---
title: "Normal characteristics; relation to standard normal"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:normal-characteristics-standard-normal-relation"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/normal-characteristics-standard-normal-relation
site: BCT Engineering
---

# Normal characteristics; relation to standard normal

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2074 Chaitra, 2+3 marks
- Video: [Z-Scores, Standardization, and the Standard Normal Distribution (5.3)](https://www.youtube.com/watch?v=2tuBREK_mgE) — Simple Learning Pro
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/normal-characteristics-standard-normal-relation
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

Write major characteristics of normal distribution. Discuss relation between Normal distribution and Standard Normal distribution.

## Short answer

Normal distribution is symmetric bell-shaped, mean=median=mode, points of inflection at μ±σ, 68-95-99.7 rule, defined by μ and σ. Standard normal Z=(X-μ)/σ has mean 0, variance 1, and all normal probabilities are obtained from its tables.

## Answer

#### Major characteristics of the normal distribution
A normal random variable X with mean μ and standard deviation σ has the pdf

f(x) = (1/(σ√(2π))) e^(−(x − μ)²/(2σ²)), −∞ < x < ∞.

1. The curve is bell shaped and symmetric about x = μ.
2. Mean = median = mode = μ.
3. The total area under the curve is 1, half on each side of μ.
4. The curve is asymptotic to the x-axis on both sides.
5. Points of inflection are at μ ± σ.
6. Skewness is 0 and kurtosis is 3.
7. Empirical rule: about 68.27 %, 95.45 % and 99.73 % of the area lies within μ ± σ, μ ± 2σ and μ ± 3σ.
8. It is fully described by two parameters, μ and σ.
9. A linear combination of independent normal variables is also normal.

#### Relation with the standard normal distribution
Every normal distribution N(μ, σ²) can be converted into the **standard normal distribution** N(0, 1) by the transformation

Z = (X − μ)/σ

- Z has mean 0 and standard deviation 1.
- P(a < X < b) = P((a − μ)/σ < Z < (b − μ)/σ), found from one standard table.
- The reverse is X = μ + σZ.

So the standard normal is simply the normal distribution measured in units of standard deviations from the mean, and all normal curves have the same shape after standardising.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The normal curve is:
   A. Bell shaped and symmetric about mu, with mean = median = mode
   B. Skewed with mode below mean
   C. Uniform
   D. Bounded by mu +/- 3 sigma
   Answer: A. Symmetry gives equal central measures. (Key point: Shape)

2. Points of inflection are at:
   A. mu +/- 2 sigma
   B. mu only
   C. mu +/- sigma
   D. 0 and 1
   Answer: C. Curvature changes at one SD from the mean. (Key point: Inflection)

3. The transformation to standard normal is:
   A. Z = X - sigma
   B. Z = (X + mu)/sigma
   C. Z = X/mu
   D. Z = (X - mu)/sigma
   Answer: D. Gives mean 0 and SD 1. (Key point: Standardising)

4. About 95.45% of the area lies within:
   A. mu +/- sigma
   B. mu +/- 2 sigma
   C. mu +/- 3 sigma
   D. mu +/- 0.5 sigma
   Answer: B. 68.27%, 95.45%, 99.73%. (Key point: Empirical rule)

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