap a preset. The green area and all four readouts update live.

Building Any Area From the Rule

The rule only names symmetric bands, but half of each band (from μ outward) is equally useful. These four values appear in exam questions constantly:

Half of 68%

P(μ < X < μ + σ) = 0.3413

Also P(μ − σ < X < μ) = 0.3413 — symmetry again.

Half of 95%

P(μ < X < μ + 2σ) = 0.4772

Tail beyond 2σ: (1 − 0.9545)/2 = 0.0228.

Beyond 3σ — the outliers

P(X > μ + 3σ) = 0.00135

Only about 1.35 in 1000 values — Six Sigma quality control is built on this number.

Combining halves

P(μ − σ < X < μ + 2σ) = 0.3413 + 0.4772 = 0.8185

Split any non-symmetric band at μ and add the pieces.

The Governing Idea

P(a<X<b)=area under the curve from a to bP(a < X < b) = \text{area under the curve from } a \text{ to } b

Trap: one-tailed questions

"What percentage scored ABOVE the average?" is not 68% — above the average alone is just 0.5. And "above μ + σ" = (1 − 0.6827)/2 = 0.1587, not 31.73%. Always sketch the band before answering.

Solved Examples

Solved Example

Problem

Marks ~ N(50, 10²). Find P(40 < X < 60).

Solution

P(40 < X < 60) = 0.6826

Solved Example

Problem

Delivery times ~ N(30, 5²) minutes. Find P(X > 45).

Solution

P(X > 45) ≈ 0.00135 — almost never happens

Solved Example

Problem

Income of a sales team ~ N(₹24000, ₹2000²). Find P(₹24000 < X < ₹28000).

Solution

P = 0.4772 (about 47.7% of the team)

Key Takeaways

Key Takeaways

  • Probability = area under the normal curve. Total area 1 means total probability 1.
  • Memorise: 0.6827 within 1σ, 0.9545 within 2σ, 0.9973 within 3σ — in both X-form and Z-form.
  • Half-bands matter too: μ to μ + σ = 0.3413, μ to μ + 2σ = 0.4772 — split any asymmetric range at μ and add the halves.
  • Symmetry is a tool: P(0 to z) = P(−z to 0), and equal-distance bands about 0 enclose equal areas — the dashed mirror line in the calculator proves it visually.
  • Sketch the range first in every question — most area mistakes come from shading the wrong slice, not from bad arithmetic.

Practice

  1. X ~ N(100, 15²). Use the rule to find P(85 < X < 130).
  2. What is P(X > μ) for any normal distribution? Explain in one line.
  3. A quality check finds σ = 2 g around a 500 g target. At least what weight puts a packet in the worst 0.13%?

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