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Class 12 Statistics Notes · GSEB

Properties of Normal Distribution

Properties of Normal Distribution — master symmetry, mean = median = mode, and the 68-95-99.7 empirical rule with a shaded-zone explorer. GSEB Class 12 Commerce Statistics notes with visual examples.

Last updated: 22 Sep 2026

Notes

Properties of the Normal Distribution

Six properties define the normal distribution completely. Click each card to see how it shows up in real data — the first and third are exam favourites.

The left half mirrors the right half exactly. Consequently mean = median = mode, all sitting at the single peak.

Real life: if the class average is 62, exactly half the class scored above 62 and half below — the median is also 62.

Highest at the centre, tapering smoothly on both sides — the familiar bell.

Real life: in a class of N(60, 12²), scores near 60 are common; 96 or 24 are rare.

The tails stretch to infinity but never actually touch the x-axis — extreme values become vanishingly rare, never impossible.

Real life: scoring 100 when the average is 60 is nearly zero-likelihood — but the curve never says the probability is exactly 0.

The complete area under the curve equals 1, i.e. 100% of the probability.

Real life: every student in the class is somewhere under this curve — nobody falls outside it.

Because of symmetry, the area on each side of μ is exactly 0.5 — so P(X < μ) = P(X > μ) = 0.5.

Real life: a median scorer in your class is, by definition, at the 50th percentile.

Once μ and σ are fixed, every detail of the curve — all probabilities, all percentages — is locked in.

Real life:knowing "average 50, SD 10" is enough to predict the whole shape of the marks distribution.

The 68–95–99.7 Rule (Empirical Rule)

The most-tested property of all: a fixed percentage of data always lies within 1, 2, and 3 standard deviations of the mean. Toggle the zones — they stack as layers.

68–95–99.7 Rule Explorer
μμ ± σμ ± σμ ± 2σμ ± 2σμ ± 3σμ ± 3σx

μ ± σ

68.27%

of all data

Layers stay honest

The zones stack like onion layers — the inner 68% sits on top of the 95%, which sits on top of 99.7%. Change μ or σ and every zone slides together: the percentages stay fixed, only the x-values move. That is what "standard" means.

μ ± 1σ → 68.27%

About 2 out of every 3 values.

μ ± 2σ → 95.45%

About 19 out of every 20 values.

μ ± 3σ → 99.73%

Almost everything — only 3 in 1000 outside.

One Sigma

P(μ−σ<X<μ+σ)=0.6827P(\mu - \sigma < X < \mu + \sigma) = 0.6827

Two Sigma

P(μ−2σ<X<μ+2σ)=0.9545P(\mu - 2\sigma < X < \mu + 2\sigma) = 0.9545

Exam shortcut

Questions like "what percentage lies between μ and μ + σ?" are traps for the unwary — that band is only half of the 68% zone, i.e. 34.13%. Always check whether the range is symmetric about μ before quoting 68/95/99.7 directly.

Mean = Median = Mode

μ − 3σμ − 2σμ − σμ = M = mμ + σμ + 2σμ + 3σμ = median = mode = 120x

Why all three coincide

Symmetry forces it: the midpoint that splits the area in half (median) and the peak where density is highest (mode) must both sit at the centre — the mean. For a class with μ = 120 marks, the median scorer and the most common score cluster are all at 120.

When they differ, the data is NOT normal

Indian monthly salaries might show mean = ₹52,000, median = ₹38,000, mode = ₹30,000 — a few CEOs pull the mean far to the right while most people cluster near ₹30,000. Mean > median > mode is the classic signature of right-skewed data. Never apply bell-curve rules to it.

Solved Examples

Solved Example

Problem

Marks are N(80, 5²). What percentage of students scored between 70 and 90?

Solution

95.45% of students

Solved Example

Problem

In a normal distribution, mean = 45. What is the median? What is P(X < 45)?

Solution

Median = 45, P(X < 45) = 0.5

Solved Example

Problem

A factory packs dals in packets labelled 500 g, σ = 10 g. What is the minimum weight that puts a packet in the top 0.13%?

Solution

About 530 g (μ + 3σ)

Key Takeaways

Key Takeaways

  • Symmetric curve → mean = median = mode, all at the centre. If the three differ in real data, the data is not normal.
  • Total area = 1, with exactly 0.5 on each side of μ — so P(X < μ) = 0.5 always.
  • The 68–95–99.7 rule: 68.27% within 1σ, 95.45% within 2σ, 99.73% within 3σ — memorise these three numbers; they carry whole exam questions.
  • The curve is asymptotic: tails approach but never touch the x-axis, so extreme values are rare, not forbidden.
  • The curve is completely determined by μ and σ — two numbers predict the entire distribution.

Practice

  1. X ~ N(200, 20²). Find the range covering the middle 95.45% of values.
  2. What percentage of a normal distribution lies between μ and μ + σ? (Hint: half of 68.27%.)
  3. A survey reports mean income ₹52,000, median ₹38,000, mode ₹30,000. Is the distribution normal? Which direction is it skewed?

Explore Further