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Class 12 Statistics Notes · GSEB

Standard Normal Variable

Standard Normal Variable — learn the Z-transformation Z = (X − μ)/σ that converts any normal variable to Z ~ N(0, 1). GSEB Class 12 Commerce Statistics notes with interactive converter.

Last updated: 22 Sep 2026

Notes

Standard Normal Variable

Standard Normal Variable (Z)
A normal variable that has been standardised to have mean 0 and standard deviation 1. Any normal variable X ~ N(μ, σ²) converts to Z ~ N(0, 1) using Z = (X − μ) / σ.

Z-score Formula

Z=X−μσZ = \frac{X - \mu}{\sigma}

Standard Normal

Z∼N(0,1)Z \sim N(0, 1)

Real-life reading of a Z-score

Think of a Z-score as the "number of toppers away from the average" you are. In a test with μ = 50, σ = 10, scoring 70 means you are two full standard deviations above the class — Z = +2. A Z-score tells your relative standing, not your raw marks.

Z = 0

Exactly at the mean

Scored precisely the class average.

Z = +1

One σ above mean

Better than about 84% of the class.

Z = −1

One σ below mean

Below average — room to improve.

Z = +2

Two σ above mean

Top ~2.3% — a topper-tier score.

Try It: X to Z Transformer

Adjust μ, σ, and X. Watch the marker slide on both curves at once — same position, different labels.

X → Z Transformation

Original: X ~ N(50, 10²)

μμ ± σμ ± σμ ± 2σμ ± 2σμ ± 3σμ ± 3σX = 70x

Z = (X − μ) / σ

Standard: Z ~ N(0, 1)

−3−2−10123Z = 2.00x

Z = (70 − 50) / 10 = 2.00

Your score is 2.00 standard deviations ABOVE the mean

One-to-one mapping

Move any slider: the rose line on the left and the green line on the right always sit at the same relative position in their curves. That is the entire trick — different scales, identical shape.

Why Standardise?

The whole point

Different exams have different averages and spreads — raw marks are not comparable. After converting to Z, every normal distribution becomes the same N(0, 1) curve, and one Z-table works for all of them.

You scored 71 in both papers. Which performance is genuinely better?

Same Marks, Different Exams
AspectMathematics (μ = 55, σ = 8)Gujarati (μ = 62, σ = 6)
Your marks7171
Z-scoreZ = (71 − 55)/8 = +2.00Z = (71 − 62)/6 = +1.50
Relative standingTop ~2.3% — outstandingTop ~6.7% — very good
Verdict✓ Far above a tough paperGood, but less exceptional

✓ Zerodha example — comparing investments

Your portfolio returned 12% this quarter. The market average was 8% with σ = 4%. Your Z = (12 − 8)/4 = +1.00— one market-σ above average. A friend's 14% in a volatile small-cap fund (μ = 10%, σ = 8%) has Z = (14 − 10)/8 = +0.50. Raw returns say your friend won; Z-scores say you beat your benchmark harder.

Solved Examples

Solved Example

Problem

Marks are N(50, 10²). Find the Z-score for a student who scored 65.

Solution

Z = +1.50

Solved Example

Problem

If Z = −1.2 and the distribution is N(60, 5²), find X.

Solution

X = 54

Solved Example

Problem

Two students score 48 and 66. Class stats: μ = 50, σ = 8. Who performed relatively better?

Solution

Student 2 (Z = +2.00 vs Z = −0.25) — far better relative performance

Key Takeaways

Key Takeaways

  • Z = (X − μ)/σ converts any normal variable into the standard normal Z ~ N(0, 1).
  • A Z-score counts how many standard deviations a value sits from the mean — it measures relative standing, not raw value.
  • Z = 0 at the mean, positive above, negative below — so two students from different exams become comparable instantly.
  • Standardisation is why ONE Z-table serves every normal distribution in the exam.
  • Invert when needed: X = μ + Zσ — this is how you go back from Z to real marks, heights, or rupees.

Practice

  1. X ~ N(70, 5²). Find Z for X = 80 and for X = 62.
  2. A student has Z = +1.75 in a test with μ = 44, σ = 8. What were her marks?
  3. Two friends invest: A returns 9% (market μ = 7%, σ = 2%), B returns 15% (μ = 12%, σ = 6%). Who beat their benchmark by more?

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